HP (Hewlett Packard) 40gs User Manual

hp40g+.book Page i Friday, December 9, 2005 1:03 AM

HP 40gs graphing calculator

user's guide

Edition1

Part Number F2225AA-90001

title.fm Page ii Friday, February 17, 2006 9:48 AM

Notice

THIS MANUAL AND ANY EXAMPLES CONTAINED HEREIN ARE

PROVIDED "AS IS" AND ARE SUBJECT TO CHANGE WITHOUT

NOTICE. HEWLETT-PACKARD COMPANY MAKES NO WAR-

RANTY OF ANY KIND WITH REGARD TO THIS MANUAL,

INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES

OF MERCHANTABILITY, NON-INFRINGEMENT AND FITNESS

FOR A PARTICULAR PURPOSE.

HEWLETT-PACKARD CO. SHALL NOT BE LIABLE FOR ANY

ERRORS OR FOR INCIDENTAL OR CONSEQUENTIAL DAMAGES

IN CONNECTION WITH THE FURNISHING, PERFORMANCE, OR

USE OF THIS MANUAL OR THE EXAMPLES CONTAINED HEREIN.

opment Company, L.P.

Reproduction, adaptation, or translation of this manual is prohibited without

prior written permission of Hewlett-Packard Company, except as allowed

under the copyright laws.

Hewlett-Packard Company

4995 Murphy Canyon Rd,

Suite 301

San Diego, CA 92123

Printing History

Edition 1

April 2005

hp40g+.book Page iv Friday, December 9, 2005 1:03 AM

Function aplet interactive analysis........................................... 3-9

Plotting a piecewise-defined function................................ 3-12

4 Parametric aplet

About the Parametric aplet .................................................... 4-1

Getting started with the Parametric aplet............................. 4-1

5 Polar aplet

Getting started with the Polar aplet ......................................... 5-1

6 Sequence aplet

About the Sequence aplet...................................................... 6-1

Getting started with the Sequence aplet.............................. 6-1

7 Solve aplet

About the Solve aplet............................................................ 7-1

Getting started with the Solve aplet.................................... 7-2

Use an initial guess............................................................... 7-5

Interpreting results ................................................................ 7-6

Plotting to find guesses.......................................................... 7-7

Using variables in equations................................................ 7-10

8 Linear Solver aplet

About the Linear Solver aplet ................................................. 8-1

Getting started with the Linear Solver aplet.......................... 8-1

9 Triangle Solve aplet

About the Triangle Solver aplet .............................................. 9-1

Getting started with the Triangle Solver aplet....................... 9-1

10 Statistics aplet

About the Statistics aplet...................................................... 10-1

Getting started with the Statistics aplet.............................. 10-1

Entering and editing statistical data ...................................... 10-6

Defining a regression model.......................................... 10-12

Computed statistics........................................................... 10-14

Plotting............................................................................ 10-15

Plot types .................................................................... 10-16

Fitting a curve to 2VAR data ......................................... 10-17

Setting up the plot (Plot setup view) ................................ 10-18

Trouble-shooting a plot ................................................. 10-19

Exploring the graph ..................................................... 10-19

Calculating predicted values ......................................... 10-20

11 Inference aplet

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About the Inference aplet .....................................................11-1

Getting started with the Inference aplet .............................11-1

Importing sample statistics from the Statistics aplet..............11-4

Hypothesis tests ..................................................................11-8

One-Sample Z-Test..........................................................11-8

Two-Sample Z-Test..........................................................11-9

One-Proportion Z-Test....................................................11-10

Two-Proportion Z-Test....................................................11-11

One-Sample T-Test........................................................11-12

Two-Sample T-Test ........................................................11-14

Confidence intervals..........................................................11-15

One-Sample Z-Interval...................................................11-15

Two-Sample Z-Interval ...................................................11-16

One-Proportion Z-Interval...............................................11-17

Two-Proportion Z-Interval ...............................................11-17

One-Sample T-Interval ...................................................11-18

Two-Sample T-Interval....................................................11-19

12 Using the Finance Solver

Background........................................................................12-1

Performing TVM calculations ................................................12-4

Calculating Amortizations................................................12-7

13 Using mathematical functions

Math functions....................................................................13-1

The MATH menu ............................................................13-1

Math functions by category ..................................................13-2

Keyboard functions.........................................................13-3

Calculus functions...........................................................13-6

Complex number functions...............................................13-7

Constants ......................................................................13-8

Conversions...................................................................13-8

Hyperbolic trigonometry..................................................13-9

List functions ................................................................13-10

Loop functions..............................................................13-10

Matrix functions ...........................................................13-11

Polynomial functions .....................................................13-11

Probability functions......................................................13-12

Real-number functions ...................................................13-14

Two-variable statistics....................................................13-17

Symbolic functions........................................................13-17

Test functions ...............................................................13-19

Trigonometry functions ..................................................13-20

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Symbolic calculations........................................................ 13-20

Finding derivatives....................................................... 13-21

Program constants and physical constants ........................... 13-24

Program constants........................................................ 13-25

Physical constants ........................................................ 13-25

14 Computer Algebra System (CAS)

What is a CAS?................................................................. 14-1

Performing symbolic calculations .......................................... 14-1

An example .................................................................. 14-2

CAS variables.................................................................... 14-4

The current variable ....................................................... 14-4

CAS modes ....................................................................... 14-5

Using CAS functions in HOME............................................. 14-7

Online Help....................................................................... 14-8

CAS functions in the Equation Writer .................................... 14-9

ALGB menu................................................................. 14-10

DIFF menu................................................................... 14-16

REWRI menu ............................................................... 14-28

SOLV menu................................................................. 14-33

TRIG menu.................................................................. 14-38

CAS Functions on the MATH menu ..................................... 14-45

Algebra menu ............................................................. 14-45

Complex menu ............................................................ 14-45

Constant menu ............................................................ 14-46

Diff & Int menu ............................................................ 14-46

Hyperb menu .............................................................. 14-46

Integer menu ............................................................... 14-46

Modular menu............................................................. 14-51

Polynomial menu ......................................................... 14-55

Real menu................................................................... 14-60

Rewrite menu .............................................................. 14-60

Solve menu ................................................................. 14-60

Tests menu .................................................................. 14-61

Trig menu ................................................................... 14-61

CAS Functions on the CMDS menu..................................... 14-62

15 Equation Writer

Using CAS in the Equation Writer ....................................... 15-1

The Equation Writer menu bar......................................... 15-1

Configuration menus ...................................................... 15-3

Entering expressions and subexpressions............................... 15-5

How to modify an expression ....................................... 15-11

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Accessing CAS functions....................................................15-12

Equation Writer variables .................................................15-16

Predefined CAS variables .............................................15-16

The keyboard in the Equation Writer ..............................15-17

16 Step-by-Step Examples

Introduction .......................................................................16-1

17 Variables and memory management

Introduction........................................................................17-1

Storing and recalling variables .............................................17-2

The VARS menu ..................................................................17-4

Memory Manager...............................................................17-9

18 Matrices

Introduction........................................................................18-1

Creating and storing matrices...............................................18-2

Working with matrices.........................................................18-4

Matrix arithmetic.................................................................18-6

Solving systems of linear equations...................................18-8

Matrix functions and commands..........................................18-10

Argument conventions...................................................18-10

Matrix functions ...........................................................18-10

Examples .........................................................................18-13

19 Lists

Displaying and editing lists...................................................19-4

Deleting lists ..................................................................19-6

Transmitting lists.............................................................19-6

List functions.......................................................................19-6

Finding statistical values for list elements................................19-9

20 Notes and sketches

Introduction........................................................................20-1

Aplet note view...................................................................20-1

Aplet sketch view ................................................................20-3

The notepad.......................................................................20-6

21 Programming

Introduction........................................................................21-1

Program catalog ............................................................21-2

Creating and editing programs.............................................21-4

Using programs ..................................................................21-7

Customizing an aplet...........................................................21-9

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Aplet naming convention .............................................. 21-10

Example ..................................................................... 21-10

Programming commands................................................... 21-13

Aplet commands.......................................................... 21-14

Branch commands ....................................................... 21-17

Drawing commands ..................................................... 21-19

Graphic commands...................................................... 21-21

Loop commands .......................................................... 21-23

Matrix commands........................................................ 21-24

Print commands ........................................................... 21-25

Prompt commands........................................................ 21-26

Stat-One and Stat-Two commands.................................. 21-29

Stat-Two commands ..................................................... 21-30

Storing and retrieving variables in programs................... 21-31

Plot-view variables ....................................................... 21-31

Symbolic-view variables................................................ 21-38

Numeric-view variables ................................................ 21-40

Note variables............................................................. 21-43

Sketch variables .......................................................... 21-43

22 Extending aplets

Creating new aplets based on existing aplets......................... 22-1

Using a customized aplet................................................ 22-3

Resetting an aplet............................................................... 22-3

Annotating an aplet with notes............................................. 22-4

Annotating an aplet with sketches......................................... 22-4

Downloading e-lessons from the web .................................... 22-4

Sending and receiving aplets............................................... 22-4

Sorting items in the aplet library menu list.............................. 22-6

Reference information

Glossary.............................................................................. R-1

Resetting the HP 40gs ........................................................... R-3

To erase all memory and reset defaults............................... R-3

If the calculator does not turn on........................................ R-4

Operating details ................................................................. R-4

Batteries ......................................................................... R-4

Variables............................................................................. R-6

Home variables ............................................................... R-6

Function aplet variables.................................................... R-7

Parametric aplet variables................................................. R-8

Polar aplet variables ........................................................ R-9

Sequence aplet variables................................................ R-10

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Solve aplet variables.......................................................R-11

Statistics aplet variables ..................................................R-12

MATH menu categories .......................................................R-13

Math functions ...............................................................R-13

Program constants ..........................................................R-15

Physical Constants..........................................................R-16

CAS functions ................................................................R-17

Program commands........................................................R-19

Status messages..................................................................R-20

Limited Warranty

Service.......................................................................... W-3

Regulatory Notices ......................................................... W-5

Index

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Preface

The HP 40gs is a feature-rich graphing calculator. It is

also a powerful mathematics learning tool, with a built-in

computer algebra system (CAS). The HP 40gs is designed

so that you can use it to explore mathematical functions

and their properties.

You can get more information on the HP 40gs from

Hewlett-Packard’s Calculators web site. You can

download customized aplets from the web site and load

them onto your calculator. Customized aplets are special

applications developed to perform certain functions, and

to demonstrate mathematical concepts.

Hewlett Packard’s Calculators web site can be found at:

http://www.hp.com/calculators

Manual conventions

The following conventions are used in this manual to

represent the keys that you press and the menu options

that you choose to perform the described operations.

•

Key presses are represented as follows:

, etc.

•

Shift keys, that is the key functions that you access by

pressing the

follows:

key first, are represented as

ACOS, etc.

CLEAR,

MODES,

•

Numbers and letters are represented normally, as

follows:

5, 7, A, B, etc.

Menu options, that is, the functions that you select

using the menu keys at the top of the keypad are

represented as follows:

•

Input form fields and choose list items are represented

as follows:

Function, Polar, Parametric

Your entries as they appear on the command line or

within input forms are represented as follows:

2*X -3X+5

P-1

Preface.fm Page 2 Friday, February 17, 2006 9:47 AM

Notice

This manual and any examples contained herein are

provided as-is and are subject to change without notice.

Except to the extent prohibited by law, Hewlett-Packard

Company makes no express or implied warranty of any

kind with regard to this manual and specifically disclaims

the implied warranties and conditions of merchantability

and fitness for a particular purpose and Hewlett-Packard

Company shall not be liable for any errors or for

incidental or consequential damage in connection with

the furnishing, performance or use of this manual and the

examples herein.

Hewlett-Packard Development Company, L.P.

The programs that control your HP 40gs are copyrighted

and all rights are reserved. Reproduction, adaptation, or

translation of those programs without prior written

permission from Hewlett-Packard Company is also

prohibited.

P-2

hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Getting started

On/off, cancel operations

To turn on

Press

to turn on the calculator.

To cancel

When the calculator is on, the

current operation.

key cancels the

To turn off

Press

OFF to turn the calculator off.

To save power, the calculator turns itself off after several

minutes of inactivity. All stored and displayed information

is saved.

If you see the ((•)) annunciator or the Low Batmessage,

then the calculator needs fresh batteries.

HOME

HOME is the calculator’s home view and is common to all

aplets. If you want to perform calculations, or you want to

quit the current activity (such as an aplet, a program, or

an editor), press

. All mathematical functions are

available in the HOME. The name of the current aplet is

displayed in the title of the home view.

Protective cover

The calculator is provided with a slide cover to protect the

display and keyboard. Remove the cover by grasping

both sides of it and pulling down.

You can reverse the slide cover and slide it onto the back

of the calculator. this will help prevent you losing the

cover while you are using the calculator.

To prolong the life of the calculator, always place the

cover over the display and keyboard when you are not

using the calculator.

Getting started

1-1

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The display

To adjust the

contrast

Simultaneously press

decrease) the contrast.

and

(or

) to increase (or

To clear the display

•

Press CANCEL to clear the edit line.

Press

CLEAR to clear the edit line and the

display history.

Parts of the

display

Title

History

Edit line

Menu key

labels

Menu key or soft key labels. The labels for the menu

keys’ current meanings.

menu key in this picture. “Press

is the label for the first

” means to press the

first menu key, that is, the leftmost top-row key on the

calculator keyboard.

Edit line. The line of current entry.

History. The HOME display (

) shows up to four

lines of history: the most recent input and output. Older

lines scroll off the top of the display but are retained in

memory.

Title. The name of the current aplet is displayed at the top

of the HOME view. RAD, GRD, DEG specify whether

Radians, Grads or Degrees angle mode is set for HOME.

The ^Tand S symbols indicate whether there is more

history in the HOME display. Press the

scroll in the HOME display.

and

1-2

Getting started

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Annunciators. Annunciators are symbols that appear

above the title bar and give you important status

information.

Annunciator

Description

Shift in effect for next keystroke.

To cancel, press

again.

Alpha in effect for next keystroke.

To cancel, press

Low battery power.

Busy.

again.

((•))

Data is being transferred.

The keyboard

Menu Key

Labels

Menu Keys

Aplet Control

Keys

Cursor

Keys

Alpha Key

Shift Key

Enter

Key

Getting started

1-3

hp40g+.book Page 4 Friday, December 9, 2005 1:03 AM

Menu keys

•

On the calculator keyboard, the top row of keys are

called menu keys. Their meanings depend on the

context—that’s why they are blank. The menu keys

are sometimes called “soft keys”.

The bottom line of the display shows the labels for the

menu keys’ current meanings.

Aplet control keys

The aplet control keys are:

Key

Meaning

Displays the Symbolic view for the

current aplet. See “Symbolic view”

on page 1-16.

Displays the Plot view for the current

aplet. See “Plot view” on page 1-16.

Displays the Numeric view for the

current aplet. See “Numeric view” on

page 1-17.

Displays the HOME view. See

“HOME” on page 1-1.

Displays the Aplet Library menu. See

“Aplet library” on page 1-16.

Displays the VIEWS menu. See

“Aplet views” on page 1-16.

1-4

Getting started

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Entry/Edit keys

The entry and edit keys are:

Key

Meaning

Cancels the current operation if the

(^CANCEL)

calculator is on by pressing

Pressing

, then OFF turns the

calculator off.

Accesses the function printed in blue

above a key.

Returns to the HOME view, for

performing calculations.

Accesses the alphabetical

characters printed in orange below

a key. Hold down to enter a string

of characters.

Enters an input or executes an

operation. In calculations,

acts like “=”. When

is present as a menu key,

acts the same as pressing

Enters a negative number. To enter

–25, press

25. Note: this is not

the same operation that the subtract

button performs ( ).

Enters the independent variable by

inserting X, T, θ, or N into the edit

line, depending on the current

active aplet.

Deletes the character under the

cursor. Acts as a backspace key if

the cursor is at the end of the line.

Clears all data on the screen. On a

settings screen, for example Plot

CLEAR

Setup,

CLEAR returns all

settings to their default values.

Moves the cursor around the

display. Press

first to move to

the beginning, end, top or bottom.

Getting started

1-5

hp40g+.book Page 6 Friday, December 9, 2005 1:03 AM

Key

Meaning (Continued)

CHARS

Displays a menu of all available

characters. To type one, use the

arrow keys to highlight it, and press

. To select multiple characters,

select each and press

press

, then

Shifted keystrokes

There are two shift keys that you use to access the

operations and characters printed above the keys:

and

Key

Description

Press the

key to access the

operations printed in blue above the

keys. For instance, to access the

Modes screen, press

, then

press

. (^MODESis labeled in

blue above the

not need to hold down

key). You do

when

you press HOME. This action is

depicted in this manual as “press

MODES.”

To cancel a shift, press

again.

The alphabetic keys are also shifted

keystrokes. For instance, to type Z,

press

Z. (The letters are

printed in orange to the lower right of

each key.)

To cancel Alpha, press

again.

For a lower case letter, press

For a string of letters, hold down

while typing.

1-6

Getting started

chapter-1.fm Page 7 Friday, December 16, 2005 2:20 PM

HELPWITH

Example

The HP 40gs built-in help is available in HOME only. It

provides syntax help for built-in math functions.

Access the HELPWITH command by pressing

SYNTAX and then the math key for which you require

syntax help.

Press

SYNTAX

Note: Remove the left parenthesis from built-in

functions such as sine, cosine, and tangent before

invoking the HELPWITH command.

Note: In the CAS system, pressing the

will show the CAS help menu.

SYNTAX

Math keys

HOME (

) is the place to do non-symbolic

calculations. (For symbolic calculations, use the computer

algebra system, referred throughout this manual as CAS).

Keyboard keys. The most common operations are

available from the keyboard, such as the arithmetic (like

) and trigonometric (like

to complete the operation:

displays 16.

) functions. Press

256

MATH menu. Press

to open the MATH

menu. The MATH menu is a

comprehensive list of math

functions that do not appear

on the keyboard. It also

includes categories for all other functions and constants.

The functions are grouped by category, ranging in

alphabetical order from Calculus to Trigonometry.

•

The arrow keys scroll through the list (

and move from the category list in the left column

to the item list in the right column ( ).

)

•

Press

to insert the selected command onto the

edit line.

Press

to dismiss the MATH menu without

selecting a command.

Getting started

1-7

hp40g+.book Page 8 Friday, December 9, 2005 1:03 AM

•

Pressing

displays the list of Program

Constants. You can use these in programs that

you develop.

Pressing

displays a menu of physical

constants from the fields of chemistry, physics,

and quantum mechanics. You can use these

constants in calculations. (pSee “Physical

constants” on page 13-25 for more information.)

•

Pressing

takes you to the beginning of the

MATH menu.

See “Math functions by category” on page 13-2 for

details of the math functions.

H I N T

When using the MATH menu, or any menu on the

HP 40gs, pressing an alpha key takes you straight to the

first menu option beginning with that alpha character.

With this method, you do not need to press

first.

Just press the key that corresponds to the command’s

beginning alpha character.

Note that when the MATH menu is open, you can also

access CAS commands. You do this by pressing

This enables you to use CAS commands on the HOME

screen, without opening CAS. See Chapter 14 for details

of CAS commands.

Program

commands

Pressing

CMDS displays the list of Program

Commands. See “Programming commands” on

page 21-13.

Inactive keys

If you press a key that does not operate in the current

context, a warning symbol like this

no beep.

appears. There is

Menus

A menu offers you a choice

of items. Menus are

displayed in one or two

columns.

•

The

arrow in the

display means more

items below.

•

The

arrow in the

display means more items above.

1-8

Getting started

hp40g+.book Page 9 Friday, December 9, 2005 1:03 AM

To search a menu

•

Press

to scroll through the list. If you press

, you’ll go all the way to

the end or the beginning of the list. Highlight the item

you want to select, then press (or ).

If there are two columns, the left column shows

general categories and the right column shows

specific contents within a category. Highlight a

general category in the left column, then highlight an

item in the right column. The list in the right column

changes when a different category is highlighted.

Press

when you have highlighted your

selection.

•

To speed-search a list, type the first letter of the word.

For example, to find the Matrix category in

press , the Alpha “M” key.

To go up a page, you can press

down a page, press

. To go

To cancel a menu

Press

(for CANCEL) or

. This cancels the

current operation.

Input forms

An input form shows several fields of information for you

to examine and specify. After highlighting the field to

edit, you can enter or edit a number (or expression). You

can also select options from a list (

forms include items to check (

examples input forms.

). Some input

). See below for

Reset input form

values

To reset a field to its default values in an input form, move

the cursor to that field and press

. To reset all default

field values in the input form, press

CLEAR.

Getting started

1-9

hp40g+.book Page 10 Friday, December 9, 2005 1:03 AM

Mode settings

You use the Modes input form to set the modes for HOME.

H I N T

Although the numeric setting in Modes affects only

HOME, the angle setting controls HOME and the current

aplet. The angle setting selected in Modes is the angle

setting used in both HOME and current aplet. To further

configure an aplet, you use the ^SETUPkeys (

and

Press

form.

MODES to access the HOME MODES input

Setting

Options

Angle

Measure

Angle values are:

Degrees. 360 degrees in a circle.

Radians. 2π radians in a circle.

Grads. 400 grads in a circle.

The angle mode you set is the angle

setting used in both HOME and the

current aplet. This is done to ensure

that trigonometric calculations done in

the current aplet and HOME give the

same result.

Number

Format

The number format mode you set is the

number format used in both HOME

and the current aplet.

Standard. Full-precision display.

Fixed. Displays results rounded to a

number of decimal places. Example:

123.456789 becomes 123.46 in

Fixed 2 format.

Scientific. Displays results with an

exponent, one digit to the left of the

decimal point, and the specified

number of decimal places. Example:

123.456789 becomes 1.23E2 in

Scientific 2 format.

1-10

Getting started

hp40g+.book Page 11 Friday, December 9, 2005 1:03 AM

Setting

Options (Continued)

Engineering. Displays result with an

exponent that is a multiple of 3, and

the specified number of significant

digits beyond the first one. Example:

123.456E7 becomes 1.23E9 in

Engineering 2 format.

Fraction. Displays results as fractions

based on the specified number of

decimal places. Examples:

123.456789 becomes 123 in

Fraction 2 format, and .333 becomes

1/3 and 0.142857 becomes 1/7.

See “Using fractions” on page 1-25.

Mixed Fraction. Displays results as

mixed fractions based on the specified

number of decimal places. A mixed

fraction has an integer part and a

fractional part. Examples:

123.456789 becomes 123+16/35

in Fraction 2 format, and 7÷ 3 returns

2+1/3. See “Using fractions” on

page 1-25.

Decimal

Mark

Dot or Comma. Displays a number

as 12456.98 (Dot mode) or as

12456,98 (Comma mode). Dot mode

uses commas to separate elements in

lists and matrices, and to separate

function arguments. Comma mode

uses periods (dot) as separators in

these contexts.

Setting a mode

This example demonstrates how to change the angle

measure from the default mode, radians, to degrees for

the current aplet. The procedure is the same for changing

number format and decimal mark modes.

1. Press

form.

MODES to open the HOME MODES input

Getting started

1-11

chapter-1.fm Page 12 Friday, December 9, 2005 1:26 AM

The cursor (highlight) is

in the first field, Angle

Measure.

2. Press

to display a

list of choices.

3. Press

to select

Degrees,and press

. The angle measure

changes to degrees.

4. Press

HOME.

to return to

H I N T

Whenever an input form has a list of choices for a field,

you can press

to cycle through them instead of using

Aplets (E-lessons)

Aplets are the application environments where you

explore different classes of mathematical operations. You

select the aplet that you want to work with.

Aplets come from a variety of sources:

•

Built-in the HP 40gs (initial purchase).

Aplets created by saving existing aplets, which have

been modified, with specific configurations. See

“Creating new aplets based on existing aplets” on

page 22-1.

•

Downloaded from HP’s Calculators web site.

Copied from another calculator.

Aplets are stored in the

Aplet library. See “Aplet

library” on page 1-16 for

further information.

You can modify

configuration settings for

the graphical, tabular, and

1-12

Getting started

hp40g+.book Page 13 Friday, December 9, 2005 1:03 AM

symbolic views of the aplets in the following table. See

“Aplet view configuration” on page 1-18 for further

information.

Aplet

name

Use this aplet to explore:

Function

Real-valued, rectangular functions y in

terms of x. Example: y = 2x²+ 3x + 5 .

Inference

Confidence intervals and Hypothesis

tests based on the Normal and

Students-t distributions.

Parametric Parametric relations x and y in terms of

t. Example: x = cos(t) and y = sin(t).

Polar

Polar functions r in terms of an angle θ.

Example: r = 2cos(4θ) .

Sequence

Sequence functions U in terms of n, or

in terms of previous terms in the same or

another sequence, such as U_{n – 1}and

U_{n – 2}. Example: U₁= 0, U₂= 1 and

U_n= U_{n – 2}+ U_{n – 1}

Solve

Equations in one or more real-valued

variables. Example: x + 1 = x²– x – 2 .

Finance

Time Value of Money (TVM)

calculations.

Linear

Solver

Solutions to sets of two or three linear

equations.

Triangle

Solver

Unknown values for the lengths and

angles of triangles.

Statistics

One-variable (x) or two-variable (x and

y) statistical data.

In addition to these aplets, which can be used in a variety

of applications, the HP 40gs is supplied with two

teaching aplets: Quad Explorer and Trig Explorer. You

cannot modify configuration settings for these aplets.

A great many more teaching aplets can be found at HP’s

web site and other web sites created by educators,

together with accompanying documentation, often with

student work sheets. These can be downloaded free of

Getting started

1-13

hp40g+.book Page 14 Friday, December 9, 2005 1:03 AM

charge and transferred to the HP 40gs using the provided

Connectivity Kit.

Quad Explorer

aplet

The Quad Explorer aplet is used to investigate the

behaviour of y = a(x + h)²+ v as the values of a, h and

v change, both by manipulating the equation and seeing

the change in the graph, and by manipulating the graph

and seeing the change in the equation.

H I N T

More detailed documentation, and an accompanying

student work sheet can be found at HP’s web site.

Press

, select Quad

Explorer, and then press

. The Quad Explorer

aplet opens in

mode, in which the arrow

keys, the

and the

and

keys,

key are used to change the shape of the

graph. This changing shape is reflected in the equation

displayed at the top right corner of the screen, while the

original graph is retained for comparison. In this mode

the graph controls the equation.

It is also possible to have the

equation control the graph.

Pressing

displays a

sub-expression of your

equation.

Pressing the

expressions, while pressing the

their values.

and

key moves between sub-

and

key changes

Pressing

allows the user to select whether all three

sub-expressions will be explored at once or only one at a

time.

button is provided to

evaluate the student’s

knowledge. Pressing

displays a target quadratic

graph. The student must

manipulate the equation’s parameters to make the

equation match the target graph. When a student feels

that they have correctly chosen the parameters a

button evaluates the answer and provide feedback. An

button is provided for those who give up!

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Trig Explorer aplet

The Trig Explorer aplet is used to investigate the

behaviour of the graph of y = asin(bx + c) + d as the

values of a, b, c and d change, both by manipulating the

equation and seeing the change in the graph, or by

manipulating the graph and seeing the change in the

equation.

Press

, select Trig

Explorer, and then press

to display the screen

shown right.

In this mode, the graph

controls the equation.

Pressing the

keys transforms the

graph, with these

transformations reflected in the equation.

and

The button labelled

a toggle between

Origin

and

is chosen, the ‘point of

control’ is at the origin (0,0)

and the and

keys control vertical and

horizontal transformations. When

. When

is chosen the

‘point of control’ is on the first extremum of the graph (i.e.

for the sine graph at (π ⁄ 2,1) .

The arrow keys change the

Extremum

amplitude and frequency of

the graph. This is most easily

seen by experimenting.

Pressing

displays the

equation at the top of the

screen. The equation is

controlled by the graph.

Pressing the

and

keys moves from parameter

to parameter. Pressing the

parameter’s values.

key changes the

The default angle setting for this aplet is radians. The

angle setting can be changed to degrees by pressing

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Aplet library

Aplets are stored in the Aplet library.

To open an aplet

Press

aplet and press

to display the Aplet library menu. Select the

From within an aplet, you can return to HOME any time

by pressing

Aplet views

When you have configured an aplet to define the relation

or data that you want to explore, you can display it in

different views. Here are illustrations of the three major

aplet views (Symbolic, Plot, and Numeric), the six

supporting aplet views (from the VIEWS menu), and the

two user-defined views (Note and Sketch).

Note: some aplets—such as the Linear Solver aplet and

the Triangle Solver aplet—only have a single view, the

Numeric view.

Symbolic view

Plot view

Press

to display the aplet’s Symbolic view.

You use this view to define

the function(s) or equation(s)

that you want to explore.

See “About the Symbolic

view” on page 2-1 for

further information.

Press

to display the aplet’s Plot view.

In this view, the functions that

you have defined are

displayed graphically.

See “About the Plot view” on

page 2-5 for further

information.

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Numeric view

Press

to display the aplet’s Numeric view.

In this view, the functions that

you have defined are

displayed in tabular format.

See “About the numeric

view” on page 2-16 for

further information.

Plot-Table view

The VIEWS menu contains the Plot-Table view.

Select Plot-Table

Splits the screen into the plot

and the data table. See

“Other views for scaling and

splitting the graph” on

page 2-13 for futher information.

Plot-Detail view

The VIEWS menu contains the Plot-Detail view.

Select Plot-Detail

Splits the screen into the plot

and a close-up.

See “Other views for scaling and splitting the graph” on

page 2-13 for further information.

Overlay Plot

view

The VIEWS menu contains the Overlay Plot view.

Select Overlay Plot

Plots the current

expression(s)withouterasing

any pre-existing plot(s).

See “Other views for scaling and splitting the graph” on

page 2-13 for further information.

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Note view

Press

NOTE to display the aplet’s note view.

This note is transferred with

the aplet if it is sent to

another calculator or to a

PC. A note view contains text

to supplement an aplet.

See “Notes and sketches” on page 20-1 for further

information.

Sketch view

Press

SKETCH to display the aplet’s sketch view.

Displays pictures to

supplement an aplet.

See “Notes and sketches” on

page 20-1 for further

information.

Aplet view configuration

You use the SETUP keys (

, and

) to configure the aplet. For example, press

SETUP-PLOT (

) to display the input form for

setting the aplet’s plot settings. Angle measure is

controlled using the ^MODESview.

Plot Setup

Press

SETUP-PLOT.

Sets parameters to plot a

graph.

Numeric Setup

Symbolic Setup

Press

SETUP-NUM. Sets

parameters for building a

table of numeric values.

This view is only available in

the Statistics aplet in

mode, where it plays an

important role in choosing

data models.

Press

SETUP-SYMB.

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To change views

Each view is a separate environment. To change a view,

select a different view by pressing

keys or select a view from the VIEWS menu. To change

to HOME, press . You do not explicitly close the

current view, you just enter another one—like passing

from one room into another in a house. Data that you

enter is automatically saved as you enter it.

To save aplet

configuration

You can save an aplet configuration that you have used,

and transfer the aplet to other HP 40gs calculators. See

“Creating new aplets based on existing aplets” on

page 22-1.

Mathematical calculations

The most commonly used math operations are available

from the keyboard. Access to other math functions is via

the MATH menu ( ). You can also CAS for symbolic

calculations. See “Computer Algebra System (CAS)” on

page 14-1 for further information.

To access programming commands, press

CMDS.

See “Programming commands” on page 21-13 for

further information.

Where to start

The home base for the calculator is the HOME view

(

). You can do all non-symbolic calculations here,

and you can access all

operations. (Symbolic

calculations are done using CAS.)

Entering

expressions

•

In the HOME view, you enter an expression in the

same left-to-right order that you would write the

expression. This is called algebraic entry. (In CAS

you enter expressions using the Equation Writer,

explained in detail in Chapter 15, “Equation

Writer”.)

•

To enter functions, select the key or MATH menu item

for that function. You can also enter a function by

using the Alpha keys to spell out its name.

Press

to evaluate the expression you have in

the edit line (where the blinking cursor is). An

expression can contain numbers, functions, and

variables.

Getting started

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hp40g+.book Page 20 Friday, December 9, 2005 1:03 AM

23²– 14 8

Example

Calculate

---------------------------

–3

ln(45)

Long results

If the result is too long to fit on the display line, or if you

want to see an expression in textbook format, press

to highlight it and then press

Negative

numbers

Type

to start a negative number or to insert a

negative sign.

To raise a negative number to a power, enclose it in

parentheses. For example, (–5) = 25, whereas –5 =

–25.

Scientific

A number like 5 × 10⁴or 3.21 × 10^–7is written in

scientific notation, that is, in terms of powers of ten. This

is simpler to work with than 50000 or 0.000000321. To

enter numbers like these, use ^EEX. (This is easier than

notation

(powers of 10)

using

(4 × 10^–13)(6 × 10²³

)

----------------------------------------------------

Example

Calculate

3 × 10^–5

EEX

Explicit and

implicit

multiplication

Implied multiplication takes place when two operands

appear with no operator in between. If you enter AB, for

example, the result is A*B.

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However, for clarity, it is better to include the

multiplication sign where you expect multiplication in an

expression. It is clearest to enter ABas A*B.

H I N T

Implied multiplication will not always work as expected.

For example, entering A(B+4)will not give A*(B+4).

Instead an error message is displayed: “Invalid User

Function”. This is because the calculator interprets

A(B+4)as meaning ‘evaluate function Aat the value

B+4’, and function Adoes not exist. When in doubt, insert

the * sign manually.

Parentheses

You need to use parentheses to enclose arguments for

functions, such as SIN(45). You can omit the final

parenthesis at the end of an edit line. The calculator

inserts it automatically.

Parentheses are also important in specifying the order of

operation. Without parentheses, the HP 40gs calculates

according to the order of algebraic precedence (the next

topic). Following are some examples using parentheses.

Entering...

Calculates...

sin (45 + π)

sin (45) + π

85 × 9

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hp40g+.book Page 22 Friday, December 9, 2005 1:03 AM

Algebraic

precedence

order of

Functions within an expression are evaluated in the

following order of precedence. Functions with the same

precedence are evaluated in order from left to right.

1. Expressions within parentheses. Nested parentheses

are evaluated from inner to outer.

evaluation

2. Prefix functions, such as SIN and LOG.

3. Postfix functions, such as !

4. Power function, ^, NTHROOT.

5. Negation, multiplication, and division.

6. Addition and subtraction.

7. AND and NOT.

8. OR and XOR.

9. Left argument of | (where).

10.Equals, =.

Largest and

smallest

numbers

The smallest number the HP 40gs can represent is

–499

1 × 10

(1E–499). A smaller result is displayed as

499

zero. The largest number is 9.99999999999 × 10

(1E499). A greater result is displayed as this number.

Clearing

numbers

•

clears the character under the cursor. When the

cursor is positioned after the last character,

deletes the character to the left of the cursor, that is, it

performs the same as a backspace key.

•

CANCEL (

) clears the edit line.

CLEAR clears all input and output in the

display, including the display history.

Using previous

results

The HOME display (

) shows you four lines of

input/output history. An unlimited (except by memory)

number of previous lines can be displayed by scrolling.

You can retrieve and reuse any of these values or

expressions.

Input

Output

Last input

Last output

Edit line

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When you highlight a previous input or result (by pressing

), the and menu labels appear.

To copy a previous

line

Highlight the line (press

) and press

. The

number (or expression) is copied into the edit line.

To reuse the last

result

Press

ANS (last answer) to put the last result from the

HOME display into an expression. ^ANSis a variable that

is updated each time you press

To repeat a

previous line

To repeat the very last line, just press

. Otherwise,

highlight the line (press

) first, and then press

The highlighted expression or number is re-entered. If the

previous line is an expression containing the ^ANS, the

calculation is repeated iteratively.

Example

See how

(50), and

ANS retrieves and reuses the last result

updates ^ANS(from 50 to 75 to 100).

You can use the last result as the first expression in the edit

line without pressing ANS. Pressing , or

, (or other operators that require a preceding

argument) automatically enters ANS before the operator.

You can reuse any other expression or value in the HOME

display by highlighting the expression (using the arrow

keys), then pressing

. See “Using previous results”

on page 1-22 for more details.

The variable ^ANSis different from the numbers in HOME’s

display history. A value in ^ANSis stored internally with the

full precision of the calculated result, whereas the

displayed numbers match the display mode.

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hp40g+.book Page 24 Friday, December 9, 2005 1:03 AM

H I N T

When you retrieve a number from ANS, you obtain the

result to its full precision. When you retrieve a number

from the HOME’s display history, you obtain exactly what

was displayed.

Pressing

whereas pressing

into the edit line.

evaluates (or re-evaluates) the last input,

ANS copies the last result (as ANS)

Storing a value

in a variable

You can save an answer in a variable and use the

variable in later calculations. There are 27 variables

available for storing real values. These are A to Z and θ.

See Chapter 17, “Variables and memory management”

for more information on variables. For example:

1. Perform a calculation.

2. Store the result in the A variable.

3. Perform another calculation using the A variable.

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Accessing the

display history

Pressing

enables the highlight bar in the display

history. While the highlight bar is active, the following

menu and keyboard keys are very useful:

Key

Function

Scrolls through the display history.

Copies the highlighted expression to

the position of the cursor in the edit line.

Displays the current expression in

standard mathematical form.

Deletes the highlighted expression from

the display history, unless there is a

cursor in the edit line.

Clears all lines of display history and

the edit line.

CLEAR

Clearing the

display history

It’s a good habit to clear the display history (

CLEAR) whenever you have finished working in HOME. It

saves calculator memory to clear the display history.

Remember that all your previous inputs and results are

saved until you clear them.

Using fractions

To work with fractions in HOME, you set the number

format to Fractionor Mixed Fraction, as follows:

Setting Fraction

mode

1. In HOME, open the HOME MODES input form.

MODES

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hp40g+.book Page 26 Friday, December 9, 2005 1:03 AM

2. Select Number Format, press

to display the

options, and highlight Fractionor Mixed

Fraction.

3. Press

to select the Number Format option, then

move to the precision value field.

4. Enter the precision value that you want to use, and

press

to HOME.

to set the precision. Press

to return

See “Setting fraction precision” below for more

information.

Setting fraction

precision

The fraction precision setting determines the precision in

which the HP 40gs converts a decimal value to a fraction.

The greater the precision value that is set, the closer the

fraction is to the decimal value.

By choosing a precision of 1 you are saying that the

fraction only has to match 0.234 to at least 1 decimal

place (3/13 is 0.23076...).

The fractions used are found using the technique of

continued fractions.

When converting recurring decimals this can be

important. For example, at precision 6 the decimal

0.6666 becomes 3333/5000 (6666/10000) whereas

at precision 3, 0.6666 becomes 2/3, which is probably

what you would want.

For example, when converting .234 to a fraction, the

precision value has the following effect:

1-26

Getting started

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•

Precision set to 1:

Precision set to 2:

Precision set to 3:

Precision set to 4

Fraction

calculations

When entering fractions:

•

You use the

key to separate the numerator part

and the denominator part of the fraction.

•

To enter a mixed fraction, for example, 1 / , you

enter it in the format (1+ / ).

For example, to perform the following calculation:

3(2 / + 5 / )

1. Set the Number format mode to Fractionor

Mixed Fractionand specify a precision value of

4.In this example, we’ll select Fractionas our

format.)

MODES

Select

Fraction

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hp40g+.book Page 28 Friday, December 9, 2005 1:03 AM

2. Enter the calculation.

Note: Ensure you are in

the HOME view.

3. Evaluate the calculation.

Note that if you had

selected Mixed

Fractioninstead of

Fractionas the

Number format, the answer would have been

expressed as 25+7/8.

Converting

decimals to

fractions

To convert a decimal value to a fraction:

1. Set the number format mode to Fraction or Mixed

Fraction.

2. Either retrieve the value from the History, or enter the

value on the command line.

3. Press

to convert the number to a fraction.

When converting a decimal to a fraction, keep the

following points in mind:

•

When converting a recurring decimal to a fraction,

set the fraction precision to about 6, and ensure that

you include more than six decimal places in the

recurring decimal that you enter.

In this example, the

fraction precision is set

to 6. The top

calculation returns the

correct result. The

bottom one does not.

•

To convert an exact decimal to a fraction, set the

fraction precision to at least two more than the

number of decimal places in the decimal.

1-28

Getting started

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In this example, the

fraction precision is set

to 6.

Complex numbers

Complex results

The HP 40gs can return a complex number as a result for

some math functions. A complex number appears as an

ordered pair (x, y), where x is the real part and y is the

imaginary part. For example, entering –1 returns (0,1).

To enter complex

numbers

Enter the number in either of these forms, where x is the

real part, y is the imaginary part, and i is the imaginary

constant, –1 :

•

(x, y) or

x + iy.

To enter i:

•

press

keys to select Constant,

to move to the right column of the menu,

select i, and

Storing complex

numbers

There are 10 variables available for storing complex

numbers: Z0 to Z9. To store a complex number in a

variable:

•

Enter the complex number, press

, enter the

variable to store the number in, and press

Z 0

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hp40g+.book Page 30 Friday, December 9, 2005 1:03 AM

Catalogs and editors

The HP 40gs has several catalogs and editors. You use

them to create and manipulate objects. They access

features and stored values (numbers or text or other items)

that are independent of aplets.

•

A catalog lists items, which you can delete or

transmit, for example an aplet.

•

An editor lets you create or modify items and

numbers, for example a note or a matrix.

Catalog/Editor

Contents

Aplet library

Aplets.

(

)

Sketch editor

Sketches and diagrams, See

Chapter 20, “Notes and

sketches”.

(

SKETCH)

Lists. In HOME, lists are

enclosed in {}. See Chapter 19,

“Lists”.

List (

LIST)

One- and two-dimensional

arrays. In HOME, arrays are

enclosed in []. See Chapter 18,

“Matrices”.

Matrix (

MATRIX)

Notes (short text entries). See

Chapter 20, “Notes and

sketches”.

Notepad (

NOTEPAD)

Programs that you create, or

associated with user-defined

aplets. See Chapter 21,

“Programming”.

Program (

PROGRM)

Equation Writer

The editor used for creating

expressions and equations in

CAS. See Chapter 15,

“Equation Writer”.

(

)

1-30

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Aplets and their views

Aplet views

This section examines the options and functionality of the

three main views for the Function, Polar, Parametric, and

Sequence aplets: Symbolic, Plot, and Numeric views.

About the Symbolic view

The Symbolic view is the defining view for the Function,

Parametric, Polar, and Sequence aplets. The other views

are derived from the symbolic expression.

You can create up to 10 different definitions for each

Function, Parametric, Polar, and Sequence aplet. You

can graph any of the relations (in the same aplet)

simultaneously by selecting them.

Defining an expression (Symbolic view)

Choose the aplet from the Aplet Library.

Press

select an aplet.

The Function,

Parametric, Polar, and Sequence aplets start in the

Symbolic view.

If the highlight is on an existing expression, scroll to

an empty line—unless you don’t mind writing over the

expression—or, clear one line (

) or all lines

(

CLEAR).

Expressions are selected (check marked) on entry. To

deselect an expression, press

expressions are plotted.

. Allselected

Aplets and their views

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hp40g+.book Page 2 Friday, December 9, 2005 1:03 AM

– For a Function

definition, enter

an expression to

define F(X). The

only independent

variable in the

expression is X.

– For a

Parametric

definition, enter

a pair of

expressions to

define X(T) and

Y(T). The only

independent variable in the expressions is T.

– For a Polar

definition, enter

an expression to

define R(θ). The

only independent

variable in the

expression is θ.

– For a Sequence

definition, either

enter the first term,

or the first and

second terms, for U

(U1, or...U9, or

U0). Then define

the nth term of the sequence in terms of N or of

the prior terms, U(N–1) and/or U(N–2). The

expressions should produce real-valued

sequences with integer domains. Or define the

nth term as a non-recursive expression in terms of

n only. In this case, the calculator inserts the first

two terms based on the expression that you

define.

–

Note: You will have to enter the second term if the

hp40gs is unable to calculate it automatically.

Typically if Ux(N) depends on Ux(N–2) then you

must enter Ux(2).

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Evaluating expressions

In aplets

In the Symbolic view, a variable is a symbol only, and

does not represent one specific value. To evaluate a

function in Symbolic view, press . If a function calls

another function, then resolves all references to

other functions in terms of their independent variable.

1. Choose the Function

aplet.

Select Function

2. Enter the expressions in the Function aplet’s Symbolic

view.

3. Highlight F3(X).

4. Press

Note how the values

for F1(X) and F2(X) are

substituted into F3(X).

Aplets and their views

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hp40g+.book Page 4 Friday, December 9, 2005 1:03 AM

In HOME

You can also evaluate any expression in HOME by

entering it into the edit line and pressing

For example, define F4 as below. In HOME, type

F4(9)and press . This evaluates the expression,

substituting 9in place of Xinto F4.

SYMB view keys The following table details the menu keys that you use to

work with the Symbolic view.

Key

Meaning

Copies the highlighted expression to

the edit line for editing. Press

when done.

Checks/unchecks the current

expression (or set of expressions).

Only checked expression(s) are

evaluated in the Plot and Numeric

views.

Enters the independent variable in the

Function aplet. Or, you can use the

key on the keyboard.

Enters the independent variable in the

Parametric aplet. Or, you can use the

key on the keyboard.

Enters the independent variable in the

Polar aplet. Or, you can use the

key on the keyboard.

Enters the independent variable in the

Sequence aplet. Or, you can use the

key on the keyboard.

Displays the current expression in text

book form.

Resolves all references to other

definitions in terms of variables and

evaluates all arithmetic expressions.

Displays a menu for entering variable

names or contents of variables.

2-4

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hp40g+.book Page 5 Friday, December 9, 2005 1:03 AM

Key

Meaning (Continued)

Displays the menu for entering math

operations.

CHARS

Displays special characters. To enter

one, place the cursor on it and press

. To remain in the CHARS menu

and enter another special character,

press

Deletes the highlighted expression or

the current character in the edit line.

CLEAR

Deletes all expressions in the list or

clears the edit line.

About the Plot view

After entering and selecting (check marking) the

expression in the Symbolic view, press

. To adjust

the appearance of the graph or the interval that is

displayed, you can change the Plot view settings.

You can plot up to ten expressions at the same time.

Select the expressions you want to be plotted together.

Setting up the plot (Plot view setup)

Press

SETUP-PLOT to define any of the settings

shown in the next two tables.

1. Highlight the field to edit.

–

If there is a number to enter, type it in and press

If there is an option to choose, press

highlight your choice, and press

As a shortcut to

change and press

options.

, just highlight the field to

to cycle through the

–

If there is an option to select or deselect, press

to check or uncheck it.

2. Press

to view more settings.

to view the new plot.

3. When done, press

Aplets and their views

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hp40g+.book Page 6 Friday, December 9, 2005 1:03 AM

Plot view

settings

The plot view settings are:

Field

Meaning

XRNG, YRNG

Specifies the minimum and

maximum horizontal (X) and

vertical (Y) values for the plotting

window.

RES

For function plots: Resolution;

“Faster” plots in alternate pixel

columns; “Detail” plots in every

pixel column.

TRNG

θRNG

Parametric aplet: Specifies the t-

values (T) for the graph.

Polar aplet: Specifies the angle (θ)

value range for the graph.

NRNG

TSTEP

θSTEP

Sequence aplet: Specifies the

index (N) values for the graph.

For Parametric plots: the increment

for the independent variable.

For Polar plots: the increment

value for the independent

variable.

SEQPLOT

For Sequence aplet: Stairstep or

Cobweb types.

XTICK

YTICK

Horizontal spacing for tickmarks.

Vertical spacing for tickmarks.

Those items with space for a checkmark are settings you

can turn on or off. Press

page.

to display the second

Field

Meaning

SIMULT

If more than one relation is being

plotted, plots them simultaneously

(otherwise sequentially).

INV. CROSS

Cursor crosshairs invert the status

of the pixels they cover.

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Field

Meaning (Continued)

CONNECT

Connect the plotted points. (The

Sequence aplet always connects

them.)

LABELS

Label the axes with XRNGand

YRNGvalues.

AXES

GRID

Draw the axes.

Draw grid points using XTICK

and YTICKspacing.

Reset plot

settings

To reset the default values for all plot settings, press

CLEAR in the Plot Setup view. To reset the default

value for a field, highlight the field, and press

Exploring the graph

Plot view gives you a selection of keys and menu keys to

explore a graph further. The options vary from aplet to

aplet.

PLOT view keys

The following table details the keys that you use to work

with the graph.

Key

Meaning

CLEAR

Erases the plot and axes.

Offers additional pre-defined views

for splitting the screen and for scaling

(“zooming”) the axes.

Moves cursor to far left or far right.

Moves cursor between relations.

Interrupts plotting.

Continues plotting if interrupted.

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Key

Meaning (Continued)

Turns menu-key labels on and off.

When the labels are off, pressing

turns them back on.

•

Pressing

once displays the

full row of labels.

•

Pressing

a second time

removes the row of labels to

display only the graph.

•

Pressing

a third time

displays the coordinate mode.

Displays the ZOOM menu list.

Turns trace mode on/off. A white box

appears over the

Opens an input form for you to enter

an X (or T or N or θ) value. Enter the

value and press

. The cursor jumps

to the point on the graph that you

entered.

Function aplet only: turns on menu list

for root-finding functions (see

“Analyse graph with FCN functions”

on page 3-4).

Displays the current, defining

expression. Press

menu.

to restore the

Trace a graph

You can trace along a function using the

key

which moves the cursor along the graph. The display also

shows the current coordinate position (x, y) of the cursor.

Trace mode and the coordinate display are automatically

set when a plot is drawn.

Note: Tracing might not appear to exactly follow your

plot if the resolution (in Plot Setup view) is set to Faster.

This is because RES: FASTER plots in only every other

column, whereas tracing always uses every column.

In Function and Sequence Aplets: You can also

scroll (move the cursor) left or right beyond the edge of

the display window in trace mode, giving you a view of

more of the plot.

To move between

relations

If there is more than one relation displayed, press

to move between relations.

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To jump directly to

a value

To jump straight to a value rather than using the Trace

function, use the

a value. Press

menu key. Press

, then enter

to jump to the value.

To turn trace on/off

If the menu labels are not displayed, press

first.

•

Turn off trace mode by pressing

Turn on trace mode by pressing

To turn the coordinate display off, press

Zoom within a

graph

One of the menu key options is

the plot on a larger or smaller scale. It is a shortcut for

changing the Plot Setup.

. Zooming redraws

The Set Factors...option enables you to set the

factors by which you zoom in or zoom out, and whether

the zoom is centered about the cursor.

ZOOM options

Press

, select an option, and press

. (If

options are

is not displayed, press

available in all aplets.

.) Not all

Option

Meaning

Center

Re-centers the plot around the

current position of the cursor without

changing the scale.

Box...

Lets you draw a box to zoom in on.

See “Other views for scaling and

splitting the graph” on page 2-13.

Divides horizontal and vertical

scales by the X-factor and Y-factor.

For instance, if zoom factors are 4,

then zooming in results in 1/4 as

many units depicted per pixel. (see

Set Factors...)

Out

Multiplies horizontal and vertical

scales by the X-factor and Y-factor

(see Set Factors...).

X-Zoom In

Divides horizontal scale only, using

X-factor.

X-Zoom Out

Multiplies horizontal scale, using

X-factor.

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Option

Meaning (Continued)

Y-Zoom In

Divides vertical scale only, using

Y-factor.

Y-Zoom Out

Square

Multiplies vertical scale only, using

Y-factor.

Changes the vertical scale to match

the horizontal scale. (Use this after

doing a Box Zoom, X-Zoom, or

Y-Zoom.)

Set

Factors...

Sets the X-Zoom and Y-Zoom factors

for zooming in or zooming out.

Includes option to recenter the plot

before zooming.

Auto Scale

Rescales the vertical axis so that the

display shows a representative

piece of the plot, for the supplied x

axis settings. (For Sequence and

Statistics aplets, autoscaling

rescales both axes.)

The autoscale process uses the first

selected function only to determine

the best scale to use.

Decimal

Rescales both axes so each pixel =

0.1 units. Resets default values for

XRNG

(–6.5 to 6.5) and YRNG (–3.1 to

3.2). (Not in Sequence or Statistics

aplets.)

Integer

Trig

Rescales horizontal axis only,

making each pixel =1 unit. (Not

available in Sequence or Statistics

aplets.)

Rescales horizontal axis so

1 pixel = π/24 radians, 7.58, or

8 / grads; rescales vertical axis

1 pixel = 0.1 unit.

(Not in Sequence or Statistics

aplets.)

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Option

Meaning (Continued)

Un-zoom

Returns the display to the previous

zoom, or if there has been only one

zoom, un-zoom displays the graph

with the original plot settings.

ZOOM examples

The following screens show the effects of zooming options

on a plot of 3sinx .

Plot of 3sinx

Zoom In:

Un-zoom:

Un-zoom

Note: Press

to move to

the bottom of the Zoom list.

Zoom Out:

Out

Now un-zoom.

X-Zoom In:

X-Zoom In

Now un-zoom.

X-Zoom Out:

X-Zoom Out

Now un-zoom.

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Y-Zoom In:

Y-Zoom In

Now un-zoom.

Y-Zoom Out:

Y-Zoom Out

Zoom Square:

Square

To box zoom

The Box Zoom option lets you draw a box around the

area you want to zoom in on by selecting the endpoints

of one diagonal of the zoom rectangle.

1. If necessary, press

labels.

to turn on the menu-key

2. Press

3. Position the cursor on one corner of the rectangle.

Press

4. Use the cursor keys

, etc.) to drag to

the opposite corner.

and select Box...

(

5. Press

to zoom in

on the boxed area.

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To set zoom factors

1. In the Plot view, press

2. Press

3. Select Set Factors...and press

4. Enter the zoom factors. There is one zoom factor for

the horizontal scale (XZOOM) and one for the vertical

scale (YZOOM).

Zooming out multiplies the scale by the factor, so that

a greater scale distance appears on the screen.

Zooming in divides the scale by the factor, so that a

shorter scale distance appears on the screen.

Other views for scaling and splitting the graph

The preset viewing options menu (

) contains

options for drawing the plot using certain pre-defined

configurations. This is a shortcut for changing Plot view

settings. For instance, if you have defined a trigonometric

function, then you could select Trigto plot your function

on a trigonometric scale. It also contains split-screen

options.

In certain aplets, for example those that you download

from the world wide web, the preset viewing options

menu can also contain options that relate to the aplet.

VIEWS menu

options

Press

, select an option, and press

Option

Meaning

Plot-

Detail

Splits the screen into the plot and a

close-up.

Plot-Table

Splits the screen into the plot and

the data table.

Overlay

Plot

Plots the current expression(s)

without erasing any pre-existing

plot(s).

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Option

Meaning (Continued)

Auto Scale

Rescales the vertical axis so that the

display shows a representative

piece of the plot, for the supplied x

axis settings. (For Sequence and

Statistics aplets, autoscaling

rescales both axes.)

The autoscale process uses the first

selected function only to determine

the best scale to use.

Decimal

Rescales both axes so each pixel =

0.1 unit. Resets default values for

XRNG

(–6.5 to 6.5) and YRNG (–3.1 to

3.2). (Not in Sequence or Statistics

aplets.)

Integer

Trig

Rescales horizontal axis only,

making each pixel=1 unit. (Not

available in Sequence or Statistics

aplets.)

Rescales horizontal axis so

1 pixel=π/24 radian, 7.58, or

8 / grads; rescales vertical axis so

1 pixel =0.1 unit.

(Not in Sequence or Statistics

aplets.)

Split the screen

The Plot-Detail view can give you two simultaneous views

of the plot.

1. Press

. Select Plot-Detailand press

The graph is plotted twice. You can now zoom in on

the right side.

2. Press

select the zoom method

and press

. This zooms the

right side. Here is an

example of split screen with Zoom In.

–

The Plot menu keys are available as for the full

plot (for tracing, coordinate display, equation

display, and so on).

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–

moves the leftmost cursor to the

screen’s left edge and moves the

rightmost cursor to the screen’s right edge.

–

The

plot.

menu key copies the right plot to the left

3. To un-split the screen, press

over the whole screen.

. The left side takes

The Plot-Table view gives you two simultaneous views of

the plot.

1. Press

. Select

Plot-Tableand

press

. The screen

displays the plot on the

left side and a table of

numbers on the right side.

2. To move up and down the table, use the

and

cursor keys. These keys move the tra.ce point left or

right along the plot, and in the table, the

corresponding values are highlighted.

3. To move between functions, use the

and

cursor keys to move the cursor from one graph to

another.

4. To return to a full Numeric (or Plot) view, press

(or

Overlay plots

If you want to plot over an existing plot without erasing

that plot, then use Overlay Plotinstead of

. Note that tracing follows only the current

functions from the current aplet.

Decimal scaling

Integer scaling

Decimal scaling is the default scaling. If you have

changed the scaling to Trig or Integer, you can change it

back with Decimal.

Integer scaling compresses the axes so that each pixel is

1 × 1 and the origin is near the screen center.

Trigonometric

scaling

Use trigonometric scaling whenever you are plotting an

expression that includes trigonometric functions.

Trigonometric plots are more likely to intersect the axis at

points factored by π.

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About the numeric view

Afterenteringandselecting

(check marking) the

expression or expressions

that you want to explore in

the Symbolic view, press

to view a table of

data values for the independent variable (X, T, θ, or N)

and dependent variables.

Setting up the table (Numeric view setup)

Press

NUM to define

any of the table settings.

Use the Numeric Setup

input form to configure the

table.

1. Highlight the field to edit. Use the arrow keys to move

from field to field.

–

If there is a number to enter, type it in and press

or . To modify an existing number,

press

–

If there is an option to choose, press

highlight your choice, and press

– Shortcut: Press the

key to copy values

from the Plot Setup into NUMSTARTand

NUMSTEP. Effectively, the

you to make the table match the pixel columns in

the graph view.

menu key allows

2. When done, press

numbers.

to view the table of

Numeric view

settings

The following table details the fields on the Numeric

Setup input form.

Field

Meaning

NUMSTART

The independent variable’s

starting value.

NUMSTEP

The size of the increment from

one independent variable value

to the next.

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Field

Meaning (Continued)

NUMTYPE

Type of numeric table: Automatic

or Build Your Own. To build your

own table, you must type each

independent value into the table

yourself.

NUMZOOM

Allows you to zoom in or out on a

selected value of the independent

variable.

Reset numeric

settings

To reset the default values for all table settings, press

CLEAR.

Exploring the table of numbers

NUM view

menu keys

The following table details the menu keys that you use to

work with the table of numbers.

Key

Meaning

Displays ZOOM menu list.

Toggles between two character

sizes.

Displays the defining function

expression for the highlighted

column. To cancel this display, press

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Zoom within a

table

Zooming redraws the table of numbers in greater or

lesser detail.

ZOOM options

The following table lists the zoom options:

Option

Meaning

Decreases the intervals for the

independent variable so a narrower

range is shown. Uses the NUMZOOM

factor in Numeric Setup.

Out

Increases the intervals for the

independent variable so that a

wider range is shown. Uses the

NUMZOOMfactor in Numeric Setup.

Decimal

Integer

Trig

Changes intervals for the

independent variable to 0.1 units.

Starts at zero. (Shortcut to changing

NUMSTARTand NUMSTEP.)

Changes intervals for the

independent variable to 1 unit.

Starts at zero. (Shortcut to changing

NUMSTEP.)

Changes intervals for independent

variable to π/24 radian or 7.5

degrees or 8 / grads. Starts at

zero.

Un-zoom

Returns the display to the previous

zoom.

The display on the right is a Zoom In of the display on the

left. The ZOOMfactor is 4.

H I N T

To jump to an independent variable value in the table,

use the arrow keys to place the cursor in the independent

variable column, then enter the value to jump to.

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Automatic

recalculation

You can enter any new value in the X column. When you

press

, the values for the dependent variables are

recalculated, and the entire table is regenerated with the

same interval between X values.

Building your own table of numbers

The default NUMTYPEis “Automatic”, which fills the table

with data for regular intervals of the independent (X, T, θ,

or N) variable. With the NUMTYPEoption set to “Build

Your Own”, you fill the table yourself by typing in the

independent-variable values you want. The dependent

values are then calculated and displayed.

Build a table

1. Start with an expression defined (in Symbolic view) in

the aplet of your choice. Note: Function, Polar,

Parametric, and Sequence aplets only.

2. In the Numeric Setup (

NUM), choose

NUMTYPE: Build Your Own.

3. Open the Numeric view (

4. Clear existing data in the table (

CLEAR).

5. Enter the independent values in the left-hand column.

Type in a number and press

to enter them in order, because the

can rearrange them. To insert a number between two

. You do not have

function

others, use

F1 and F2

entries are

generated

automatically

You enter

numbers into

the X column

Clear data

Press

CLEAR,

to erase the data from a table.

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“Build Your Own” menu keys

Key

Meaning

Puts the highlighted independent

value (X, T, θ, or N) into the edit

line. Pressing

replaces

this variable with its current value.

Inserts a zero value at the position

of the highlight. Replace a zero

by typing the number you want

and pressing

Sorts the independent variable

values into ascending or

descending order. Press

and select the ascending or

descending option from the

menu, and press

Toggles between two character

sizes.

Displays the defining function

expression for the highlighted

column.

Deletes the highlighted row.

Clears all data from the table.

CLEAR

Example: plotting a circle

Plot the circle, x + y = 9. First rearrange it to read

y = ± 9 – x².

To plot both the positive and negative y values, you need

to define two equations as follows:

y = 9 – x²and y = – 9 – x²

1. In the Function aplet, specify the functions.

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Select

Function

2. Reset the graph setup to the default settings.

SETUP-PLOT

CLEAR

3. Plot the two functions

and hide the menu so

that you can see all the

circle.

4. Reset the numeric setup to the default settings.

SETUP-NUM

CLEAR

5. Display the functions in numeric form.

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Function aplet

About the Function aplet

The Function aplet enables you to explore up to 10

real-valued, rectangular functions y in terms of x. For

example y = 2x + 3 .

Once you have defined a function you can:

•

create graphs to find roots, intercepts, slope, signed

area, and extrema

•

create tables to evaluate functions at particular

values.

This chapter demonstrates the basic tools of the Function

aplet by stepping you through an example. See “Aplet

views” on page 2-1 for further information about the

functionality of the Symbolic, Numeric, and Plot views.

Getting started with the Function aplet

The following example involves two functions: a linear

function y = 1 – x and a quadratic equation

y = (x + 3)²– 2 .

Open the

Function aplet

1. Open the Function aplet.

Select Function

The Function aplet starts

in the Symbolic view.

The Symbolic view is the defining view for Function,

Parametric, Polar, and Sequence aplets. The other

views are derived from the symbolic expression.

Function aplet

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Define the

expressions

2. There are 10 function definition fields on the Function

aplet’s Symbolic view screen. They are labeled F1(X)

to F0(X). Highlight the function definition field you

want to use, and enter an expression. (You can press

to delete an existing line, or

clear all lines.)

CLEAR to

Set up the plot

You can change the scales of the x and y axes, graph

resolution, and the spacing of the axis ticks.

3. Display plot settings.

SETUP-PLOT

Note: For our example, you can leave the plot

settings at their default values since we will be using

the Auto Scale feature to choose an appropriate y

axis for our x axis settings. If your settings do not

match this example, press

default values.

CLEAR to restore the

4. Specify a grid for the graph.

Plot the

functions

5. Plot the functions.

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Change the

scale

6. You can change the scale to see more or less of your

graphs. In this example, choose Auto Scale. (See

“VIEWS menu options” on page 2-13 for a

description of Auto Scale).

Select Auto

Scale

Trace a graph

7. Trace the linear function.

6 times

Note: By default, the

tracer is active.

8. Jump from the linear function to the quadratic

function.

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Analyse graph

with FCN

functions

9. Display the Plot view menu.

From the Plot view menu, you can use the functions

on the FCN menu to find roots, intersections, slopes,

and areas for a function defined in the Function aplet

(and any Function-based aplets). The FCN functions

act on the currently selected graph. See “FCN

functions” on page 3-10 for further information.

To find a root of the

quadratic function

10.Move the cursor to the graph of the quadratic

equation by pressing the

the cursor so that it is near x = –1 by pressing the

or key.

key. Then move

SelectRoot

The root value is

displayed at the

bottom of the screen.

Note: If there is more

than one root (as in our

example), the

coordinates of the root closest to the current cursor

position are displayed.

To find the

11.Find the intersection of the two functions.

intersection of the

two functions

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12.Choose the linear function whose intersection with the

quadratic function you wish to find.

The coordinates of the

intersection point are

displayed at the

bottom of the screen.

Note: If there is more

than one intersection

(as in our example), the coordinates of the

intersection point closest to the current cursor position

are displayed.

To find the slope of

the quadratic

function

13.Find the slope of the quadratic function at the

intersection point.

Select Slope

The slope value is

displayed at the

bottom of the screen.

To find the signed

area of the two

functions

14.To find the area between the two functions in the

range –2 ≤ x ≤ –1, first move the cursor to

F1(x) = 1 – x and select the signed area option.

Select Signedarea

15.Move the cursor to x = –2 by pressing the

key.

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16.Press

to accept using F2(x) = (x + 3) – 2 as the

other boundary for the integral.

17. Choose the end value

for x.

The cursor jumps to

x = –1 on the linear

function.

18.Display the numerical

value of the integral.

Note: See “Shading

area” on page 3-11

for another method of

calculating area.

To find the

extremum of the

quadratic

19.Move the cursor to the quadratic equation and find

the extremum of the quadratic.

Select Extremum

The coordinates of the

extremum are

displayed at the

bottom of the screen.

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H I N T

The Root and Extremum functions return one value only

even if the function has more than one root or extremum.

The function finds the value closest to the position of the

cursor. You need to re-locate the cursor to find other roots

or extrema that may exist.

Display the

numeric view

20.Display the numeric view.

Set up the table

21.Display the numeric setup.

SETUP-NUM

See “Setting up the table (Numeric view setup)” on

page 2-16 for more information.

22.Match the table settings to the pixel columns in the

graph view.

Explore the

table

23.Display the table of values.

Function aplet

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To navigate around

a table

24.Move to X = –5.9.

6 times

To go directly to a

value

25.Move directly to X = 10.

1 0

To access the zoom

options

26.Zoom in on X = 10 by a factor of 4. Note: NUMZOOM

has a setting of 4.

To change font size

27. Display table numbers in large font.

To display the

symbolic definition

of a column

28.Display the symbolic definition for the F1 column.

The symbolic definition of

F1 is displayed at the

bottom of the screen.

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Function aplet interactive analysis

From the Plot view (

), you can use the functions on

the FCN menu to find roots, intersections, slopes, and

areas for a function defined in the Function aplet (and any

Function-based aplets). See “FCN functions” on page 3-

10. The FCN operations act on the currently selected

graph.

The results of the FCN functions are saved in the following

variables:

•

Area

Extremum

Isect

Root

Slope

For example, if you use the Root function to find the root

of a plot, you can use the result in calculations in HOME.

Access FCN

variables

The FCN variables are contained on the VARS menu.

To access FCN variables in HOME:

Select Plot FCN

to choose a

variable

To access FCN variable in the Function aplet’s Symbolic

view:

Select Plot FCN

to choose a variable

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FCN functions

The FCN functions are:

Function

Description

Root

Select Rootto find the root of the

current function nearest the

cursor. If no root is found, but only

an extremum, then the result is

labeled EXTR:instead of ROOT:.

(The root-finder is also used in the

Solve aplet. See also “Interpreting

results” on page 7-6.) The cursor

is moved to the root value on the

x-axis and the resulting x-value is

saved in a variable named

ROOT.

Extremum

Select Extremumto find the

maximum or minimum of the

current function nearest the

cursor. This displays the

coordinate values and moves the

cursor to the extremum. The

resulting value is saved in a

variable named EXTREMUM.

Slope

Select Slopeto find the numeric

derivative at the current position

of the cursor. The result is saved in

a variable named SLOPE.

Signed area

Select Signed areato find the

numeric integral. (If there are two

or more expressions

checkmarked, then you will be

asked to choose the second

expression from a list that

includes the x-axis.) Select a

starting point, then move the

cursor to selection ending point.

The result is saved in a variable

named AREA.

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Function

Description (Continued)

Intersection

Select Intersectionto find the

intersection of two graphs nearest

the cursor. (You need to have at

least two selected expressions in

Symbolic view.) Displays the

coordinate values and moves the

cursor to the intersection. (Uses

Solve function.) The resulting x-

value is saved in a variable

named ISECT.

Shading area

You can shade a selected area between functions. This

process also gives you an approximate measurement of

the area shaded.

1. Open the Function aplet. The Function aplet opens in

the Symbolic view.

2. Select the expressions whose curves you want to

study.

3. Press

4. Press

to plot the functions.

to position the cursor at the starting

point of the area you want to shade.

5. Press

6. Press

, then select Signed areaand press

7. Press

, choose the function that will act as the

boundary of the shaded area, and press

8. Press the

9. Press

key to shade in the area.

to calculate the area. The area

measurement is displayed near the bottom of the

screen.

To remove the shading, press

to re-draw the plot.

Function aplet

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Plotting a piecewise-defined function

Suppose you wanted to plot the following piecewise-

defined function.

⎧

⎪

⎨

⎪

⎩

x + 2 ;x ≤ –1

x²

;–1 < x ≤ 1

f(x) =

4 – x ;x ≥ 1

1. Open the Function

aplet.

Select

Function

2. Highlight the line you want to use, and enter the

expression. (You can press

to delete an existing

line, or CLEAR to clear all lines.)

CHARS ≤

CHARS >

AND

CHARS ≤ 1

CHARS > 1

Note: You can use the

menu key to assist in the

entry of equations. It has the same effect as pressing

3-12

Function aplet

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Parametric aplet

About the Parametric aplet

The Parametric aplet allows you to explore parametric

equations. These are equations in which both x and y are

defined as functions of t. They take the forms x = f(t)

and y = g(t) .

Getting started with the Parametric aplet

The following example uses the parametric equations

x(t) = 3sint

y(t) = 3cost

Note: This example will produce a circle. For this

example to work, the angle measure must be set to

degrees.

Open the

Parametric aplet

1. Open the Parametric aplet.

Select

Parametric

Define the

expressions

2. Define the expressions.

Parametric aplet

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hp40g+.book Page 2 Friday, December 9, 2005 1:03 AM

Set angle

measure

3. Set the angle measure to degrees.

MODES

Select Degrees

Set up the plot

4. Display the graphing options.

PLOT

The Plot Setup input form has two fields not included

in the Function aplet, TRNGand TSTEP. TRNG

specifies the range of t values. TSTEPspecifies the

step value between t values.

5. Set the TRNGand TSTEPso that t steps from 0° to

360° in 5° steps.

360

Plot the

expression

6. Plot the expression.

7. To see all the circle, press

twice.

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Overlay plot

8. Plot a triangle graph over the existing circle graph.

PLOT

120

Select OverlayPlot

A triangle is displayed

rather than a circle (without changing the equation)

because the changed value of TSTEPensures that

points being plotted are 120° apart instead of nearly

continuous.

You are able to explore the graph using trace, zoom,

split screen, and scaling functionality available in the

Function aplet. See “Exploring the graph” on page 2-

7 for further information.

Display the

numbers

9. Display the table of values.

You can highlight a

t-value, type in a

replacement value,

and see the table jump

to that value. You can also zoom in or zoom out on

any t-value in the table.

You are able to explore the table using

, build your own table, and split screen

functionality available in the Function aplet. See

“Exploring the table of numbers” on page 2-17 for

further information.

Parametric aplet

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Polar aplet

Getting started with the Polar aplet

Open the Polar

aplet

1. Open the Polar aplet.

Select Polar

Like the Function aplet,

the Polar aplet opens

in the Symbolic view.

Define the

expression

2. Define the polar equation r = 2πcos(θ ⁄ 2)cos(θ)².

Specify plot

settings

3. Specify the plot settings. In this example, we will use

the default settings, except for the θRNGfields.

SETUP-PLOT

CLEAR

Plot the

expression

4. Plot the expression.

Polar aplet

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Explore the

graph

5. Display the Plot view menu key labels.

The Plot view options

available are the same

as those found in the

Function aplet. See

“Exploring the graph”

on page 2-7 for further information.

Display the

numbers

6. Display the table of values for θ and R1.

The Numeric view

options available are

the same as those

found in the Function

aplet. See “Exploring the table of numbers” on

page 2-17 for further information.

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Sequence aplet

About the Sequence aplet

The Sequence aplet allows you to explore sequences.

You can define a sequence named, for example, U1:

•

in terms of n

in terms of U1(n–1)

in terms of U1(n–2)

in terms of another sequence, for example, U2(n)

in any combination of the above.

The Sequence aplet allows you to create two types of

graphs:

–

A Stairsteps graph plots n on the horizontal

axis and U on the vertical axis.

–

A Cobweb graph plots U

on the horizontal

n–1

axis and U on the vertical axis.

Getting started with the Sequence aplet

The following example defines and then plots an

expression in the Sequence aplet. The sequence

illustrated is the well-known Fibonacci sequence where

each term, from the third term on, is the sum of the

preceding two terms. In this example, we specify three

sequence fields: the first term, the second term and a rule

for generating all subsequent terms.

However, you can also define a sequence by specifying

just the first term and the rule for generating all

subsequent terms. You will, though,have to enter the

second term if the hp40gs is unable to calculate it

automatically. Typically if the nth term in the sequence

depends on n–2, then you must enter the second term.

Sequence aplet

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Open the

Sequence aplet

1. Open the Sequence aplet.

Select

Sequence

The Sequence aplet

starts in the Symbolic

view.

Define the

expression

2. Define the Fibonacci sequence, in which each term

(after the first two) is the sum of the preceding two

terms:

U₁= 1 , U₂= 1 ,U_n= U_{n – 1}+ U_{n – 2}for n > 3 .

In the Symbolic view of the Sequence aplet, highlight

the U1(1) field and begin defining your sequence.

Note: You can use the

, and

equations.

menu keys to assist in the entry of

Specify plot

settings

3. In Plot Setup, first set the SEQPLOToption to

Stairstep. Reset the default plot settings by

clearing the Plot Setup view.

SETUP-PLOT

CLEAR

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Sequence aplet

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Plot the

sequence

4. Plot the Fibonacci

sequence.

5. In Plot Setup, set the SEQPLOT option to Cobweb.

SETUP-PLOT

Select Cobweb

Display the table 6. Display the table of values for this example.

Sequence aplet

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Solve aplet

About the Solve aplet

The Solve aplet solves an equation or an expression for

its unknown variable. You define an equation or

expression in the symbolic view, then supply values for all

the variables except one in the numeric view. Solve works

only with real numbers.

Note the differences between an equation and an

expression:

•

An equation contains an equals sign. Its solution is a

value for the unknown variable that makes both sides

have the same value.

•

An expression does not contain an equals sign. Its

solution is a root, a value for the unknown variable

that makes the expression have a value of zero.

You can use the Solve aplet to solve an equation for any

one of its variables.

When the Solve aplet is started, it opens in the Solve

Symbolic view.

•

In Symbolic view, you specify the expression or

equation to solve. You can define up to ten equations

(or expressions), named E0 to E9. Each equation can

contain up to 27 real variables, named A to Z and θ.

•

In Numeric view, you specify the values of the known

variables, highlight the variable that you want to

solve for, and press

You can solve the equation as many times as you want,

using new values for the knowns and highlighting a

different unknown.

Note: It is not possible to solve for more than one variable

at once. Simultaneous linear equations, for example,

should be solved using the Linear Solver aplet,matrices or

graphs in the Function aplet.

Solve aplet

7-1

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Getting started with the Solve aplet

Suppose you want to find the acceleration needed to

increase the speed of a car from 16.67 m/sec (60 kph)

to 27.78 m/sec (100 kph) in a distance of 100 m.

The equation to solve is:

V²= U²+ 2AD

Open the Solve

aplet

1. Open the Solve aplet.

Select Solve

The Solve aplet starts in

the symbolic view.

Define the

equation

2. Define the equation.

Note: You can use the

entry of equations.

menu key to assist in the

Enter known

variables

3. Display the Solve numeric view screen.

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4. Enter the values for the known variables.

2 7 7 8

1 6 6 7

1 0 0

H I N T

If the Decimal Mark setting in the Modes input form

(

MODES) is set to Comma, use instead of

Solve the

unknown

variable

5. Solve for the unknown variable (A).

Therefore, the acceleration needed to increase the

speed of a car from 16.67 m/sec (60 kph) to 27.78

m/sec

(100 kph) in a distance of 100 m is approximately

2.47 m/s .

Because the variable A in the equation is linear we

know that we need not look for any other solutions.

Plot the

equation

The Plot view shows one graph for each side of the

selected equation. You can choose any of the

variables to be the independent variable.

The current equation isV²= U²+ 2AD.

One of these is Y = V², with V = 27.78 , that is,

Y = 771.7284 . This graph will be a horizontal line.

The other graph will beY = U²+ 2AD, with

U = 16.67 and D = 100 , that is,

Y = 200A + 277.8889 . This graph is also a line. The

desired solution is the value of A where these two

lines intersect.

Solve aplet

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6. Plot the equation for variable A.

Select Auto

Scale

7. Trace along the graph

representing the left

side of the equation

until the cursor nears

the intersection.

20 times

Note the value of A displayed near the bottom left

corner of the screen.

The Plot view provides a convenient way to find an

approximation to a solution instead of using the

Numeric view Solve option. See “Plotting to find

guesses” on page 7-7 for more information.

Solve aplet’s NUM view keys

The Solve aplet’s NUM view keys are:

Key

Meaning

Copies the highlighted value to the

edit line for editing. Press

done.

when

Displays a message about the

solution (see “Interpreting results” on

page 7-6).

Displays other pages of variables, if

any.

Displays the symbolic definition of the

current expression. Press

done.

when

Finds a solution for the highlighted

variable, based on the values of the

other variables.

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Key

Meaning (Continued)

Clears highlighted variable to zero or

deletes current character in edit line,

if edit line is active.

CLEAR

Resets all variable values to zero or

clears the edit line, if cursor is in edit

line.

Use an initial guess

You can usually obtain a faster and more accurate

solution if you supply an estimated value for the unknown

variable before pressing

a solution at the initial guess.

. Solve starts looking for

Before plotting, make sure the unknown variable is

highlighted in the numeric view. Plot the equation to help

you select an initial guess when you don’t know the range

in which to look for the solution. See “Plotting to find

guesses” on page 7-7 for further information.

H I N T

An initial guess is especially important in the case of a

curve that could have more than one solution. In this case,

only the solution closest to the initial guess is returned.

Number format

You can change the number format for the Solve aplet in

the Numeric Setup view. The options are the same as in

HOME MODES: Standard, Fixed, Scientific,

Engineering, Fraction and Mixed Fraction. For all except

Standard, you also specify how many digits of accuracy

you want. See “Mode settings” on page 1-10 for more

information.

You might find it handy to set a different number format

for the Solve aplet if, for example, you define equations

to solve for the value of money. A number format of

Fixed2would be appropriate in this case.

Solve aplet

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Interpreting results

After Solve has returned a solution, press

in the

Numeric view for more information. You will see one of

the following three messages. Press

message.

to clear the

Message

Condition

Zero

The Solve aplet found a point where

both sides of the equation were

equal, or where the expression was

zero (a root), within the calculator's

12-digit accuracy.

Sign Reversal

Solve found two points where the

difference between the two sides of

the equation has opposite signs, but

it cannot find a point in between

where the value is zero. Similarly,

for an expression, where the value

of the expression has different signs

but is not precisely zero. This might

be because either the two points are

neighbours (they differ by one in the

twelfth digit), or the equation is not

real-valued between the two points.

Solve returns the point where the

value or difference is closer to zero.

If the equation or expression is

continuously real, this point is

Solve’s best approximation of an

actual solution.

Extremum

Solve found a point where the value

of the expression approximates a

local minimum (for positive values)

or maximum (for negative values).

This point may or may not be a

solution.

Or: Solve stopped searching at

9.99999999999E499, the largest

number the calculator can

represent.

Note that the value returned is

probably not valid.

7-6

Solve aplet

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If Solve could not find a solution, you will see one of the

following two messages.

Message

Condition

Bad Guess(es)

The initial guess lies outside the

domain of the equation.

Therefore, the solution was not a

real number or it caused an error.

Constant?

The value of the equation is the

same at every point sampled.

H I N T

It is important to check the information relating to the

solve process. For example, the solution that the Solve

aplet finds is not a solution, but the closest that the

function gets to zero. Only by checking the information

will you know that this is the case.

The Root-Finder

at work

You can watch the process of the root-finder calculating

and searching for a root. Immediately after pressing

to start the root-finder, press any key except

You will see two intermediate guesses and, to the left, the

sign of the expression evaluated at each guess. For

example:

+ 2 2.219330555745

– 1 21.31111111149

You can watch as the root-finder either finds a sign

reversal or converges on a local extrema or does not

converge at all. If there is no convergence in process, you

might want to cancel the operation (press

over with a different initial guess.

) and start

Plotting to find guesses

The main reason for plotting in the Solve aplet is to help

you find initial guesses and solutions for those equations

that have difficult-to-find or multiple solutions.

Consider the equation of motion for an accelerating

body:

AT ²

X =V₀T +

Solve aplet

7-7

hp40g+.book Page 8 Friday, December 9, 2005 1:03 AM

where X is distance, V is initial velocity, T is time, and A

is acceleration. This is actually two equations, Y = X and

Y = V T + (AT ) / 2.

Since this equation is quadratic for T, there can be both

a positive and a negative solution. However, we are

concerned only with positive solutions, since only positive

distance makes sense.

1. Select the Solve aplet and enter the equation.

Select Solve

2. Find the solution for T (time) when X=30, V=2, and

A=4. Enter the values for X, V, and A; then highlight

the independent variable, T.

to highlight T

3. Use the Plot view to find an initial guess for T. First set

appropriate X and Y ranges in the Plot Setup. With

equation X = V x T + A x T /2, the plot will produce

two graphs: one for Y = X and one for

X = V x T + A x T /2. Since we have set X = 30 in

this example, one of the graphs will be Y = 30 .

Therefore, make the YRNG–5 to 35. Keep the XRNG

default of –6.5 to 6.5.

SETUP-PLOT

4. Plot the graph.

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5. Move the cursor near the positive (right-side)

intersection. This cursor value will be an initial guess

for T.

Press

until the

cursor is at the

intersection.

The two points of

intersection show that

there are two solutions for this equation. However,

only positive values for X make sense, so we want to

find the solution for the intersection on the right side

of the y-axis.

6. Return to the Numeric

view.

Note: the T-value is filled in with the position of the

cursor from the Plot view.

7. Ensure that the T value is highlighted, and solve the

equation.

Use this equation to solve for another variable, such as

velocity. How fast must a body’s initial velocity be in

order for it to travel 50 m within 3 seconds? Assume the

same acceleration, 4 m/s . Leave the last value of V as

the initial guess.

Solve aplet

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Using variables in equations

You can use any of the real variable names, A to Z and

θ. Do not use variable names defined for other types,

such as M1 (a matrix variable).

Home variables

All home variables (other than those for aplet settings, like

Xminand Ytick) are global, which means they are

shared throughout the different aplets of the calculator. A

value that is assigned to a home variable anywhere

remains with that variable wherever its name is used.

Therefore, if you have defined a value for T (as in the

above example) in another aplet or even another Solve

equation, that value shows up in the Numeric view for this

Solve equation. When you then redefine the value for T

in this Solve equation, that value is applied to T in all

other contexts (until it is changed again).

This sharing allows you to work on the same problem in

different places (such as HOME and the Solve aplet)

without having to update the value whenever it is

recalculated.

H I N T

As the Solve aplet uses existing variable values, be sure

to check for existing variable values that may affect the

solve process. (You can use

CLEAR to reset all

values to zero in the Solve aplet’s Numeric view if you

wish.)

Aplet variables

Functions defined in other aplets can also be referenced

in the Solve aplet. For example, if, in the Function aplet,

you define F1(X)=X +10, you can enter F1(X)=50in

the Solve aplet to solve the equation X +10=50.

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Linear Solver aplet

About the Linear Solver aplet

The Linear Solver aplet allows you to solve a set of linear

equations. The set can contain two or three linear

equations.

In a two-equation set, each equation must be in the form

ax + by = k . In a three-equation set, each equation must

be in the form ax + by + cz = k .

You provide values for a, b, and k (and c in three-

equation sets) for each equation, and the Linear Solver

aplet will attempt to solve for x and y (and z in three-

equation sets).

The hp40gs will alert you if no solution can be found, or

if there is an infinite number of solutions.

Note that the Linear Solver aplet only has a numeric view.

Getting started with the Linear Solver aplet

The following example defines a set of three equations

and then solves for the unknown variables.

Open the

Linear Solver

aplet

1. Open the Linear Sequence aplet.

Select Linear

Solver

The Linear Equation

Solver opens.

Choose the

equation set

2. If the last time you used

the Linear Solver aplet

you solved for two

equations, the two-

equation input form is

displayed (as in the

Linear Solver aplet

8-1

hp40g+.book Page 2 Friday, December 9, 2005 1:03 AM

example in the previous step). To solve a three-

equation set, press

displays three equations.

. Now the input form

If the three-equation input form is displayed and you want

to solve a two-equation set, press

In this example, we are going to solve the following

equation set:

6x + 9y + 6z = 5

7x + 10y + 8z = 10

6x + 4y = 6

Hence we need the three-equation input form.

Define and

solve the

equations

3. You define the equations you want to solve by

entering the co-efficients of each variable in each

equation and the constant term. Notice that the cursor

is immediately positioned at the co-efficient of x in the

first equation. Enter that co-efficient and press

4. The cursor moves to the next co-efficient. Enter that co-

efficient, press

, and continue doing

likewise until you have defined all the equations.

Note: you can enter the name of a variable for any

co-efficient or constant. Press

entering the name. The

and begin

menu key appears.

Press that key to lock alphabetic entry mode. Press it

again to cancel the lock.

Once you have entered

enough values for the

solver to be able to

generate solutions,

those solutions appear

on the display. In the

example at the right,

the solver was able to find solutions for x, y, and z as

soon as the first co-efficient of the last equation was

entered.

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As you enter each of

the remaining known

values, the solution

changes. The example

at the right shows the

final solution once all

the co-efficients and

constants are entered for the set of equations we set

out to solve.

Linear Solver aplet

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Triangle Solve aplet

About the Triangle Solver aplet

The Triangle Solver aplet allows you to determine the

length of a side of a triangle, or the angle at the vertex of

a triangle, from information you supply about the other

lengths and/or other angles.

You need to specify at least three of the six possible

values—the lengths of the three sides and the size of the

three angles—before the solver can calculate the other

values. Moreover, at least one value you specify must be

a length. For example, you could specify the lengths of

two sides and one of the angles; or you could specify two

angles and one length; or all three lengths. In each case,

the solver will calculate the remaining lengths or angles.

The HP 40gs will alert you if no solution can be found, or

if you have provided insufficient data.

If you are determining the properties of a right-angled

triangle, a simpler input form is available by pressing the

menu key.

Note that the Triangle Solver aplet only has a numeric

view.

Getting started with the Triangle Solver aplet

The following example solves for the unknown length of

the side of a triangle whose two known sides—of lengths

4 and 6—meet at an angle of 30 degrees.

Before you begin: You should make sure that your angle

measure mode is appropriate. If the angle information

you have is in degrees (as in this example) and your

current angle measure mode is radians or grads, change

the mode to degrees before running the solver. (See

“Mode settings” on page 1-10 for instructions.) Because

the angle measure mode is associated with the aplet, you

should start the aplet first and then change the setting.

Triangle Solve aplet

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Open the

Triangle

Solver aplet

1. Open the Triangle Solver aplet.

Select

Triangle Solver

The Triangle Solver

aplet opens.

Note: if you have already used the Triangle Solver,

the entries and results from the previous use will still

be displayed. To start the Triangle Solver afresh, clear

the previous entries and results by pressing

CLEAR.

Choose the

triangle type

2. If the last time you used

the Triangle Solver

aplet you used the

right-angled triangle

input form, that input

form is displayed

again (as in the

example at the right). If the triangle you are

investigating is not a right-angled triangle, or you are

not sure what type it is, you should use the general

input form (illustrated in the previous step). To switch

to the general input form, press

If the general input form is displayed and you are

investigating a right-angled triangle, press

display the simpler input form.

Specify the

known values

3. Using the arrow keys, move to a field whose value

you know, enter the value and press

Repeat for each known value.

Note that the lengths of

the sides are labeled

A, B, and C, and the

angles are labeled α,

β, and δ. It is important

that you enter the

known values in the

appropriate fields. In our example, we know the

length of two sides and the angle at which those

sides meet. Hence if we specify the lengths of sides A

and B, we must enter the angle as δ (since δ is the

angle where A and B meet). If instead we entered the

9-2

Triangle Solve aplet

hp40g+.book Page 3 Friday, December 9, 2005 1:03 AM

lengths as B and C, we would need to specify the

angle as α. The illustration on the display will help

you determine where to enter the known values.

Note: if you need to change the angle neasure mode,

press

MODES, change the mode, and then

to return to the aplet.

4. Press

. The solver

calculates the values of

the unknown variables

and displays. As the

illustration at the right

shows, the length of

the unknown side in our example is 3.2296. (The

other two angles have also been calculated.)

Note: if two sides and

an adjacent acute

angle are entered and

there are two solutions,

only one will be

displayed initially.

In this case, an

menu key is displayed

(as in this example).

You press

display the second

solution, and

again to return to the

first solution.

Errors

No solution with

given data

If you are using the general

input form and you enter

more than 3 values, the

values might not be

consistent, that is, no

triangle could possibly have all the values you specified.

In these cases, No sol with given dataappears on

the screen.

The situation is similar if you are using the simpler input

form (for a right-angled triangle) and you enter more than

two values.

Triangle Solve aplet

9-3

hp40g+.book Page 4 Friday, December 9, 2005 1:03 AM

Not enough data

If you are using the general

input form, you need to

specify at least three values

for the Triangle Solver to

be able to calculate the

remaining attributes of the

triangle. If you specify less than three, Not enough

dataappears on the screen.

If you are using the simplified input form (for a right-

angled triangle), you must specify at least two values.

In addition, you cannot specify only angles and no

lengths.

hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Statistics aplet

About the Statistics aplet

The Statistics aplet can store up to ten data sets at one

time. It can perform one-variable or two-variable

statistical analysis of one or more sets of data.

The Statistics aplet starts with the Numeric view which is

used to enter data. The Symbolic view is used to specify

which columns contain data and which column contains

frequencies.

You can also compute statistics values in HOME and

recall the values of specific statistics variables.

The values computed in the Statistics aplet are saved in

variables, and many of these variables are listed by the

function accessible from the Statistics aplet’s

Numeric view screen.

Getting started with the Statistics aplet

The following example asks you to enter and analyze the

advertising and sales data (in the table below), compute

statistics, fit a curve to the data, and predict the effect of

more advertising on sales.

Advertising minutes

(independent, x)

Resulting Sales ($)

(dependent, y)

1400

920

1100

2265

2890

2200

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Open the

Statistics aplet

1. Open the Statistics aplet and clear existing data by

pressing

Select Statistics

The Statistics aplet

starts in the Numerical

view.

1VAR/2VAR

menu key label

At any time the Statistics aplet is configured for only

one of two types of statistical explorations: one-

variable (

) or two-variable (

). The 5th

menu key label in the Numeric view toggles between

these two options and shows the current option.

2. Select

You need to select

because in this example

we are analyzing a dataset comprising two

variables: advertising minutes and resulting sales.

Enter data

3. Enter the data into the columns.

to move to the next

column

1400

1100

2890

920

2265

2200

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Choose fit and

data columns

4. Select a fit in the Symbolic setup view.

SETUP-SYMB

Select Linear

You can create up to five explorations of two-variable

data, named S1to S5. In this example, we will create

just one: S1.

5. Specify the columns that hold the data you want to

analyze.

You could have entered

your data into columns

other than C1and C2.

Explore statistics 6. Find the mean advertising time (MEANX) and the

mean sales (MEANY).

MEANXis 3.3 minutes

and MEANYis about

$1796.

7. Scroll down to display the value for the correlation

coefficient (CORR). The CORRvalue indicates how

well the linear model fits the data.

9 times

The value is .8995.

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Setup plot

8. Change the plotting range to ensure all the data

points are plotted (and select a different point mark, if

you wish).

SETUP-PLOT

100

4000

Plot the graph

9. Plot the graph.

Draw the

regression curve

10.Draw the regression curve (a curve to fit the data

points).

This draws the

regression line for the

best linear fit.

Display the

equation for

best linear fit

11.Return to the Symbolic view.

12.Display the equation for the best linear fit.

to move to the

FIT1field

The full FIT1

expression is shown.

The slope (m) is 425.875. The y-intercept (b) is

376.25.

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Predict values

13.To find the predicted sales figure if advertising were

to go up to 6 minutes:

S (to highlight

Stat-Two)

(to highlight

PREDY)

14.Return to the Plot view.

15.Jump to the indicated point on the regression line.

Observe the predicted

y-value in the left

bottom corner of the

screen.

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Entering and editing statistical data

The Numeric view (

) is used to enter data into the

Statistics aplet. Each column represents a variable named

C0to C9. After entering the data, you must define the

data set in the Symbolic view (

H I N T

A data column must have at least four data points to

provide valid two-variable statistics, or two data points

for one-variable statistics.

You can also store statistical data values by copying lists

from HOME into Statistics data columns. For example, in

HOME, L1

C1stores a copy of the list L1into the

data-column variable C1.

Statistics aplet’s NUM view keys

The Statistics aplet’s Numeric view keys are:

Key

Meaning

Copies the highlighted item into the

edit line.

Inserts a zero value above the

highlighted cell.

Sorts the specified independent

data column in ascending or

descending order, and rearranges

a specified dependent (or

frequency) data column

accordingly.

Switches between larger and

smaller font sizes.

A toggle switch to select one-

variable or two-variable statistics.

This setting affects the statistical

calculations and plots. The label

indicates which setting is current.

Computes descriptive statistics for

each data set specified in Symbolic

view.

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Key

Meaning (Continued)

Deletes the currently highlighted

value.

CLEAR

Clears the current column or all

columns of data. Pregss

CLEAR to display a menu list,

then select the current column or all

columns option, and press

Moves to the first or last row, or first

cursor key or last column.

Example

You are measuring the height of students in a classroom

to find the mean height. The first five students have the

following measurements 160cm, 165cm, 170cm,

175cm, 180cm.

1. Open the Statistics aplet.

Select

Statistics

2. Enter the measurement

data.

160

165

170

175

180

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3. Find the mean of the

sample.

Ensure the

menu key label

. Press

reads

to see the

statistics calculated from the sample data in C1.

Note that the title of the

column of statistics is

H1. There are 5 data

set definitions available

for one-variable

statistics: H1–H5. If

data is entered in C1, H1is automatically set to use

C1for data, and the frequency of each data point is

set to 1. You can select other columns of data from

the Statistics Symbolic setup view.

4. Press

to close the

statistics window and

press

key to see

the data set definitions.

The first column

indicates the associated column of data for each data

set definition, and the second column indicates the

constant frequency, or the column that holds the

frequencies.

The keys you can use from this window are:

Key

Meaning

Copies the column variable (or

variable expression) to the edit line

for editing. Press

when done.

Checks/unchecks the current data

set. Only the checkmarked data

set(s) are computed and plotted.

Typing aid for the column variables

(

) or for the Fit expressions ( ).

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Key

Meaning (Continued)

Displays the current variable

expression in standard

mathematical form. Press

done.

when

Evaluates the variables in the

highlighted column (C1, etc.)

expression.

Displays the menu for entering

variable names or contents of

variables.

Displays the menu for entering math

operations.

Deletes the highlighted variable or

the current character in the edit line.

CLEAR

Resets default specifications for the

data sets or clears the edit line (if it

was active).

Note: If

CLEAR is used the

data sets will need to be selected

again before re-use.

To continue our example, suppose that the heights of the

rest of the students in the class are measured, but each

one is rounded to the nearest of the five values first

recorded. Instead of entering all the new data in C1, we

shall simply add another column, C2, that holds the

frequencies of our five data points in C1.

Height

(cm)

Frequency

160

165

170

175

180

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5. Move the highlight bar

into the right column of

the H1definition and

replace the frequency

value of 1 with the

name C2.

6. Return to the numeric view.

7. Enter the frequency data shown in the above table.

8. Display the computed

statistics.

The mean height is

approximately

167.63cm.

9. Setup a histogram plot for the data.

SETUP-PLOT

Enter set up information

appropriate to your

data.

10.Plot a histogram of the data.

Save data

The data that you enter is automatically saved. When you

are finished entering data values, you can press a key for

another Statistics view (like

another aplet or HOME.

), or you can switch to

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Edit a data set

Delete data

In the Numeric view of the Statistics aplet, highlight the

data value to change. Type a new value and

press

, or press

to copy the value to the edit

after modifying the

line for modification. Press

value on the edit line.

•

To delete a single data item, highlight it and press

. The values below the deleted cell will scroll up

one row.

•

To delete a column of data, highlight an entry in that

column and press

name.

CLEAR. Select the column

•

To delete all columns of data, press

CLEAR.

Select All columns.

Insert data

Highlight the entry following the point of insertion. Press

, then enter a number. It will write over the zero that

was inserted.

Sort data

values

1. In Numeric view, highlight the column you want to

sort, and press

2. Specify the Sort Order. You can choose either

Ascendingor Descending.

3. Specify the INDEPENDENTand DEPENDENTdata

columns. Sorting is by the independent column. For

instance, if Age is C1and Income is C2and you

want to sort by Income, then you make C2the

independent column for the sorting and C1the

dependent column.

–

To sort just one column, choose None for the

dependent column.

–

For one-variable statistics with two data columns,

specify the frequency column as the dependent

column.

4. Press

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Defining a regression model

The Symbolic view includes an expression (Fit1 through

Fit5) that defines the regression model, or “fit”, to use for

the regression analysis of each two-variable data set.

There are three ways to select a regression model:

•

Accept the default option to fit the data to a straight

line.

Select one of the available fit options in Symbolic

Setup view.

Enter your own mathematical expression in Symbolic

view. This expression will be plotted, but it will not be

fitted to the data points.

Angle Setting

You can ignore the angle measurement mode unless your

Fit definition (in Symbolic view) involves a trigonometric

function. In this case, you should specify in the mode

screen whether the trigonometric units are to be

interpreted in degrees, radians, or grads.

To choose the fit 1. In Numeric view, make sure

is set.

2. Press

SETUP-SYMB to display the Symbolic Setup

view. Highlight the Fit number (S1FITto S5FIT) you

want to define.

3. Press

and select from the list. Press

when

done. The regression formula for the fit is displayed in

Symbolic view.

Fit models

Ten fit models are available:

Fit model

Meaning

Linear

(Default.) Fits the data to a

straight line, y = mx+b. Uses a

least-squares fit.

Logarithmic

Exponential

Power

Fits to a logarithmic curve,

y = m lnx + b.

Fits to an exponential curve,

y = be .

Fits to a power curve, y = bx .

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Fit model

Meaning (Continued)

Quadratic

Fits to a quadratic curve,

y = ax +bx+c. Needs at least

three points.

Cubic

Fits to a cubic curve,

y = ax +bx +cx+d. Needs at least

four points.

Logistic

Fits to a logistic curve,

--------------------------

y =

1 + ae^(–bx)

where L is the saturation value for

growth. You can store a positive

real value in L, or—if L=0—let L

be computed automatically.

Exponent

Fits to an exponent curve,

y = ab^x.

Trigonometric

Fits to a trigonometric curve,

y = a ⋅ sin(bx + c) + d . Needs

at least three points.

User Defined

Define your own expression (in

Symbolic view.)

To define your

own fit

1. In Numeric view, make sure

2. Display the Symbolic view.

is set.

3. Highlight the Fit expression (Fit1, etc.) for the

desired data set.

4. Type in an expression and press

The independent variable must be X, and the

expression must not contain any unknown variables.

Example: 1.5 × cosx + 0.3 × sinx .

This automatically changes the Fit type (S1FIT, etc.) in

the Symbolic Setup view to User Defined.

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Computed statistics

One-variable

Statistic

NΣ

Definition

Number of data points.

TOTΣ

Sum of data values (with their

frequencies).

MEANΣ

PVARΣ

SVARΣ

PSDEV

Mean value of data set.

Population variance of data set.

Sample variance of data set.

Population standard deviation of

data set.

SSDEV

Sample standard deviation of data

set.

MINΣ

Minimum data value in data set.

First quartile: median of values to

left of median.

MEDIAN

Median value of data set.

Third quartile: median of values to

right of median.

MAXΣ

Maximum data value in data set.

When the data set contains an odd number of values, the

data set’s median value is not used when calculating Q1

and Q3 in the table above. For example, for the following

data set:

{3,5,7,8,15,16,17}

only the first three items, 3, 5, and 7 are used to calculate

Q1, and only the last three terms, 15, 16, and 17 are

used to calculate Q3.

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Two-variable

Statistic

MEANX

ΣX

Definition

Mean of x- (independent) values.

Sum of x-values.

ΣX2

Sum of x -values.

MEANY

ΣY

Mean of y- (dependent) values.

Sum of y-values.

ΣY2

Sum of y -values.

ΣXY

Sum of each xy.

SCOV

Sample covariance of independent

and dependent data columns.

PCOV

CORR

Population covariance of

independent and dependent data

columns

Correlation coefficient of the

independent and dependent data

columns for a linear fit only

(regardless of the Fit chosen).

Returns a value from 0 to 1, where

1 is the best fit.

RELERR

The relative error for the selected

fit. Provides a measure of accuracy

for the fit.

Plotting

You can plot:

•

histograms (

)

box-and-whisker plots (

scatter plots ( ).

)

Once you have entered your data (

data set ( ), and defined your Fit model for two-

variable statistics ( SETUP-SYMB), you can plot your

), defined your

data. You can plot up to five scatter or box-and-whisker

plots at a time. You can plot only one histogram at a time.

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To plot statistical

data

1. In Symbolic view (

sets you want to plot.

), select (

) the data

2. For one-variable data (

), select the plot type in

Plot Setup (

SETUP-PLOT). Highlight STATPLOT,

press

, select either Histogramor

BoxWhisker, and press

3. For any plot, but especially for a histogram, adjust the

plotting scale and range in the Plot Setup view. If you

find histogram bars too fat or too thin, you can adjust

them by adjusting the HWIDTHsetting.

4. Press

. If you have not adjusted the Plot Setup

select Auto Scale

yourself, you can try

Auto Scale can be relied upon to give a good starting

scale which can then be adjusted in the Plot Setup view.

Plot types

Histogram

One-variable statistics.

The numbers below the plot

mean that the current bar

(where the cursor is) starts at

0 and ends at 2 (not

including 2), and the

frequency for this column, (that is, the number of data

elements that fall between 0 and 2) is 1. You can see

information about the next bar by pressing the

key.

Box and

Whisker Plot

One-variable statistics.

The left whisker marks the

minimum data value. The

box marks the first quartile,

the median (where the cursor

is), and the third quartile.

The right whisker marks the maximum data value. The

numbers below the plot mean that this column has a

median of 13.

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Scatter Plot

Two-variable statistics.

The numbers below the plot

indicate that the cursor is at

the first data point for S2, at

(1, 6). Press

to move to

the next data point and

display information about it.

To connect the data points as

they are plotted, checkmark

CONNECTin the second

page of the Plot Setup. This is

not a regression curve.

Fitting a curve to 2VAR data

In the Plot view, press

. This draws a curve to fit the

checked two-variable data set(s). See “To choose the fit”

on page 10-12.

The expression in Fit2

shows that the

slope=1.98082191781

and the y-

intercept=2.2657.

Correlation

coefficient

The correlation coefficient is stored in the CORRvariable.

It is a measure of fit to a linear curve only. Regardless of

the Fit model you have chosen, CORR relates to the linear

model.

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Relative Error

The relative error is a measure of the error between

predicted values and actual values based on the specified

Fit. A smaller number means a better fit.

The relative error is stored in a variable named RELERR.

The relative error provides a measure of fit accuracy for

all fits, and it does depend on the Fit model you have

chosen.

H I N T

In order to access the CORRand RELERRvariables after

you plot a set of statistics, you must press

to access

to display the

the numeric view and then

correlation values. The values are stored in the variables

when you access the Symbolic view.

Setting up the plot (Plot setup view)

The Plot Setup view (

SETUP-PLOT) sets most of the

same plotting parameters as it does for the other built-in

aplets.

See “Setting up the plot (Plot view setup)” on page 2-5.

Settings unique to the Statistics aplet are as follows:

Plot type (1VAR)

Histogram width

Histogram range

STATPLOTenables you to specify either a histogram or

a box-and-whisker plot for one-variable statistics (when

is set). Press

to change the highlighted

setting

HWIDTHenables you to specify the width of a histogram

bar. This determines how many bars will fit in the display,

as well as how the data is distributed (how many values

each bar represents).

HRNGenables you to specify the range of values for a set

of histogram bars. The range runs from the left edge of the

leftmost bar to the right edge of the rightmost bar. You

can limit the range to exclude any values you suspect are

outliers.

Plotting mark

(2VAR)

S1MARKthrough S5MARKenables you to specify one of

five symbols to use to plot each data set. Press

change the highlighted setting.

Connected points

(2VAR)

CONNECT(on the second page), when checkmarked,

connects the data points as they are plotted. The resulting

line is not the regression curve. The order of plotting is

according to the ascending order of independent values.

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For instance, the data set (1,1), (3,9), (4,16), (2,4) would

be plotted and traced in the order (1,1), (2,4), (3,9),

(4,16).

Trouble-shooting a plot

If you have problems plotting, check that you have the

following:

•

The correct

view).

menu label on (Numeric

The correct fit (regression model), if the data set is

two-variable.

Only the data sets to compute or plot are

checkmarked (Symbolic view).

•

The correct plotting range. Try using

Auto

Scale (instead of ), or adjust the plotting

parameters (in Plot Setup) for the ranges of the axes

and the width of histogram bars (HWIDTH).

mode, ensure that both paired columns contain

data, and that they are the same length.

mode, ensure that a paired column of frequency

values is the same length as the data column that it refers

to.

Exploring the graph

The Plot view has menu keys for zooming, tracing, and

coordinate display. There are also scaling options under

. These options are described in“Exploring the

graph” on page 2-7.

Statistics aplet’s PLOT view keys

Key

Meaning

CLEAR

Erases the plot.

Offers additional pre-defined views

for splitting the screen, overlaying

plots, and autoscaling the axes.

Moves cursor to far left or far right.

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Key

Meaning (Continued)

Displays ZOOM menu.

Turns trace mode on/off. The white

box appears next to the option when

Trace mode is active.

Turns fit mode on or off. Turning

on draws a curve to fit the data points

according to the current regression

model.

(2var

Enables you to specify a value on the

statistics only) line of best fit to jump to or a data

point number to jump to.

Displays the equation of the

regression curve.

Hides and displays the menu key

labels. When the labels are hidden,

any menu key displays the (x,y)

coordinates. Pressing

redisplays the menu labels.

Calculating predicted values

The functions PREDXand PREDYestimate (predict) values

for X or Y given a hypothetical value for the other. The

estimation is made based on the curve that has been

calculated to fit the data according to the specified fit.

Find predicted

values

1. In Plot view, draw the regression curve for the data

set.

2. Press

3. Press

to move to the regression curve.

and enter the value of X. The cursor

jumps to the specified point on the curve and the

coordinate display shows X and the predicted value

of Y.

In HOME:

•

Enter PREDX(y-value)

to find the predicted

value for the independent variable given a

hypothetical dependent value.

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•

Enter PREDY(x-value) to find the predicted value of the

dependent variable given a hypothetical independent

variable.

You can type PREDXand PREDYinto the edit line, or you

can copy these function names from the MATH menu

under the Stat-Two category.

H I N T

In cases where more than one fit curve is displayed, the

PREDY function uses the most recently calculated curve. In

order to avoid errors with this function, uncheck all fits

except the one that you want to work with, or use the Plot

View method.

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Inference aplet

About the Inference aplet

The Inference capabilities include calculation of

confidence intervals and hypothesis tests based on the

Normal Z-distribution or Student’s t-distribution.

Based on the statistics from one or two samples, you can

test hypotheses and find confidence intervals for the

following quantities:

•

mean

proportion

difference between two means

difference between two proportions

Example data

When you first access an input form for an Inference test,

by default, the input form contains example data. This

example data is designed to return meaningful results that

relate to the test. It is useful for gaining an understanding

of what the test does, and for demonstrating the test. The

calculator’s on-line help provides a description of what

the example data represents.

Getting started with the Inference aplet

This example describes the Inference aplet’s options and

functionality by stepping you through an example using

the example data for the Z-Test on 1 mean.

Open the

Inference aplet

1. Open the Inference aplet.

Select Inference

The Inference aplet

opens in the Symbolic

view.

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Inference aplet’s SYMB view keys

The table below summarizes the options available in

Symbolic view.

Hypothesis

Tests

Confidence Intervals

Z: 1 μ, the Z-Test

on 1 mean

Z-Int: 1 μ, the confidence

interval for 1 mean, based on

the Normal distribution

Z: μ – μ , the

Z-Int: μ – μ , the confidence

Z-Test on the

difference of two

means

interval for the difference of

two means, based on the

Normal distribution

Z: 1 π, the Z-Test

on 1 proportion

Z-Int: 1 π, the confidence

interval for 1 proportion,

based on the Normal

distribution

Z: π1 – π2, the

Z-Test on the

difference in two

proportions

Z-Int: π1 – π2, the confidence

interval for the difference of

two proportions, based on the

Normal distribution

T: 1 μ, the T-Test on

1 mean

T-Int: 1 μ, the confidence

interval for 1 mean, based on

the Student’s t-distribution

T: μ – μ , the T-

T-Int: μ – μ , the confidence

Test on the

difference of two

means

interval for the difference of

two means, based on the

Student’s t-distribution

If you choose one of the hypothesis tests, you can choose

the alternative hypothesis to test against the null

hypothesis. For each test, there are three possible choices

for an alternative hypothesis based on a quantitative

comparison of two quantities. The null hypothesis is

always that the two quantities are equal.Thus, the

alternative hypotheses cover the various cases for the two

quantities being unequal: <, >, and ≠.

In this section, we will use the example data for the Z-Test

on 1 mean to illustrate how the aplet works and what

features the various views present.

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Select the

inferential

method

2. Select the HypothesisTest inferential method.

Select HYPOTH TEST

3. Define the type of test.

Z–Test:1 μ

4. Select an alternative hypothesis.

μ< μ0

Enter data

5. Enter the sample statistics and population

parameters.

setup-NUM

The table below lists the fields in this view for our current

Z-Test:1 μ example.

Field

Definition

name

μ0

Assumed population mean

Population standard deviation

Sample mean

Sample size

Alpha level for the test

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By default, each field already contains a value.

These values constitute the example database and

are explained in the

feature of this aplet.

Display on-line

help

6. To display the on-line

help, press

7. To close the on-line help,

press

Display test

results in

numeric format

8. Display the test results in numeric format.

The test distribution value

and its associated

probability are

displayed, along with

the critical value(s) of the test and the associated

critical value(s) of the statistic.

Note: You can access the on-line help in Numeric

view.

Plot test results

9. Display a graphic view of the test results.

Horizontal axes are

presented for both the

distribution variable and

the test statistic. A

generic bell curve represents the probability

distribution function. Vertical lines mark the critical

value(s) of the test, as well as the value of the test

statistic. The rejection region is marked

and the

test numeric results are displayed between the

horizontal axes.

Importing sample statistics from the Statistics aplet

The Inference aplet supports the calculation of confidence

intervals and the testing of hypotheses based on data in

the Statistics aplet. Computed statistics for a sample of

data in a column in any Statistics-based aplet can be

imported for use in the Inference aplet. The following

example illustrates the process.

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A calculator produces the following 6 random numbers:

0.529, 0.295, 0.952, 0.259, 0.925, and 0.592

Open the

Statistics aplet

1. Open the Statistics aplet and reset the current

settings.

Select

Statistics

The Statistics aplet opens in the Numeric view.

Enter data

2. In the C1 column, enter the random numbers

produced by the calculator.

529

295

952

259

925

592

H I N T

If the Decimal Mark setting in the Modes input form

(

modes) is set to Comma, use

instead of

3. If necessary, select 1-variable statistics. Do this by

pressing the fifth menu key until

as its menu label.

is displayed

Calculate

statistics

4. Calculate statistics.

The mean of 0.592

seems a little large

compared to the

expected value of 0.5. To see if the difference is

statistically significant, we will use the statistics

computed here to construct a confidence interval for

the true mean of the population of random numbers

and see whether or not this interval contains 0.5.

5. Press

to close the computed statistics window.

Inference aplet

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Open Inference

aplet

6. Open the Inference aplet and clear current settings.

Select

Inference

Select inference

method and

type

7. Select an inference method.

Select CONF INTERVAL

8. Select a distribution statistic type.

Select T-Int: 1μ

Set up the

interval

calculation

9. Set up the interval calculation. Note: The default

values are derived from sample data from the on-line

help example.

Setup-NUM

11-6

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Import the data

10.Import the data from the Statistics aplet. Note: The

data from C1 is displayed by default.

Note: Press

to see

the statistics before

importing them into the

Numeric Setup view.

Also, if there is more than one aplet based on the

Statistics aplet, you are prompted to choose one.

11.Specify a 90%

confidence interval in the C: field.

to move to

the C: field

0.9

DisplayNumeric 12.Display the confidence interval in the Numeric view.

Note: The interval setting is 0.5.

view

Display Plot

view

13.Display the confidence interval in the Plot view.

You can see, from the

second text row, that the

mean is contained within the 90% confidence

interval (CI) of 0.3469814 to 0.8370186.

Note: The graph is a simple, generic bell-curve. It is

not meant to accurately represent the t-distribution

with 5 degrees of freedom.

Inference aplet

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Hypothesis tests

You use hypothesis tests to test the validity of hypotheses

that relate to the statistical parameters of one or two

populations. The tests are based on statistics of samples

of the populations.

The HP 40gs hypothesis tests use the Normal

Z-distribution or Student’s t-distribution to calculate

probabilities.

One-Sample Z-Test

Menu name

Z-Test: 1 μ

On the basis of statistics from a single sample, the

One-Sample Z-Test measures the strength of the evidence

for a selected hypothesis against the null hypothesis. The

null hypothesis is that the population mean equals a

specified value Η : μ = μ .

You select one of the following alternative hypotheses

against which to test the null hypothesis:

H₁:μ₁< μ₂

H₁:μ₁> μ₂

H₁:μ₁≠ μ₂

Inputs

The inputs are:

Field name

Definition

Sample mean.

Sample size.

Hypothetical population mean.

Population standard deviation.

Significance level.

11-8

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Results

The results are:

Result

Test Z

Prob

Description

Z-test statistic.

Probability associated with the

Z-Test statistic.

Critical Z

Boundary values of Z

associated with the α level that

you supplied.

Boundary values of x required

by the α value that you

supplied.

Critical x

Two-Sample Z-Test

Menu name

Z-Test: μ1–μ2

On the basis of two samples, each from a separate

population, this test measures the strength of the evidence

for a selected hypothesis against the null hypothesis. The

null hypothesis is that the mean of the two populations are

equal (H : μ1= μ2).

You select one of the following alternative hypotheses

against which to test the null hypothesis:

H₁:μ₁< μ₂

H₁:μ₁> μ₂

H₁:μ₁≠ μ₂

Inputs

The inputs are:

Field name

Definition

Sample 1 mean.

Sample 2 mean.

σ1

Sample 1 size.

Sample 2 size.

Population 1 standard

deviation.

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Field name

Definition

σ2

Population 2 standard

deviation.

Significance level.

Results

The results are:

Result

Test Z

Prob

Description

Z-Test statistic.

Probability associated with the

Z-Test statistic.

Critical Z

Boundary value of Z

associated with the α level that

you supplied.

One-Proportion Z-Test

Menu name

Z-Test: 1π

On the basis of statistics from a single sample, this test

measures the strength of the evidence for a selected

hypothesis against the null hypothesis. The null hypothesis

is that the proportion of successes in the two populations

is equal: H :π =π

You select one of the following alternative hypotheses

against which to test the null hypothesis:

H₁:π < π₀

H₁:π > π₀

H₁:π ≠ π₀

11-10

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Inputs

The inputs are:

Field name

Definition

Number of successes in the sample.

Sample size.

Population proportion of successes.

Significance level.

Results

The results are:

Result

Test P

Test Z

Prob

Description

Proportion of successes in the sample.

Z-Test statistic.

Probability associated with the Z-Test

statistic.

Critical Z

Boundary value of Z associated with

the level you supplied.

Two-Proportion Z-Test

Menu name

Z-Test: π1 – π2

On the basis of statistics from two samples, each from a

different population, the Two-Proportion Z-Test measures

the strength of the evidence for a selected hypothesis

against the null hypothesis. The null hypothesis is that the

proportion of successes in the two populations is equal

H0: π = π .

You select one of the following alternative hypotheses

against which to test the null hypothesis:

H₁:π₁< π₂

H₁:π₁> π₂

H₁:π₁≠ π₂

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Inputs

The inputs are:

Field name

Definition

Sample 1 mean.

Sample 2 mean.

Sample 1 size.

Sample 2 size.

Significance level.

Results

The results are:

Result

Description

Test π1–π2

Difference between the

proportions of successes in the

two samples.

Test Z

Prob

Z-Test statistic.

Probability associated with the

Z-Test statistic.

Critical Z

Boundary values of Z

associated with the α level that

you supplied.

One-Sample T-Test

Menu name

T-Test: 1 μ

The One-sample T-Test is used when the population

standard deviation is not known. On the basis of statistics

from a single sample, this test measures the strength of the

evidence for a selected hypothesis against the null

hypothesis. The null hypothesis is that the sample mean

has some assumed value,

:μ = μ

You select one of the following alternative hypotheses

against which to test the null hypothesis:

H₁:μ < μ₀

H₁:μ > μ₀

H₁:μ ≠ μ₀

11-12

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Inputs

The inputs are:

Field name

Definition

Sample mean.

Sample standard deviation.

Sample size.

μ0

Hypothetical population mean.

Significance level.

Results

The results are:

Result

Test T

Prob

Description

T-Test statistic.

Probability associated with the

T-Test statistic.

Critical T

Boundary value of T associated

with the α level that you

supplied.

Boundary value of x required

by the α value that you

supplied.

Critical x

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Two-Sample T-Test

Menu name

T-Test: μ1 – μ2

The Two-sample T-Test is used when the population

standard deviation is not known. On the basis of statistics

from two samples, each sample from a different

population, this test measures the strength of the evidence

for a selected hypothesis against the null hypothesis. The

null hypothesis is that the two populations means are

equal H : μ = μ .

You select one of the following alternative hypotheses

against which to test the null hypothesis

H₁:μ₁< μ₂

H₁:μ₁> μ₂

H₁:μ₁≠ μ₂

Inputs

The inputs are:

Field

name

Definition

Sample 1 mean.

Sample 2 mean.

Sample 1 standard deviation.

Sample 2 standard deviation.

Sample 1 size.

Sample 2 size.

Significance level.

_Pooled? Check this option to pool samples

based on their standard deviations.

11-14

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Results

The results are:

Result

Test T

Prob

Description

T-Test statistic.

Probability associated with the T-Test

statistic.

Critical T

Boundary values of T associated with

the α level that you supplied.

Confidence intervals

The confidence interval calculations that the HP 40gs can

perform are based on the Normal Z-distribution or

Student’s t-distribution.

One-Sample Z-Interval

Menu name

Z-INT: μ 1

This option uses the Normal Z-distribution to calculate a

confidence interval for m, the true mean of a population,

when the true population standard deviation, s, is known.

Inputs

The inputs are:

Field

Definition

name

Sample mean.

Population standard deviation.

Sample size.

Confidence level.

Inference aplet

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Results

The results are:

Result

Critical Z

μ min

Description

Critical value for Z.

Lower bound for μ.

Upper bound for μ.

μ max

Two-Sample Z-Interval

Menu name

Z-INT: μ1– μ2

This option uses the Normal Z-distribution to calculate a

confidence interval for the difference between the means

of two populations, μ –μ , when the population standard

deviations, σ and σ , are known.

Inputs

The inputs are:

Field

Definition

name

Sample 1 mean.

Sample 2 mean.

σ1

σ2

Sample 1 size.

Sample 2 size.

Population 1 standard deviation.

Population 2 standard deviation.

Confidence level.

Results

The results are:

Result

Description

Critical Z

Critical value for Z.

Lower bound for μ – μ .

Δ μ Min

Upper bound for μ – μ .

Δ μ Max

11-16

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One-Proportion Z-Interval

Menu name

Z-INT: 1 π

This option uses the Normal Z-distribution to calculate a

confidence interval for the proportion of successes in a

population for the case in which a sample of size, n, has

a number of successes, x.

Inputs

The inputs are:

Field

Definition

name

Sample success count.

Sample size.

Confidence level.

Results

The results are:

Result

Critical Z

π Min

Description

Critical value for Z.

Lower bound for π.

Upper bound for π.

π Max

Two-Proportion Z-Interval

Menu name

Z-INT: π1 – π2

This option uses the Normal Z-distribution to calculate a

confidence interval for the difference between the

proportions of successes in two populations.

Inputs

The inputs are:

Field

Definition

name

Sample 1 success count.

Sample 2 success count.

Inference aplet

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Field

Definition (Continued)

name

Sample 1 size.

Sample 2 size.

Confidence level.

Results

The results are:

Result

Description

Critical Z

Critical value for Z.

Lower bound for the difference between

the proportions of successes.

Δ π Min

Upper bound for the difference between

the proportions of successes.

Δ π Max

One-Sample T-Interval

Menu name

T-INT: 1 μ

This option uses the Student’s t-distribution to calculate a

confidence interval for m, the true mean of a population,

for the case in which the true population standard

deviation, s, is unknown.

Inputs

The inputs are:

Field

Definition

name

Sample mean.

Sample standard deviation.

Sample size.

Confidence level.

11-18

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Results

The results are:

Result

Critical T

μ Min

Description

Critical value for T.

Lower bound for μ.

Upper bound for μ.

μ Max

Two-Sample T-Interval

Menu name

T-INT: μ1 – μ2

This option uses the Student’s t-distribution to calculate a

confidence interval for the difference between the means

of two populations, μ1 – μ2, when the population

standard deviations, s1and s2, are unknown.

Inputs

The inputs are:

Field

Definition

name

Sample 1 mean.

Sample 2 mean.

Sample 1 standard deviation.

Sample 2 standard deviation.

Sample 1 size.

Sample 2 size.

Confidence level.

_Pooled

Whether or not to pool the samples

based on their standard deviations.

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Results

The results are:

Result

Description

Critical T

Critical value for T.

Lower bound for μ – μ .

Δ μ Min

Upper bound for μ – μ .

Δ μ Max

11-20

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Using the Finance Solver

The Finance Solver, or Finance aplet, is available by

using the APLET key in your calculator. Use the up and

down arrow keys to select the Finance aplet. Your screen

should look as follows:

Press the

key or the

soft menu key to

activate the aplet. The resulting screen shows the different

elements involved in the solution of financial problems

with your HP 40gs calculator.

Background information on and applications of financial

calculations are provided next.

Background

The Finance Solver application provides you with the

ability of solving time-value-of-money (TVM) and

amortization problems. These problems can be used for

calculations involving compound interest applications as

well as amortization tables.

Compound interest is the process by which earned

interest on a given principal amount is added to the

principal at specified compounding periods, and then the

Using the Finance Solver

12-1

hp40g+.book Page 2 Friday, December 9, 2005 1:03 AM

combined amount earns interest at a certain rate.

Financial calculations involving compound interest

include savings accounts, mortgages, pension funds,

leases, and annuities.

Time Value of Money (TVM) calculations, as the name

implies, make use of the notion that a dollar today will be

worth more than a dollar sometime in the future. A dollar

today can be invested at a certain interest rate and

generate a return that the same dollar in the future cannot.

This TVM principle underlies the notion of interest rates,

compound interest and rates of return.

TVM transactions can be represented by using cash flow

diagrams. A cash flow diagram is a time line divided into

equal segments representing the compounding periods.

Arrows represent the cash flows, which could be positive

(upward arrows) or negative (downward arrows),

depending on the point of view of the lender or borrower.

The following cash flow diagram shows a loan from a

borrower's point of view:

Present value (PV)

(Loan)

Money

Equal periods

received is

a positive

number

(PMT)

Payment Payment Payment

(PMT) (PMT) (PMT)

Payment

(PMT)

Money

Future value

(FV)

paid out is

a negative

number

Equal payments

On the other hand, the following cash flow diagram

shows a load from the lender's point of view:

Equal payments

PMT

PMT PMT

PMT

Loan

}

Equal periods

In addition, cash flow diagrams specify when payments

occur relative to the compounding periods: at the

beginning of each period or at the end. The Finance

Solver application provides both of these payment

modes: Begin mode and End mode. The following cash

12-2

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hp40g+.book Page 3 Friday, December 9, 2005 1:03 AM

flow diagram shows lease payments at the beginning of

each period.

Capitalized

value of

lease

}

PMT

PMT PMT PMT

PMT

The following cash flow diagram shows deposits into an

account at the end of each period.

PMT PMT PMT PMT

PMT

As these cash-flow diagrams imply, there are five TVM

variables:

The total number of compounding periods

or payments.

I%YR

The nominal annual interest rate (or

investment rate). This rate is divided by

the number of payments per year (P/YR)

to compute the nominal interest rate per

compounding period -- which is the

interest rate actually used in TVM

calculations.

The present value of the initial cash flow.

To a lender or borrower, PV is the amount

of the loan; to an investor, PV is the initial

investment. PV always occurs at the

beginning of the first period.

Using the Finance Solver

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The periodic payment amount. The

payments are the same amount each

period and the TVM calculation assumes

that no payments are skipped. Payments

can occur at the beginning or the end of

each compounding period -- an option

you control by setting the Payment mode

to Beg or End.

PMT

The future value of the transaction: the

amount of the final cash flow or the

compounded value of the series of

previous cash flows. For a loan, this is the

size of the final balloon payment (beyond

any regular payment due). For an

investment this is the cash value of an

investment at the end of the investment

period.

Performing TVM calculations

1. Launch the Financial Solver as indicated at the

beginning of this section.

2. Use the arrow keys to highlight the different fields and

enter the known variables in the TVM calculations,

pressing the

soft-menu key after entering each

known value. Be sure that values are entered for at

least four of the five TVM variables (namely, N, I%YR,

PV, PMT, and FV).

3. If necessary, enter a different value for P/YR (default

value is 12, i.e., monthly payments).

4. Press the key

to change the Payment mode (Beg

or End) as required.

5. Use the arrow keys to highlight the TVM variable you

wish to solve for and press the

soft-menu key.

12-4

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Example 1 - Loan calculations

Suppose you finance the purchase of a car with a 5-year

loan at 5.5% annual interest, compounded monthly. The

purchase price of the car is $19,500, and the down

payment is $3,000. What are the required monthly

payments? What is the largest loan you can afford if your

maximum monthly payment is $300? Assume that the

payments start at the end of the first period.

Solution. The following cash flow diagram illustrates the

loan calculations:

FV = 0

l%YR = 5.5

PV = $16,500

N = 5 x 12 = 60

P/YR = 12; End mode

PMT = ?

Start the Finance Solver, selecting P/YR = 12 and End

payment option.

•

Enter the known TVM variables as shown in the

diagram above. Your input form should look as

follows:

•

Highlighting the PMT field, press the

menu key to obtain a payment of -315.17 (i.e., PMT

= -$315.17).

soft

To determine the maximum loan possible if the

monthly payments are only $300, type the value

–300 in the PMT field, highlight the PV field, and

press the soft menu key. The resulting value is

PV = $15,705.85.

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Example 2 - Mortgage with balloon payment

Suppose you have taken out a 30-year, $150,000 house

mortgage at 6.5% annual interest. You expect to sell the

house in 10 years, repaying the loan in a balloon

payment. Find the size of the balloon payment, the value

of the mortgage after 10 years of payment.

Solution. The following cash flow diagram illustrates the

case of the mortgage with balloon payment:

l%YR = 6.5

PV = $150,000

N = 30 x 12 = 360 (for PMT)

N = 10 x 12 = 120 (for balloon payment)

P/YR = 12; End mode

PMT = ?

Balloon payment,

FV = ?

•

Start the Finance Solver, selecting P/YR = 12 and

End payment option.

Enter the known TVM variables as shown in the

diagram above. Your input form, for calculating

monthly payments for the 30-yr mortgage, should

look as follows:

•

Highlighting the PMT field, press the

menu key to obtain a payment of -948.10 (i.e., PMT

= -$948.10)

soft

To determine the balloon payment or future value (FV)

for the mortgage after 10 years, use N = 120,

highlight the FV field, and press the

soft menu

key. The resulting value is FV = -$127,164.19. The

negative value indicates a payment from the

homeowner. Check that the required balloon

payments at the end of 20 years (N=240) and 25

years (N = 300) are -$83,497.92 and

-$48,456.24, respectively.

12-6

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Calculating Amortizations

Amortization calculations, which also use the TVM

variables, determine the amounts applied towards

principal and interest in a payment or series of payments.

To calculate amortizations:

1. Start the Finance Solver as indicated at the beginning

of this section.

2. Set the following TVM variables:

Number of payments per year (P/YR)

Payment at beginning or end of periods

3. Store values for the TVM variables I%YR, PV, PMT,

and FV, which define the payment schedule.

4. Press the

soft menu key and enter the

number of payments to amortize in this batch.

5. Press the

soft menu key to amortize a batch of

payments. The calculator will provide for you the

amount applied to interest, to principal, and the

remaining balance after this set of payments have

been amortized.

Example 3 - Amortization for home mortgage

For the data of Example 2 above, find the amortization of

the loan after the first 10 years (12x10 = 120 payments).

Pressing the

screen to the left. Enter 120 in the PAYMENTS field, and

press the soft menu key to produce the results

shown to the right.

soft menu key produces the

To continue amortizing the loan:

1. Press the

soft menu key to store the new

balance after the previous amortization as PV.

2. Enter the number of payments to amortize in the new

batch.

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3. Press the

soft menu key to amortize the new

batch of payments. Repeat steps 1 through 3 as

often as needed.

Example 4 - Amortization for home mortgage

For the results of Example 3, show the amortization of the

next 10 years of the mortgage loan. First, press the

soft menu key. Then, keeping 120 in the PAYMENTS

field, press the

shown below.

soft menu key to produce the results

To amortize a series of future payments starting at payment p:

1. Calculate the balance of the loan at payment p-1.

2. Store the new balance in PV using the

menu key.

soft

3. Amortize the series of payments starting at the new

PV.

The amortization operation reads the values from the

TVM variables, rounds the numbers it gets from PV and

PMT to the current display mode, then calculates the

amortization rounded to the same setting. The original

variables are not changed, except for PV, which is

updated after each amortization.

12-8

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Using mathematical functions

Math functions

The HP 40gs contains many math functions. The functions

are grouped in categories. For example, the Matrix

category contains functions for manipulating matrices.

The Probability category (shown as Prob.on the MATH

menu) contains functions for working with probability.

To use a math function in HOME view, you enter the

function onto the command line, and include the

arguments in parentheses after the function. You can also

select a math function from the MATH menu.

Note that this chapter covers only the use of mathematical

functions in HOME view. The use of mathematical

functions in CAS is described in Chapter14, “Computer

Algebra System (CAS)”.

The MATH menu

The MATH menu provides access to math functions,

physical constants, and programming constants. You can

also access CAS commands.

The MATH menu is organized by category. For each

category of functions on the left, there is a list of function

names on the right. The highlighted category is the

current category.

•

When you press

, you see the menu list of

Math categories in the left column and the

corresponding functions of the highlighted category

in the right column. The menu key

indicates

that the MATH FUNCTIONS menu list is active.

Using mathematical functions

13-1

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To select a function

1. Press

to display the MATH menu. The

categories appear in alphabetical order.

2. Press

to scroll through the categories. To

jump directly to a category, press the first letter of the

category’s name. Note: You do not need to press

first.

3. The list of functions (on the right) applies to the

currently highlighted category (on the left). Use

and

to switch between the category list and the

function list.

4. Highlight the name of the function you want and

press

. This copies the function name (and an

initial parenthesis, if appropriate) to the edit line.

N O T E

If you press

while the MATH menu is open, CAS

functions and commands are displayed. You can select a

CAS function or command in the same way that you

select a function from the MATH menu (by pressing the

arrow keys and then

). The function or command

selected appears on the edit line in HOME (and with an

initial parenthesis, if appropriate).

Function categories (MATH menu)

•

Calculus

•

Loop

•

Symbolic

Tests

Complex

numbers

Matrix

Polynomial

Probability

Trigonometry

(Trig)

•

Constant

Convert

Real numbers

(Real)

Hyperbolic

trigonometry

(Hyperb.)

•

Two-variable

statistics

(Stat-Two)

•

Lists

Math functions by category

Syntax

Each function’s definition includes its syntax, that is, the

exact order and spelling of a function’s name, its

delimiters (punctuation), and its arguments. Note that the

syntax for a function does not require spaces.

13-2

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Functions common to keyboard and menus

These functions are common to the keyboard and MATH

menu.

For a description, see “p” on

page 13-8.

For a description, see “ARG” on

page 13-7.

ARG

For a description, see “ ” on

∂

page 11-7.

For a description, see “AND” on

page 13-19.

AND

For a description, see

“COMB(5,2) returns 10. That is,

there are ten different ways that

five things can be combined two

at a time.!” on page 13-12.

For a description, see “S” on

page 13-11.

∑

For a description, see “Scientific

notation (powers of 10)” on

page 1-20.

EEX

∫

For a description, see “ ” on

page 11-7.

The multiplicative inverse

function finds the inverse of a

square matrix, and the

x^–1

multiplicative inverse of a real or

complex number. Also works on

a list containing only these object

types.

Keyboard functions

The most frequently used functions are available directly

from the keyboard. Many of the keyboard functions also

accept complex numbers as arguments.

Using mathematical functions

13-3

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Add, Subtract, Multiply, Divide. Also accepts complex

numbers, lists and matrices.

value1+ value2, etc.

Natural exponential. Also accepts complex numbers.

e^value

Example

e^5returns 148.413159103

Natural logarithm. Also accepts complex numbers.

LN(value)

Example

LN(1)returns 0

Exponential (antilogarithm). Also accepts complex

numbers.

10^value

Example

10^3 returns 1000

Common logarithm. Also accepts complex numbers.

LOG(value)

Example

LOG(100) returns 2

Sine, cosine, tangent. Inputs and outputs depend on the

current angle format (Degrees, Radians, or Grads).

SIN(value)

COS(value)

TAN(value)

Example

TAN(45) returns 1 (Degrees mode).

–1

ASIN

Arc sine: sin x. Output range is from –90° to 90°, –π/2

to π/2, or –100 to 100 grads. Inputs and outputs depend

on the current angle format. Also accepts complex

numbers.

ASIN(value)

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Example

ASIN(1) returns 90 (Degrees mode).

–1

ACOS

Arc cosine: cos x. Output range is from 0° to 180°, 0 to

π, or 0 to 200 grads. Inputs and outputs depend on the

current angle format. Also accepts complex numbers.

Output will be complex for values outside the normal

COS domain of –1 ≤ x ≤ 1 .

ACOS(value)

Example

ACOS(1)returns 0(Degrees mode).

–1

ATAN

Arc tangent: tan x. Output range is from –90° to 90°,

2π/2 to π/2, or –100 to 100 grads. Inputs and outputs

depend on the current angle format. Also accepts

complex numbers.

ATAN(value)

Example

ATAN(1)returns 45(Degrees mode).

Square. Also accepts complex numbers.

value

Example

18 returns 324

Square root. Also accepts complex numbers.

value

Example

324 returns 18

Negation. Also accepts complex numbers.

–value

Example

-(1,2) returns (-1,-2)

Power (x raised to y). Also accepts complex numbers.

value^power

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Example

2^8 returns 256

ABS

Absolute value. For a complex number, this is x²+ y².

ABS(value)

ABS((x,y))

Example

ABS(–1) returns 1

ABS((1,2))returns 2.2360679775

Takes the nth root of x.

root NTHROOT value

Example

3NTHROOT8 returns 2

Calculus functions

The symbols for differentiation and integration are

available directly form the keyboard—

respectively—as well as from the MATH menu.

and S

Differentiates expression with respect to the variable of

differentiation. From the command line, use a formal

name (S1, etc.) for a non-numeric result. See “Finding

derivatives” on page 13-21.

∂

variable(expression)

∂

Example

s1(s1 +3*s1)returns 2*s1+3

∂

∫

Integrates expression from lower to upper limits with

respect to the variable of integration. To find the definite

integral, both limits must have numeric values (that is, be

numbers or real variables). To find the indefinite integral,

one of the limits must be a formal variable (s1, etc).

∫

(lower, upper, expression, variable)

See “Using formal variables” on page 13-20 for

further details.

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Example

∫

(0,s1,2*X+3,X)

finds the indefinite result 3*s1+2*(s1^2/2)

See “To find the indefinite integral using formal

variables” on page 13-23 for more information on

finding indefinite integrals.

TAYLOR

Calculates the nth order Taylor polynomial of expression

at the point where the given variable = 0.

TAYLOR(expression, variable, n)

Example

TAYLOR(1 + sin(s1) ,s1,5)with Radians

angle measure and Fraction number format (set in

MODES) returns 1+s1^2+-(1/3)*s1^4.

Complex number functions

These functions are for complex numbers only. You can

also use complex numbers with all trigonometric and

hyperbolic functions, and with some real-number and

keyboard functions. Enter complex numbers in the form

(x,y), where x is the real part and y is the imaginary part.

ARG

Argument. Finds the angle defined by a complex number.

Inputs and outputs use the current angle format set in

Modes.

ARG((x, y))

Example

ARG((3,3)) returns 45 (Degrees mode)

CONJ

Complex conjugate. Conjugation is the negation (sign

reversal) of the imaginary part of a complex number.

CONJ((x, y))

Example

CONJ((3,4)) returns (3,-4)

Imaginary part, y, of a complex number, (x, y).

IM ((x, y))

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Example

IM((3,4)) returns 4

Real part x, of a complex number, (x, y).

RE((x, y))

Example

RE((3,4)) returns 3

Constants

The constants available from the MATH FUNCTIONS

menu are mathematical constants. These are described in

this section. The HP 40gs has two other menus of

constants: program constants and physical constants.

These are described in “Program constants and physical

constants” on page 13-24.

Natural logarithm base. Internally represented as

2.71828182846.

Imaginary value for –1 , the complex number (0,1).

MAXREAL

Maximum real number. Internally represented as

499

9.99999999999 x10

MAXREAL

MINREAL

Minimum real number. Internally represented as

-499

1x10

MINREAL

Internally represented as 3.14159265359.

Conversions

The conversion functions are found on the Convert

menu. They enable you to make the following

conversions.

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→C

Convert from Fahrenheit to Celcius.

Example

→C(212) returns 100

→F

Convert from Celcius to Fahrenheit.

Example

→F(0) returns 32

→CM

→IN

Convert from inches to centimeters.

Convert from centimeters to inches.

Convert from US gallons to liters.

Convert from liters to US gallons.

Convert from pounds to kilograms.

Convert from kilograms to pounds.

Convert from miles to kilometers.

Convert from kilometers to miles.

Convert from radians to degrees.

Convert from degrees to radians.

→L

→LGAL

→KG

→LBS

→KM

→MILE

→DEG

→RAD

Hyperbolic trigonometry

The hyperbolic trigonometry functions can also take

complex numbers as arguments.

–1

ACOSH

ASINH

ATANH

Inverse hyperbolic cosine : cosh x.

ACOSH(value)

–1

Inverse hyperbolic sine : sinh x.

ASINH(value)

–1

Inverse hyperbolic tangent : tanh x.

ATANH(value)

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COSH

SINH

Hyperbolic cosine

COSH(value)

Hyperbolic sine.

SINH(value)

TANH

ALOG

Hyperbolic tangent.

TANH(value)

Antilogarithm (exponential). This is more accurate than

10^xdue to limitations of the power function.

ALOG(value)

EXP

Natural exponential. This is more accurate than e^xdue

to limitations of the power function.

EXP(value)

EXPM1

LNP1

Exponent minus 1 : e^x– 1 . This is more accurate than

EXP when x is close to zero.

EXPM1(value)

Natural log plus 1 : ln(x+1). This is more accurate than

the natural logarithm function when xis close to zero.

LNP1(value)

List functions

These functions work on list data. See “List functions” on

page 19-6.

Loop functions

The loop functions display a result after evaluating an

expression a given number of times.

ITERATE

Repeatedly for #times evaluates an expression in terms of

variable. The value for variable is updated each time,

starting with initialvalue.

ITERATE(expression, variable, initialvalue,

#times)

Example

ITERATE(X ,X,2,3) returns 256

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RECURSE

Provides a method of defining a sequence without using

the Symbolic view of the Sequence aplet. If used with |

(“where”), RECURSE will step through the evaluation.

RECURSE(sequencename, term , term , term )

Example

RECURSE(U,U(N-1)*N,1,2)

U1(N)

Stores a factorial-calculating function named U1.

When you enter U1(5), for example, the function

calculates 5! (120).

Summation. Finds the sum of expression with respect to

variable from initialvalue to finalvalue.

Σ(variable=initialvalue, finalvalue, expression)

Example

Σ(C=1,5,C )returns 55.

Matrix functions

These functions are for matrix data stored in matrix

variables. See “Matrix functions and commands” on

page 18-10.

Polynomial functions

Polynomials are products of constants (coefficients) and

variables raised to powers (terms).

POLYCOEF

Polynomial coefficients. Returns the coefficients of the

polynomial with the specified roots.

POLYCOEF([roots])

Example

To find the polynomial with roots 2, –3, 4, –5:

POLYCOEF([2,-3,4,-5]) returns[1,2,-25,

-26,120], representing x +2x –25x –26x+120.

POLYEVAL

Polynomial evaluation. Evaluates a polynomial with the

specified coefficients for the value of x.

POLYEVAL([coefficients], value)

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Example

For x +2x –25x –26x+120:

POLYEVAL([1,2,-25,-26,120],8)returns

3432.

POLYFORM

POLYROOT

Polynomial form. Creates a polynomial in variable1 from

expression.

POLYFORM(expression, variable1)

Example

POLYFORM((X+1)^2+1,X)returns X^2+2*X+2.

Polynomial roots. Returns the roots for the nth-order

polynomial with the specified n+1 coefficients.

POLYROOT([coefficients])

Example

For x +2x –25x –26x+120:

POLYROOT([1,2,-25,-26,120])returns

[2,-3,4,-5].

H I N T

The results of POLYROOT will often not be easily seen in

HOME due to the number of decimal places, especially if

they are complex numbers. It is better to store the results

of POLYROOT to a matrix.

For example, POLYROOT([1,0,0,-8]

M1will

store the three complex cube roots of 8 to matrix M1 as

a complex vector. Then you can see them easily by going

to the Matrix Catalog. and access them individually in

calculations by referring to M1(1), M1(2) etc.

Probability functions

COMB

Number of combinations (without regard to order) of n

things taken r at a time: n!/(r!(n-r)).

COMB(n, r)

Example

COMB(5,2) returns 10. That is, there are ten

different ways that five things can be combined two

at a time.!

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Factorial of a positive integer. For non-integers, ! = Γ(x +

1). This calculates the gamma function.

value!

PERM

Number of permutations (with regard to order) of n things

taken r at a time: n!/(r!(n-r)!

PERM(n, r)

Example

PERM(5,2) returns 20. That is, there are 20

different permutations of five things taken two at a

time.

RANDOM

Random number (between zero and 1). Produced by a

pseudo-random number sequence. The algorithm used in

the RANDOM function uses a seed number to begin its

sequence. To ensure that two calculators must produce

different results for the RANDOM function, use the

RANDSEED function to seed different starting values

before using RANDOM to produce the numbers.

RANDOM

H I N T

The setting of Time will be different for each calculator, so

using RANDSEED(Time) is guaranteed to produce a set of

numbers which are as close to random as possible. You

can set the seed using the command RANDSEED.

UTPC

UTPF

Upper-Tail Chi-Squared Probability given degrees of

freedom, evaluated at value. Returns the probability that

a χ random variable is greater than value.

UTPC(degrees, value)

Upper-Tail Snedecor’s F Probability given numerator

degrees of freedom and denominator degrees of freedom

(of the F distribution), evaluated at value. Returns the

probability that a Snedecor's F random variable is

greater than value.

UTPF(numerator, denominator, value)

UTPN

Upper-Tail Normal Probability given mean and variance,

evaluated at value. Returns the probability that a normal

random variable is greater than value for a normal

distribution. Note: The variance is the square of the

standard deviation.

UTPN(mean, variance, value)

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UTPT

Upper-Tail Student’s t-Probability given degrees of

freedom, evaluated at value. Returns the probability that

the Student's t- random variable is greater than value.

UTPT(degrees, value)

Real-number functions

Some real-number functions can also take complex

arguments.

CEILING

Smallest integer greater than or equal to value.

CEILING(value)

Examples

CEILING(3.2) returns 4

CEILING(-3.2) returns -3

DEG→RAD

Degrees to radians. Converts value from Degrees angle

format to Radians angle format.

DEG→RAD(value)

Example

DEG→RAD(180) returns 3.14159265359, the

value of π.

FLOOR

Greatest integer less than or equal to value.

FLOOR(value)

Example

FLOOR(-3.2) returns -4

FNROOT

Function root-finder (like the Solve aplet). Finds the value

for the given variable at which expression most nearly

evaluates to zero. Uses guess as initial estimate.

FNROOT(expression, variable, guess)

Example

FNROOT(M*9.8/600-1,M,1) returns

61.2244897959.

FRAC

Fractional part.

FRAC(value)

Example

FRAC(23.2) returns .2

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HMS→

Hours-minutes-seconds to decimal. Converts a number or

expression in H.MMSSs format (time or angle that can

include fractions of a second) to x.x format (number of

hours or degrees with a decimal fraction).

HMS→(H.MMSSs)

Example

HMS→(8.30) returns 8.5

→HMS

Decimal to hours-minutes-seconds. Converts a number or

expression in x.xformat (number of hours or degrees

with a decimal fraction) to H.MMSSs format (time or

angle up to fractions of a second).

→HMS(x.x)

Example

→HMS(8.5) returns 8.3

INT

Integer part.

INT(value)

Example

INT(23.2) returns 23

MANT

MAX

Mantissa (significant digits) of value.

MANT(value)

Example

MANT(21.2E34) returns 2.12

Maximum. The greater of two values.

MAX(value1, value2)

Example

MAX(210,25) returns 210

MIN

Minimum. The lesser of two values.

MIN(value1, value2)

Example

MIN(210,25)returns 25

MOD

Modulo. The remainder of value1/value2.

value1 MODvalue2

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Example

9MOD 4 returns 1

xpercent of y; that is, x/100*y.

%(x, y)

Example

%(20,50) returns 10

%CHANGE

%TOTAL

RAD→DEG

Percent change from x to y, that is, 100(y–x)/x.

%CHANGE(x, y)

Example

%CHANGE(20,50) returns 150

Percent total : (100)y/x. What percentage of x, is y.

%TOTAL(x, y)

Example

%TOTAL(20,50) returns 250

Radians to degrees. Converts value from radians to

degrees.

RAD→DEG(value)

Example

RAD→DEG(π) returns 180

ROUND

Rounds value to decimal places. Accepts complex

numbers.

ROUND(value, places)

Round can also round to a number of significant digits as

showed in example 2.

Examples

ROUND(7.8676,2) returns 7.87

ROUND (0.0036757,-3) returns 0.00368

SIGN

Sign of value. If positive, the result is 1. If negative, –1. If

zero, result is zero. For a complex number, this is the unit

vector in the direction of the number.

SIGN(value)

SIGN((x, y))

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Examples

SIGN (–2)returns –1

SIGN((3,4)) returns (.6,.8)

TRUNCATE

XPON

Truncates value to decimal places. Accepts complex

numbers.

TRUNCATE(value, places)

Example

TRUNCATE(2.3678,2) returns 2.36

Exponent of value.

XPON(value)

Example

XPON(123.4) returns 2

Two-variable statistics

These are functions for use with two-variable statistics.

See “Two-variable” on page 10-15.

Symbolic functions

The symbolic functions are used for symbolic

manipulations of expressions. The variables can be

formal or numeric, but the result is usually in symbolic

form (not a number). You will find the symbols for the

symbolic functions = and | (where) in the CHARS menu

(

CHARS) as well as the MATH menu.

= (equals)

ISOLATE

Sets an equality for an equation. This is not a logical

operator and does not store values. (See “Test functions”

on page 13-19.)

expression1=expression2

Isolates the first occurrence of variable in expression=0

and returns a new expression, where

variable=newexpression. The result is a general solution

that represents multiple solutions by including the (formal)

variables S1 to represent any sign and n1 to represent

any integer.

ISOLATE(expression, variable)

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Examples

ISOLATE(2*X+8,X) returns -4

ISOLATE(A+B*X/C,X) returns -(A*C/B)

LINEAR?

QUAD

Tests whether expression is linear for the specified

variable. Returns 0(false) or 1(true).

LINEAR?(expression, variable)

Example

LINEAR?((X^2-1)/(X+1),X) returns 0

Solves quadratic expression=0 for variable and returns

a new expression, where variable=newexpression. The

result is a general solution that represents both positive

and negative solutions by including the formal variable

S1 to represent any sign: + or – .

QUAD(expression, variable)

Example

QUAD((X-1) -7,X) returns (2+s1*(2*√7))/2

QUOTE

Encloses an expression that should not be evaluated

numerically.

QUOTE(expression)

Examples

QUOTE(SIN(45))

expression SIN(45) rather than the value of SIN(45).

F1(X) stores the

Another method is to enclose the expression in single

quotes.

For example, X^3+2*X

F1(X)puts the

expression X^3+2*X into F1(X)in the Function

aplet.

| (where)

Evaluates expression where each given variable is set to

the given value. Defines numeric evaluation of a symbolic

expression.

expression|(variable1=value1, variable2=value2,...)

Example

3*(X+1)|(X=3) returns 12.

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Test functions

The test functions are logical operators that always return

either a 1 (true) or a 0 (false).

Less than. Returns 1 if true, 0 if false.

value1<value2

≤

Less than or equal to. Returns 1 if true, 0 if false.

value1≤value2

= =

Equals (logical test). Returns 1 if true, 0 if false.

value1==value2

≠

Not equal to. Returns 1 if true, 0 if false.

value1≠value2

Greater than. Returns 1 if true, 0 if false.

value1>value2

≥

Greater than or equal to. Returns 1 if true, 0 if false.

value1≥value2

AND

Compares value1 and value2. Returns 1 if they are both

non-zero, otherwise returns 0.

value1 AND value2

IFTE

If expression is true, do the trueclause; if not, do the

falseclause.

IFTE(expression, trueclause, falseclause)

Example

2 3

IFTE(X>0,X ,X )

NOT

Returns 1 if value is zero, otherwise returns 0.

NOT value

Returns 1 if either value1 or value2 is non-zero, otherwise

returns 0.

value1 OR value2

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XOR

Exclusive OR. Returns 1 if either value1 or value2—but

not both of them—is non-zero, otherwise returns 0.

value1 XOR value2

Trigonometry functions

The trigonometry functions can also take complex

numbers as arguments. For SIN, COS, TAN, ASIN,

ACOS, and ATAN, see the Keyboard category.

ACOT

ACSC

ASEC

COT

Arc cotangent.

ACOT(value)

Arc cosecant.

ACSC(value)

Arc secant.

ASEC(value)

Cotangent: cosx/sinx.

COT(value)

CSC

Cosecant: 1/sinx

CSC(value)

SEC

Secant: 1/cosx.

SEC(value)

Symbolic calculations

Although CAS provides the richest environment for

performing symbolic calculations, you can perform some

symbolic calculations in HOME and with the Function

aplet. CAS functions that you can perform in HOME (such

as DERVX and INTVX) are discussed in “Using CAS

functions in HOME” on page 14-7.

In HOME

When you perform calculations that contain normal

variables, the calculator substitutes values for any

variables. For example, if you enter A+B on the command

line and press

, the calculator retrieves the values

for A and B from memory and substitutes them in the

calculation.

Using formal

variables

To perform symbolic calculations, for example symbolic

differentiations and integrations, you need to use formal

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names. The HP 40gs has six formal names available for

use in symbolic calculations. These are S1 to S5. When

you perform a calculation that contains a formal name,

the HP 40gs does not carry out any substitutions.

You can mix formal names and real variables. Evaluating

(A+B+S1) will evaluate A+B, but not S1.

If you need to evaluate an expression that contains formal

names numerically, you use the | (where) command,

listed in the Math menu under the Symbolic category.

For example to evaluate (S1*S2) when S1=2 and

S2=4, you would enter the calculation as follows:

(The | symbol is in the CHARS menu: press

CHARS.

The = sign is listed in the MATH menu under Symbolic

functions.)

Symbolic

You can perform symbolic operations in the Function

aplet’s Symbolic view. For example, to find the derivative

of a function in the Function aplet’s Symbolic view, you

define two functions and define the second function as a

derivative of the first function. You then evaluate the

second function. See “To find derivatives in the Function

aplet’s Symbolic view” on page 13-22 for an example.

calculations in

the Function

aplet

Finding derivatives

The HP 40gs can perform symbolic differentiation on

some functions. There are two ways of using the HP 40gs

to find derivatives.

•

You can perform differentiations in HOME by using

the formal variables, S1 to S5.

•

You can perform differentiations of functions of X in

the Function aplet.

To find derivatives

in HOME

To find the derivative of the function in HOME, use a

formal variable in place of X. If you use X, the

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differentiation function substitutes the value that X holds,

and returns a numeric result.

For example, consider the function:

dx(sin(x²) + 2cos(x))

1. Enter the differentiation function onto the command

line, substituting S1 in place of X.

2. Evaluate the function.

3. Show the result.

To find derivatives

in the Function

aplet’s Symbolic

view

To find the derivative of the function in the Function aplet’s

Symbolic view, you define two functions and define the

second function as a derivative of the first function. For

example, to differentiate sin(x²) + 2cosx :

1. Access the Function aplet’s Symbolic view and define

F1.

2. Define F2(X) as the

derivative of F(1).

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3. Select F2(X) and

evaluate it.

4. Press

to display

the result. Note: Use

the arrow keys to view

the entire function.

You could also just define

F1(x)= dx(sin(x²) + 2cos(x)) .

To find the

For example, to find the indefinite integral of

3x²– 5dx use:

indefinite integral

using formal

variables

∫

(

0, S1, 3 X − 5, X

)

1. Enter the function.

2. Show the result format.

3. Press

to close the

show window.

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4. Copy the result and

evaluate.

Thus, substituting X for S1, it can be seen that:

x³

⎛

⎜

⎝

⎞

⎟

⎠

----

3x²– 5dx= – 5x + 3

---------------

∫

∂

(X)

∂X

This result is derived from substituting X=S1 and X=0 into

the original expression found in step 1. However,

substituting X=0 will not always evaluate to zero and may

result in an unwanted constant.

(x – 2)⁵

To see this, consider: (x – 2)⁴dx=

-------------------

∫

The ‘extra’ constant of

32/5 results from the

substitution of x = 0 into

(x – 2) /5, and should be

disregarded if an

indefinite integral is

required.

Program constants and physical constants

When you press

, three menus of functions and

constants become available:

•

the math functions menu (which appears by default)

the program constants menu, and

the physical constants menu.

The math functions menu is described extensively earlier

in this chapter.

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Program constants

The program constants are numbers that have been

assigned to various calculator settings to enable you to

test for or specify such a setting in a program. For

example, the various display formats are assigned the

following numbers:

1 Standard

2 Fixed

3 Scientific

4 Engineering

5 Fraction

6 Mixed fraction

In a program, you could store the constant number of a

particular format into a variable and then subsequently

test for that particular format.

To access the menu of program constants:

1. Press

2. Press

3. Use the arrow keys to navigate through the options.

4. Click and then to display the number

assigned to the option you selected in the previous

step.

The use of program constants is illustrated in more detail

in “Programming” on page 21-1

Physical constants

There are 29 physical constants—from the fields of

chemistry, physics and quantum mechanics—that you

can use in calculations. A list of all these constants can be

found in “Physical Constants” on page R-16.

To access the menu of physical constants:

1. Press

2. Press

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3. Use the arrow keys to navigate through the options.

4. To see the symbol and value of a selected constant,

press . (Click to close the information

window that appears.)

The following example shows the information

available about the speed of light (one of the physics

constants).

5. To use the selected constant in a calculation, press

. The constant appears at the position of the

cursor on the edit line.

Example

Suppose you want to know the potential energy of a mass

of 5 units according to the equation E = mc².

1. Enter 5

2. Press

and then press

13-26

Using mathematical functions

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3. Select light s...from the Physics menu.

4. Press

. The menu closes and the value of the

selected constant is copied to the edit line.

5. Complete the equation as you would normally and

press

to get the result.

Using mathematical functions

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Computer Algebra System (CAS)

What is a CAS?

A computer algebra system (hereafter CAS) enables you

to perform symbolic calculations. With a CAS you

manipulate mathematical equations and expressions in

symbolic form, rather than manipulating approximations

of the numerical quantities represented by those symbols.

In other words, a CAS works in exact mode, giving you

infinite precision. On the other hand, non-CAS

calculations, such as those performed in HOME view or

by an aplet, are numerical calculations and are limited by

the precision of the calculator (to 10^–12in the case of the

HP 40gs).

For example, with Standard as your numerical format,

1/2 + 1/6 returns 0.6666666666667 if you are

working in the HOME screen; however, 1/2 + 1/6

returns 2/3 if you are working with CAS. HOME

calculations are restricted to approximate (or numeric)

mode, while CAS calculations always work in exact

mode (unless you specifically change the default CAS

modes).

Each mode has advantages and disadvantages. For

example, in exact mode there is no rounding error, but

some calculations will take much longer to complete and

require more memory than equivalent calculations in

numeric mode.

Performing symbolic calculations

You perform CAS calculations with a special tool known

as the Equation Writer. Some computer algebra

operations can also be done in the HOME screen, as

long as you take certain precautions (see “Using CAS

functions in HOME” on page 14-7). Moreover, some

computer algebra operations can only be done in the

HOME screen; for example, symbolic linear algebra

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using vectors and matrices. (Vectors and matrices cannot

be entered using the Equation Writer).

To open the Equation

Writer, press the

soft-

key on the menu bar of the

HOME screen.

The illustration at the right

shows an expression being

written in the Equation

Writer. The soft keys on the

menu bar provide access to

CAS functions and

commands.

To leave the Equation Writer, press

to return to

the HOME screen. Note that expressions written in the

Equation Writer (and the results of evaluating an

expression) are not automatically copied to the HOME

history when you leave the Equation Writer. (You can,

however, manually copy them to HOME: see page 14-8).

CAS functions are described in detail in “CAS functions

in the Equation Writer” on page 14-9. Chapter 15,

“Equation Writer”, explains in detail how to enter an

expression in the Equation Writer and contains numerous

worked examples of CAS in operation.

An example

To give you an idea of how CAS works, let’s consider a

simple example. Suppose you want to convert C to the

form d ⋅ 5 where C is 2 45 – 20 and d is a whole

number.

1. Open the Equation Writer by pressing the

key on the HOME screen.

soft-

2. Enter the expression for

[Hint: use the keys on

the keyboard as you

would if entering the

expression in HOME. Press the

key twice to select

the entire first term before entering the second term.]

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3. Press

and

select just the 20 in the

20 term.

4. Press the

key and choose FACTOR.

Then press

Note that the FACTOR

function is added to the

selected term.

5. Press

to factor

the selected term.

6. Press

to select the

entire second term, and

then press

simplify it.

7. Press

to select the 45

in the first term.

8. As you did earlier, press

the menu key and

choose FACTOR. Then

press and

factor the selected term.

9. Press

to select the

entire second term, and

then press

simplify it.

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10.Press

select the entire

expression and then

three times to

press

to simplify

it to the form required.

CAS variables

When you use the symbolic calculation functions, you are

working with symbolic variables (variables that do not

contain a permanent value). In the HOME screen, a

variable of this kind must have a name like S1…S5,

s1…s5, n1…n5, but not X, which is assigned to a real

value. (By default, X is assigned to 0). To store symbolic

expressions, you must use the variables E0, E1…E9.

In the Equation Writer, all the variables may, or may not

be, assigned. For example, X is not assigned to a real

value by default, so computing X + X will return 2X.

Moreover, Equation Writer variables can have long

names, like XY or ABC, unlike in HOME where implied

multiplication is assumed. (For example ABC is

interpreted as A × B × C in HOME.) For these reasons,

variables used in the Equation Writer cannot be used in

HOME, and vice versa.

Using the PUSHcommand, you can transfer expressions

from the HOME screen history to CAS history (see

page 14-8). Likewise, you can use the POPcommand to

transfer expressions from CAS history to the HOME

screen history (see page 14-8).

The current variable

In the Equation Writer, the current variable is the name of

the symbolic variable contained in VX. It is almost always

X. (The current variable is always S1 in HOME.)

Some CAS functions depend on a current variable; for

example, the function DERVX calculates the derivative

with respect to the current variable. Hence in the Equation

Writer, DERVX(2*X+Y) returns 2 if VX = X, but 1 if VX

= Y. However, in the HOME screen, DERVX(2*S1+S2)

returns 2, but DERIV(2*S1+S2,S2) returns 1.

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CAS modes

The modes that determine

how CAS operates can be

set on CAS MODES screen.

To display CAS MODES

screen, press:

·To navigate through the options in CAS MODES screen,

press the arrow keys.

To select or deselect a mode, navigate to the appropriate

field and press

until the correct setting is displayed

(indicated by a check mark in the field). For some settings

(such as ^{INDEP VAR}and MODULO), you will need to press

to be able to change the setting.

Press

to close CAS MODES screen.

N O T E

You can also set CAS modes from within the Equation

Writer. See “Configuration menus” on page 15-3 for

information.

Selecting the

independent

variable

Many of the functions provided by CAS use a pre-

determined independent variable. By default, that

variable is the letter X (upper case) as shown in CAS

MODES screen above. However, you can change this

variable to any other letter, or combination of letters and

numbers, by editing the ^{INDEP VAR}field in CAS MODES

screen. To change the setting, press

value and then press

, enter a new

The variable VX in the calculator's {HOME CASDIR}

directory takes, by default, the value of 'X'. This is the

name of the preferred independent variable for algebraic

and calculus applications. If you use another independent

variable name, some functions (for example, HORNER)

will not work properly.

Selecting the

modulus

The MODULO option on CAS MODES screen lets you

specify the modulo you want to use in modular arithmetic.

The default value is 13.

Approximate vs.

Exact mode

When the APPROX mode is selected, symbolic operations

(for example, definite integrals, square roots, etc.), will be

calculated numerically. When this mode is unselected,

exact mode is active, hence symbolic operations will be

Computer Algebra System (CAS)

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calculated as closed-form algebraic expressions,

whenever possible. [Default: unselected.]

Num. Factor mode

When the NUM FACTOR setting is selected, approximate

roots are used when factoring. For example, x⁵+ 5x + 1

is irreducible over the integers but has approximate roots

over the reals. With ^{NUM FACTOR}set, the approximate roots

are returned. [Default: unselected.]

Complex vs. Real

mode

When COMPLEX is selected and an operation results in a

complex number, the result will be shown in the form a +

bi or in the form of an ordered pair (a,b). If ^COMPLEXmode

is not selected and an operation results in a complex

number, you will be asked to switch to ^COMPLEXmode. If

you decline, the calculator will report an error. [Default:

unselected.]

When in ^COMPLEXmode, CAS is able to perform a wider

range of operations than in non-complex (or real) mode,

but it will also be considerably slower. Thus, it is

recommended that you don’t select ^COMPLEXmode unless

requested by the calculator in the performance of a

particular operation.

Verbose vs. non-

verbose mode

When VERBOSE is selected, certain calculus applications

are provided with comment lines in the main display. The

comment lines will appear in the top lines of the display,

but only while the operation is being calculated. [Default:

unselected.]

Step-by-step mode

When STEP/STEP is selected, certain operations will be

shown one step at a time in the display. You press

to show each step in turn. [Default: selected.]

Increasing-powers

mode

When INCR POW is selected, polynomials will be listed so

that the terms will have increasing powers of the

independent variable (which is the opposite to how

polynomials are normally written). [Default: unselected.]

Rigorous setting

When RIGOROUS is selected, any algebraic expression of

the form |X|, i.e., the absolute value of X, is not

simplified to X. [Default: selected.]

Simplify non-

rational setting

When SIMP NON-RATIONAL is selected, non-rational

expressions will be automatically simplified. [Default:

selected.]

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Using CAS functions in HOME

You can use many computer algebra functions directly in

the HOME screen, as long as you take certain

precautions. CAS functions that take matrices as an

argument work only from HOME.

CAS functions can be accessed by pressing

MATH menu is displayed. You can also directly type a

function name when you are in alpha mode.

when

Note that certain calculations will be performed in

approximate mode because numbers are interpreted as

reals instead of integers in HOME. To do exact

calculations, you should use the XQ command. This

command converts an approximate argument into an

exact argument.

For example, if Radians is your angle setting, then:

ARG(XQ(1 + i)) = π/4 but

ARG(1 + i) = 0.7853...

Similarly:

FACTOR(XQ(45)) = 3²× 5 but

FACTOR(45) = 45

Note too that the symbolic HOME variable S1 serves as

the current variable for CAS functions in HOME. For

example:

DERVX(S1²+ 2 × S1) = 2 × S1 + 2

The result 2 × S1 + 2 does not depend on the Equation

Writer variable, VX.

Some CAS functions cannot work in HOME because they

require a change to the current variable.

Remember that you must use S1,S2,…S5, s1,s2,…s5,

and n1,n2,…n5 for symbolic variables and E0, E1,…E9

to store symbolic expressions. For example, if you type:

S1²– 4 × S2

then you get:

DERVX(E1) = S1 × 2

DERIV(E1, S2) = –4

INTVX(E1) = 1/3 S1³– 4 × (S2 × S1)

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Symbolic matrices are stored as a list of lists and therefore

must be stored in L0, L1…L9 (whereas numeric matrices

are stored in M0, M1,…M9). CAS linear algebra

instructions accept lists of lists as input.

For example, if you type in HOME:

XQ({{S2 + 1, 1}, { 2 , 1}})

then you have:

TRAN(L1) = {{S2 + 1, 2 }, {1, 1}}

Some numeric linear algebra commands do not directly

work on a list of lists, but will do so after a conversion by

AXL. For example, if you enter:

DET(AXL(L1))

you get:

S2–(–1 + 2 )

Send expressions

from HOME to CAS

history

In the HOME screen, you can use the PUSHcommand to

send expressions to CAS history. For example, if you

enter PUSH(S1+1), S1+1 is written to CAS history.

Send expressions

from CAS to HOME

history

In the HOME screen, you can use the POPcommand to

retrieve the last expression written to CAS history. For

example, if S1+1 is the last expression written to CAS

history and you enter POP in the HOME screen, S1+1 is

written to the HOME screen history (and S1+1 is removed

from CAS history).

Online Help

When you are working with

the Equation Writer, you can

display online help about

any CAS command. To

display the contents of the

online help, press

Press

to navigate to the

command you want help

with and then press

You can also get CAS help

from the HOME screen. Type

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HELPand press

appears.

. The menu of help topics

Each help topic includes the required syntax, along with

real sample values. You can copy the syntax, with the

sample values, to the HOME screen or to the Equation

Writer, by pressing

T I P

If you highlight a CAS command and then press

2, help about the highlighted command is displayed.

You can display the online help in French rather than

English. For instructions, see “Online Help language” on

page 15-4.

CAS functions in the Equation Writer

You can display a menu of CAS functions in four ways:

•

by displaying the MATH menu from HOME and then

pressing , or

•

opening the Equation Writer and pressing

opening the Equation Writer and selecting a function

from a soft-key menu, or

•

opening the Equation Writer and pressing

You can also directly type the name of a CAS function

when you are in ALPHA mode.

Note that in this section, CAS functions available from the

sot-key menus in the Equation Writer are described. CAS

functions available from the MATH menu are described in

“CAS Functions on the MATH menu” on page 14-45.

N O T E

When using CAS, you should be aware that the required

syntax will vary depending on whether you are applying

the command to an expression or a function. All CAS

commands are designed to work with expressions; that is,

they take expressions as arguments. If you are going to

use a function—for example, F—you need to specify an

expression made from this function, such as F(x), where x

is the independent variable.

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For example, suppose you have stored the expression x²

in G, and have defined the function F(x) as x². Suppose

now you want to calculate INTVX(X²). You could:

•

enter INTVX(X²)directly, or

enter INTVX(G), or

enter INTVX(F(X)).

Note that you can apply the command directly to an

expression or to a variable that holds an expression (the

first two cases above). But where you want to apply it to

a defined function, you need to specify the full function

name, F(X), as in the third case above.

ALGB menu

COLLECT

Factors over the integers

COLLECT combines like terms and factors the expression

over the integers.

Example

To factor x²– 4 over the integers you would type:

COLLECT(X²–4)

which gives in real mode:

(x + 2) ⋅ (x – 2)

Example

To factor x²– 2 over the integers you would type:

COLLECT(X²–2)

which gives:

x²– 2

DEF

Define a function

For its argument, DEF takes an equality between:

1. the name of a function (with parentheses containing

the variable), and

2. an expression defining the function.

DEF defines this function and returns the equality.

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Typing:

DEF(U(N) = 2N+1)

produces the result:

U(N) = 2N+1

Typing:

U(3)

then returns:

Example

Calculate the first six Fermat numbers F1...F6 and

determine whether they are prime.

So, you want to calculate:

F(k) = 2²+ 1 for k = 1...6

Typing the formula:

2²+ 1

gives a result of 17. You can then invoke the

ISPRIME?()command, which is found in the MATH

key’s Integermenu. The response is 1, which means

TRUE. Using the history (which you access by pressing the

SYMBkey), you put the expression 2²+ 1 into the

Equation Writer with ECHO, and change it to:

2²+ 1

Or better, define a function F(K) by selecting DEFfrom the

ALGBmenu on the menu bar and type:

DEF(F(K) = 2²+ 1)

The response is 2²+ 1 and F is now listed amongst the

variables (which you can verify using the VARSkey).

For K=5, you then type:

F(5)

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which gives

4294967297

You can factor F(5) with FACTOR, which you’ll find in the

ALGBmenu on the menu bar.

Typing:

FACTOR(F(5))

gives:

641·6700417

Typing:

F(6)

gives:

18446744073709551617

Using FACTORto factor it, then yields:

274177·67280421310721

EXPAND

Distributivity

EXPAND expands and simplifies an expression.

Example

Typing:

XPAND((X²+ 2 ⋅ X + 1) ⋅ (X²– 2 ⋅ X + 1)

gives:

x⁴+ 1

FACTOR

Factorization

FACTOR factors an expression.

Example

To factor:

x⁴+ 1

type:

FACTOR(X +1)

FACTORis located in the ALGBmenu.

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In real mode, the result is:

(x²+ 2 ⋅ x + 1) ⋅ (x²– 2 ⋅ x + 1)

In complex mode (using CFG), the result is:

-----

⋅ (2x + (1 + i) ⋅ 2) ⋅ (2x–(1 + i) ⋅ 2) ⋅ (2x + (1 – i) ⋅ 2)

⋅ (2x–(1 – i) ⋅ 2)

PARTFRAC

Partial fraction expansion

PARTFRAC has a rational fraction as an argument.

PARTFRAC returns the partial fraction decomposition of

this rational fraction.

Example

To perform a partial fraction decomposition of a rational

function, such as:

x⁵– 2 ⋅ x³+ 1

------------------------------------------------------------------------

x⁴– 2 ⋅ x³+ 2 ⋅ x²– (2 ⋅ x + 1)

you use the PARTFRACcommand.

In real and direct mode, this produces:

x – 3

2 ⋅ x²+ 2

–1

2 ⋅ x – 2

--------------------- ------------------

x + 2 +

In complex mode, this produces:

1 – 3i

–1

1 + 3i

-------------

-----

-------------

------------- ----------- --------------

x + 2 +

x + i x – 1 x – i

QUOTE

Quoted expression

QUOTE(expression) is used to prevent an expression

from being evaluated or simplified.

Example 1

Typing:

⎛

⎞

---

im QUOTE((2X – 1) ⋅ EXP( – 1), X = +∞

⎝

⎠

gives:

+∞

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Example 2

Typing:

SUBST(QUOTE(CONJ(Z)),Z=1+i)

gives:

CONJ(1+i)

STORE

Store an object in a variable

STORE stores an object in a variable.

STOREis found in the ALGBmenu or the Equation Writer

menu bar.

Example

Type:

STORE(X -4,ABC)

or type:

X -4

then select it and call STORE, then type ABC, then press

ENTER to confirm the definition of the variable ABC.

To clear the variable, press VARSin the Equation Writer

(then choose PURGEon the menu bar), or select

UNASSIGNon the ALGBmenu by typing, for example,

UNASSIGN(ABC)

Substitute a value for a variable

| is an infix operator used to substitute a value for a

variable in an expression (similar to the function SUBST).

| has two parameters: an expression dependent on a

parameter, and an equality (parameter=substitute value).

| substitutes the specified value for the variable in the

expression.

Typing:

X²– 1

X = 2

gives:

2²– 1

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SUBST

Substitute a value for a variable

SUBST has two parameters: an expression dependent on

a parameter, and an equality (parameter=substitute

value).

SUBST substitutes the specified value for the variable in

the expression.

Typing:

SUBST(A +1,A=2)

gives:

2²+ 1

TEXPAND

Develop in terms of sine and cosine

TEXPAND has a trigonometric expression or

transcendental function as an argument.

TEXPAND develops this expression in terms of sin(x) and

cos(x).

Example

Typing:

TEXPAND(COS(X+Y))

gives:

cos(y) ⋅ cos(x) – sin(y) ⋅ sin(x)

Example

Typing:

TEXPAND(COS(3·X))

gives:

4 ⋅ cos(x)³– 3 ⋅ cos(x)

UNASSIGN

Clear a variable

UNASSIGN is used to clear a variable, for example:

UNASSIGN(ABC)

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DIFF menu

DERIV

Derivative and partial derivative

DERIV has two arguments: an expression (or a function)

and a variable.

DERIV returns the derivative of the expression (or the

function) with respect to the variable given as the second

parameter (used for calculating partial derivatives).

Example

Calculate:

∂(x ⋅ y²⋅ z³+ x ⋅ y)

---------------------------------------------

∂z

Typing:

2 3

DERIV(X·Y ·Z + X·Y,Z)

gives:

3 ⋅ x ⋅ y²⋅ z²

DERVX

Derivative

DERVX has one argument: an expression. DERVX

calculates the derivative of the expression with respect to

the variable stored in VX.

For example, given:

x + 1

x – 1

⎛

⎝

⎞

⎠

-------------

-----------

f(x) =

+ ln

x²– 1

calculate the derivative of f.

Type:

X + 1

X – 1

⎛

⎝

⎛

⎝

⎞

⎠

--------------

------------

DERVX

+ LN

X²– 1

Or, if you have stored the definition of f(x) in F, that is, if

you have typed:

X + 1

X – 1

⎛

⎝

⎛

⎝

⎞

⎠

⎞

--------------

------------

TORE

+ LN

X²– 1

⎠

then type:

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DERVX(F)

Or, if you have defined F(X) using DEF, that is, if you have

typed:

X + 1

X – 1

⎛

⎝

⎞⎞

⎠⎠

--------------

------------

DEF(F(X) =

+ LN

X²– 1

then type:

DERVX(F(X))

Simplify the result to get:

3 ⋅ x²– 1

--------------------------------

–

x⁴– 2 ⋅ x²+ 1

DIVPC

Division in increasing order by exponent

DIVPC has three arguments: two polynomials A(X) and

B(X) (where B(0) ≠0), and a whole number n.

DIVPC returns the quotient Q(X) of the division of A(X) by

B(X), in increasing order by exponent, and with deg(Q)

<= n or Q = 0.

Q[X] is then the limited nth-order expansion of:

A[X]

-----------

B[X]

in the vicinity of X= 0.

Typing:

2 3

DIVPC(1+X +X ,1+X ,5)

gives:

1 + x³– x⁵

N O T E :

When the calculator displays a request to change to

increasing powers mode, respond yes.

FOURIER

Fourier coefficients

FOURIER has two parameters: an expression f(x) and a

whole number N.

FOURIER returns the Fourier coefficient c of f(x),

considered to be a function defined over interval [0, T]

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and with period T (T being equal to the contents of the

variable PERIOD).

If f(x) is a discrete series, then:

2iNxπ

+ ∞

---------------

f(x) =

c_Ne

∑

N = –∞

Example

Determine the Fourier coefficients of a periodic function f

with period 2π and defined over interval [0, 2π] by

f(x)=x .

Typing:

STORE(2π,PERIOD)

FOURIER(X ,N)

The calculator does not know that N is a whole number,

so you have to replace EXP(2∗ i∗N∗π) with 1 and then

simplify the expression. We get

2 ⋅ i ⋅ N ⋅ π + 2

----------------------------------

N²

So if N ≠ 0 , then:

2 ⋅ i ⋅ N ⋅ π + 2

----------------------------------

c_N

N²

Typing:

FOURIER(X ,0)

gives:

4 ⋅ π²

------------

so if N = 0 , then:

4 ⋅ π²

------------

c₀

IBP

Partial integration

IBP has two parameters: an expression of the form

u(x) ⋅ v'(x) and v(x) .

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IBP returns the AND of u(x) ⋅ v(x) and of –v(x) ⋅ u'(x)

that is, the terms that are calculated when performing a

partial integration.

It remains then to calculate the integral of the second term

of the AND, then add it to the first term of the AND to

obtain a primitive of u(x) ⋅ v'(x) .

Typing:

IBP(LN(X),X)

gives:

X·LN(X) AND - 1

The integration is completed by calling INTVX:

INTVX(X·LN(X)AND - 1)

which produces the result:

X·LN(X) - X

N O T E :

If the first IBP (or INTVX) parameter is an AND of two

elements, IBP concerns itself only with the second element

of the AND, adding the integrated term to the first element

of the AND (so that you can perform multiple IBP in

succession).

INTVX

Primitive and defined integral

INTVX has one argument: an expression.

INTVX calculates a primitive of its argument with respect

to the variable stored in VX.

Example

Calculate a primitive of sin(x) × cos(x).

Typing:

INTVX(SIN(X)·COS(X))

gives in step-by-step mode:

COS(X)·SIN(X)

Int[u’∗F(u)] with u=SIN(X)

Pressing OK then sends the result to the Equation Writer:

sin(x)²

-----------------

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Example

Given:

x + 1

⎛

-----------

+ LN

⎞

-------------

f(x) =

x²– 1

⎝

⎠

x – 1

calculate a primitive of f.

Type:

X + 1

X – 1

⎛

⎝

⎛

⎝

⎞⎞

⎠⎠

---------------

------------

NTVX

+ LN

X + 1

Or, if you have stored f(x) in F, that is, if you have already

typed:

X + 1

X – 1

⎛

⎝

⎛

⎝

⎞

⎠

⎞

--------------

------------

TORE

+ LN

X²– 1

⎠

then type:

INTVX(F)

Or, if you have used DEFto define f(x), that is, if you have

already typed:

X + 1

X – 1

⎛

⎝

⎞⎞

⎠⎠

--------------

------------

DEF(F(X) =

+ LN

X²– 1

then type:

INTVX(F(X))

The result in all cases is equivalent to:

X + 1

X – 1

⎛

⎝

⎞

⎠

------------

⋅ LN( X – 1 ) + ⋅ LN( X + 1

X ⋅ LN

You will obtain absolute values only in Rigorous mode.

(See “CAS modes” on page 14-5 for instructions on

setting and changing modes.)

Example

Calculate:

-----------------------------------

∫

x⁶+ 2 ⋅ x⁴+ x²

Typing:

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⎛

⎝

⎞

⎠

--------------------------------------

NTVX

X + 2 ⋅ X + X

gives a primitive:

-- --------------

–

–3 ⋅ atan(x) –

x²+ 1

--------------------------------------

dX which gives the

N o t e

You can also type

∫

₁X⁶+ 2 ⋅ X⁴+ X²

primitive which is zero for x = 1

3 ⋅ π + 10

⎛

⎞

⎠

-- -------------- ----------------------

–3 ⋅ atan(x) –

–

x²+ 1

⎝

Example

Calculate:

--------------------------------------------

∫

sin(x) + sin(2 ⋅ x)

Typing:

⎛

⎝

⎞

⎠

---------------------------------------------------

NTVX

SIN(X) + SIN(2 ⋅ X)

gives the result:

⋅ LN( cos(X) – 1 ) + ⋅ LN( cos(X) + 1 ) +

–2

-----

⋅ LN( 2cos(X) + 1 )

N O T E :

If the argument to INTVX is the AND of two elements,

INTVX concerns itself only with the second element of the

AND, and adds the result to the first argument.

lim

Calculate limits

LIMIT or lim has two arguments: an expression dependent

on a variable, and an equality (a variable = the value to

which you want to calculate the limit).

You can omit the name of the variable and the sign =,

when this name is in VX).

It is often preferable to use a quoted expression:

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QUOTE(expression), to avoid rewriting the expression in

normal form (i.e., not to have a rational simplification of

the arguments) during the execution of the LIMIT

command.

Example

Typing:

⎛

⎝

⎞⎞

⎠⎠

------------

lim(QUOTE((2X – 1) ⋅ EXP

,X = + ∞)

X – 1

gives:

+∞

To find a right limit, for example, type:

⎛

⎝

⎞

------------

lim

, QUOTE(1 + 0)

⎠

X – 1

gives (if X is the current variable):

+∞

To find a left limit, for example, type:

⎛

⎝

⎞

------------

lim

, QUOTE(1 – 0)

⎠

X – 1

gives (if X is the current variable):

–∞

It is not necessary to quote the second argument when it

is written with =, for example:

⎛

⎝

⎞

------------

lim

, (X = 1 + 0)

⎠

X – 1

gives:

+∞

Example

For n > 2 in the following expression, find the limit as x

approaches 0:

n ⋅ tan(x) – tan(n ⋅ x)

----------------------------------------------------

sin(n ⋅ x) – n ⋅ sin(x)

You can use the LIMITcommand to do this.

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Typing:

N ⋅ TAN(X) – TAN(N ⋅ X)

SIN(N ⋅ X) – N ⋅ SIN(X)

⎛

⎝

⎞

----------------------------------------------------------------

lim

, 0

⎠

gives:

–

NOTE: To find the limit as x approaches a (resp a ), the

second argument is written:

X=A+0(resp X=A-0)

For the following expression, find the limit as x

approaches +∞:

x + x + x –

Typing:

⎛

⎞

lim X + X + X – X, + ∞

⎝

⎠

produces (after a short wait):

NOTE: the symbol ∞ is obtained by typing SHIFT 0.

To obtain –∞:

(–)∞

To obtain +∞:

(–)(–)∞

You can also find the symbol ∞ in the MATHkey’s

Constantmenu.

PREVAL

Evaluate a primitive

PREVAL has three parameters: an expression F(VX)

dependent on the variable contained in VX, and two

expressions A and B.

For example, if VX contains X, and if F is a function,

PREVAL (F(X),A,B)returns F(B)-F(A).

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PREVAL is used for calculating an integral defined from a

primitive: it evaluates this primitive between the two limits

of the integral.

Typing:

PREVAL(X +X,2,3)

gives:

RISCH

Primitive and defined integral

RISCH has two parameters: an expression and the name

of a variable.

RISCH returns a primitive of the first parameter with

respect to the variable specified in the second parameter.

Typing:

RISCH((2·X +1)·EXP(X +1),X)

gives:

X·EXP(X +1)

N O T E :

If the RISCH parameter is the AND of two elements,

RISCH concerns itself only with the second element of the

AND, and adds the result to the first argument.

SERIES

Limited nth-order expansion

SERIES has three arguments: an expression dependent on

a variable, an equality (the variable x = the value a to

which you want to calculate the expansion) and a whole

number (the order n of the limited expansion).

You can omit the name of the variable and the = sign

when this name is in VX).

SERIES returns the limited nth-order expansion of the

expression in the vicinity of x = a.

•

Example — Expansion in the vicinity of x=a

Give a limited 4th-order expansion of cos(2 · x) in the

vicinity of x =

For this you use the SERIEScommand.

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Typing:

⎛

⎞

SERIES COS(2 ⋅ X) , X = , 4

⎝

⎠

gives:

8 3

---------

h⁵

⎛

⎝

⎞

⎠

h – h + 0

-----

⟨ – 3h + 2h +

⟩

h = X –

•

Example — Expansion in the vicinity of x=+∞

or x=–∞

Example 1

Give a 5th-order expansion of arctan(x) in the vicinity of

x=+∞, taking as infinitely small h =

Typing:

SERIES(ATAN(X),X =+∞,5)

gives:

h³h⁵

---- ----

π ⋅ h⁶

⎛

⎝

⎛

⎝

⎞⎞

⎠⎠

------------

– h +

–

+ 0

h =

Example 2

-----------

Give a 2nd-order expansion of (2x – 1)e^{x – 1}in the

vicinity of x=+∞, taking as infinitely small h =

⎛

⎝

⎞

------------

SERIES((2X – 1) ⋅ EXP

, X = + ∞, 3)

⎠

X – 1

gives:

12 + 6h + 12h²+ 17h³

------------------------------------------------------

+ 0(2 ⋅ h )

h =

6 ⋅ h

•

Unidirectional expansion

To perform an expansion in the vicinity of x = a where

x > a, use a positive real (such as 4.0) for the order.

To perform an expansion in the vicinity of x = a where

x < a, use a negative real (such as –4.0) for the order.

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You must be in Rigorous (not Sloppy) mode to apply

SERIES with unidirectional expansion. (See “CAS modes”

on page 14-5 for instructions on setting and changing

modes.

Example 1

Give a 3rd-order expansion of x²+ x³in the vicinity of

x = 0 .

Typing:

SERIES( X + X ,X= 0, 3.0)

gives:

–1

-----

⋅ h + ⋅ h + h + 0(h ) (h = x)

⋅ h +

Example 2

Give a 3rd-order expansion of x²+ x³in the vicinity of

–

x = 0 .

Typing:

SERIES( X + X ,X= 0, –3.0)

gives:

–1

-----

–1

-----

⋅ h +

⋅ h + h + 0(h ) (h = –x)

Note that h = –x is positive as x → 0^–.

Example 3

If you enter the order as an integer rather than a real, as

in:

SERIES( X + X ,X= 0, 3)

you will get the following error:

SERIES Error: Unable to find sign.

Note that if you had been in Sloppy rather than Rigorous

mode, all three examples above would have returned the

same answer as you got when exploring in the vicinity of

x = 0 :

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–1

-----

⋅ h + ⋅ h + h + 0(h ) (h = x)

⋅ h +

TABVAR

Variation table

TABVAR has as a parameter an expression with a

rational derivative.

TABVAR returns the variation table for the expression in

terms of the current variable.

Typing:

TABVAR(3X²-8X-11)

gives, in step-by-step mode:

F = (3 ⋅ x²– 8 ⋅ x – 11)

F' = (3 ⋅ 2 ⋅ x – 8)

→ (2 ⋅ (3 ⋅ x–4))

Variation table:

–∞

–

+∞

–49

---------

↓

↑

The arrows indicate whether the function is increasing or

decreasing during the specified interval. This particular

variation table indicates that the function F(x) decreases

–49

---------

for x in the interval [–∞, ₃], reaching a minimum of

at x = . It then increases in the interval [₃, +∞], reaching

a max³imum of +∞.

Note that “?” appearing in the variation table indicates

that the function is not defined in the corresponding

interval.

TAYLOR0

Limited expansion in the vicinity of 0

TAYLOR0 has a single argument: the function of x to

expand. It returns the function’s limited 4th-relative-order

expansion in the vicinity of x=0 (if x is the current

variable).

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Typing:

TAN(P⋅ X) – SIN(P⋅ X)

TAN(Q⋅ X) – SIN(Q⋅ X)

⎛

⎝

⎞

⎠

----------------------------------

TAYLOR0

gives:

P³P⁵– Q²⋅ P³

------ -----------------------------

⋅ x

Q³

4 ⋅ Q³

N o t e

‘th-order’ means that the numerator and the denominator

are expanded to the 4th relative order (here, the 5th

absolute order for the numerator, and for the

denominator, which is given in the end, the 2nd order (5−

3), seeing that the exponent of the denominator is 3).

TRUNC

Truncate at order n - 1

TRUNC enables you to truncate a polynomial at a given

order (used to perform limited expansions).

TRUNC has two arguments: a polynomial and X .

TRUNC returns the polynomial truncated at order n−1;

that is, the returned polynomial has no terms with

exponents ≥n.

Typing:

⎠

⎛⎛

⎞

TRUNC 1+X+-⋅ X ,X

⎝⎝

gives:

4x + x + 3x + 1

REWRI menu

The REWRI menu contains functions that enable you to

rewrite an expression in another form.

DISTRIB

Distributivity of multiplication

DISTRIB enables you to apply the distributivity of

multiplication in respect to addition in a single instance.

DISTRIB enables you, when you apply it several times, to

carry out the distributivity step by step.

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Typing:

DISTRIB((X+1)·(X+2)·(X+3))

gives:

x ⋅ (x + 2) ⋅ (x + 3) + 1 ⋅ (x + 2) ⋅ (x + 3)

EPSX0

Disregard small values

EPSX0 has as a parameter an expression in X, and returns

the same expression with the values less than EPS

replaced by zeroes.

Typing:

EPSX0(0.001 + X)

gives, if EPS=0.01:

0 + x

or, if EPS=0.0001:

.001 + x

EXPLN

Transform a trigonometric expression into complex

exponentials

EXPLN takes as an argument a trigonometric expression.

It transforms the trigonometric function into exponentials

and logarithms without linearizing it.

EXPLN puts the calculator into complex mode.

Typing:

EXPLN(SIN(X))

gives:

----------------------

exp(i ⋅ x) –

exp(i ⋅ x)

----------------------------------------------------

2 ⋅ i

EXP2POW

Transform exp(n∗ln(x)) as a power of x

EXP2POW transforms an expression of the form

exp(n × ln(x)), rewriting it as a power of x.

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Typing:

EXP2POW(EXP(N · LN(X)))

gives:

xⁿ

FDISTRIB

Distributivity

FDISTRIB has an expression as argument.

FDISTRIB enables you to apply the distributivity of

multiplication with respect to addition all at once.

Typing:

FDISTRIB((X+1)·(X+2)·(X+3))

gives:

x·x·x + 3·x·x + x·2·x + 3·2·x + x·x·1 + 3·x·1 + x·2·1

+ 3·2·1

After simplification (by pressing ENTER):

x + 6·x + 11·x + 6

LIN

Linearize the exponentials

LIN has as an argument an expression containing

exponentials and trigonometric functions. LIN does not

linearize trigonometric expressions (as does TLIN) but

converts a trigonometric expression to exponentials and

then linearizes the complex exponentials.

LIN puts the calculator into complex mode when dealing

with trigonometric functions.

Example 1

Typing:

LIN((EXP(X)+1) )

gives:

3·exp(x) + 1 + 3·exp(2·x) + exp(3·x)

Example 2

Typing:

LIN(COS(X) )

gives:

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-- --

⋅ exp(2 ⋅ i ⋅ x)

⋅ exp(–(2 ⋅ i ⋅ x)) +

Example 3

Typing:

LIN(SIN(X))

gives:

⋅ expi ⋅ x + ⋅ exp(–(i ⋅ x))

–

LNCOLLECT

Regroup the logarithms

LNCOLLECT has as an argument an expression

containing logarithms.

LNCOLLECT regroups the terms in the logarithms. It is

therefore preferable to use an expression that has already

been factored (using FACTOR).

Typing:

LNCOLLECT(LN(X+1)+LN(X-1))

gives:

ln((x+1)(x−1))

POWEXPAND

Transform a power

POWEXPAND writes a power in the form of a product.

Typing:

POWEXPAND((X+1) )

gives:

(x+1) · (x+1) · (x+1)

This allows you to do the development of (x + 1) in step

by step, using DISTRIBseveral times on the preceding

result.

SINCOS

Transform the complex exponentials into sin and cos

SINCOS takes as an argument an expression containing

complex exponentials.

SINCOS then rewrites this expression in terms of sin(x)

and cos(x).

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Typing:

SINCOS(EXP(i·X))

gives after turning on complex mode, if necessary:

cos(x) + i · sin(x)

SIMPLIFY

Simplify

SIMPLIFY simplifies an expression automatically.

Typing:

SIN(3⋅ X) +SIN(7⋅ X)

⎛

⎝

⎞

⎠

----------------------------------

SIMPLIFY

SIN(5⋅ X)

gives, after simplification:

4 · cos(x) − 2

XNUM

Evaluation of reals

XNUM has an expression as a parameter.

XNUM puts the calculator into approximate mode and

returns the numeric value of the expression.

Typing:

XNUM(√2)

gives:

1.41421356237

Rational approximation

XQ has a real numeric expression as a parameter.

XQ puts the calculator into exact mode and gives a

rational or real approximation of the expression.

Typing:

XQ(1.41421)

gives:

66441

--------------

46981

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Typing:

XQ(1.414213562)

gives:

√2

SOLV menu

The SOLV menu contains functions that enable you to

solve equations, linear systems, and differential

equations.

DESOLVE

Solve differential equations

DESOLVE enables you to solve differential equations. (For

linear differential equations having constant coefficients,

it is better to use LDEC.)

DESOLVE has two arguments:

1. the differential equation where y' is written as d1Y(X)

(or the differential equation and the initial conditions

separated by AND),

2. the unknown Y(X).

The mode must be set to real.

Example 1

Solve:

y” + y = cos(x)

y(0)=c y’(0) =c

Typing:

DESOLVE(d1d1Y(X)+Y(X) = COS(X),Y(X))

gives:

x + 2 ⋅ cC1

--------------------------

Y(X) = cC0 ⋅ cos(x) +

⋅ sin(x)

cC0 and cC1 are integration constants (y(0) = cC0 y’(0)

= cC1).

You can then assign values to the constants using the

SUBSTcommand.

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To produce the solutions for y(0) = 1, type:

SUBST(Y(X) =

X+2⋅ cC1

cC0⋅ COS(X) +----------------⋅ SIN(X),cC0 = 1)

which gives:

2 ⋅ cos(x) + (x + 2 ⋅ cC1) ⋅ sin(x)

---------------------------------------------------------------------------------

y(x) =

Example 2

Solve:

y” + y = cos(x)

y(0) = 1 y’(0) = 1

It is possible to solve for the constants from the outset.

Typing:

DESOLVE((d1d1Y(X)+Y(X)=COS(X))

AND (Y(0)=1) AND (d1Y(0)=1),Y(X))

gives:

2 + x

-----------

Y(x) = cosx +

⋅ sin(x)

ISOLATE

The zeros of an expression

ISOLATE returns the values that are the zeros of an

expression or an equation.

ISOLATE has two parameters: an expression or equation,

and the name of the variable to isolate (ignoring

REALASSUME).

Typing:

ISOLATE(X -1=3,X)

gives in real mode:

(x = √2) OR (x = −√2)

and in complex mode:

(x = √2 · i) OR (x = −√2) OR

(x = −(√2 · i)) OR (x = √2)

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LDEC

Linear differential equations having constant

coefficients

LDEC enables you to directly solve linear differential

equations having constant coefficients.

The parameters are the second member and the

characteristic equation.

Solve:

3·x

y” − 6 · y’ + 9 · y = x · e

Typing:

LDEC(X·EXP(3·X),X −6·X+9)

gives:

(18 ⋅ x – 6) ⋅ cC0 – (6 ⋅ x ⋅ cC1 + x³)

⎛

⎝

⎞

----------------------------------------------------------------------------------------

⋅ exp(3 ⋅ x)

⎠

cC0 and cC1 are integration constants (y(0) = cC0 and

y’(0) = cC1).

LINSOLVE

Solve linear system

LINSOLVE enables you to solve a system of linear

equations.

It is assumed that the various equations are of the form

expression = 0.

LINSOLVE has two arguments: the first members of the

various equations separated by AND, and the names of

the various variables separated by AND.

Example 1

Typing:

LINSOLVE(X+Y+3 AND X-Y+1, X AND Y)

gives:

(x = −2) AND (y = −1)

or, in Step-by-step mode (CFG, etc.):

L2=L2−L1

1 1 3

1 –1 1

ENTER

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L1=2L1+L2

1 1 3

0 –2 –2

ENTER

Reduction Result

2 0 4

0 –2 –2

then press ENTER. The following is then written to the

Equation Writer:

(x = −2) AND (y = −1)

Example 2

Type:

(2·X+Y+Z=1)AND(X+Y+2·Z=1)AND(X+2·Y+Z=4)

Then, invoke LINSOLVEand type the unknowns:

X AND Y AND Z

and press the ENTER key.

The following result is produced if you are in Step-by-step

mode (CFG, etc.):

L2=2L2−L1

2 1 1 –1

1 1 2 –1

1 2 1 –4

ENTER

L3=2L3−L1

2 1 1 –1

0 1 3 –1

1 2 1 –4

and so on until, finally:

Reduction Result

8 0 0

0 8 0 –20

0 0 –8 –4

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then press ENTER. The following is then written to the

Equation Writer:

⎛

⎝

⎞

⎛

⎞

⎠

⎛

⎞

⎠

x = – AND y =

AND z = –

⎠

⎝

SOLVE

Solve equations

SOLVE has as two parameters:

(1) either an equality between two expressions, or a

single expression (in which case = 0 is implied), and

(2) the name of a variable.

SOLVE solves the equation in R in real mode and in C in

complex mode (ignoring REALASSUME).

Typing:

SOLVE(X -1=3,X)

gives, in real mode:

(x = −√2) OR (x = √2)

or, in complex mode:

(x = −√2) OR (x = √2) OR (x = −i · √2) OR (x = i√2)

Solve systems

SOLVE also enables you to solve a system of non-linear

equations, if they are polynomials. (If they are not

polynomials, use MSOLV in the HOME screen to get a

numerical solution.)

It is assumed that the various equations are of the form

expression = 0.

SOLVE has as arguments, the first members of the various

equations separated by AND, and the names of the

various variables separated by AND.

Typing:

2 2

SOLVE(X +Y -3 AND X-Y +1,X AND Y)

gives:

(x = 1) AND (y = −√2) OR (x = 1) AND (y = √2)

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SOLVEVX

Solve equations

SOLVEVX has as a parameter either:

(1) an equality between two expressions in the variable

contained in VX, or

(2) a single such expression (in which case = 0 is

implied).

SOLVEVX solves the equation.

Example 1

Typing:

SOLVEVX(X -1=3)

gives, in real mode:

(x = −√2) OR (x = √2)

or, in complex mode, even if you have chosen X as real:

(x = −√2) OR (x = √2) OR (x = −i · √2) OR (x = i√2)

Example 2

Typing:

SOLVEVX(2X²+X)

gives, in real mode:

(x = −1/2) OR (x = 0)

TRIG menu

The TRIG menu contains functions that enable you to

transform trigonometric expressions.

ACOS2S

Transform the arccos into arcsin

ACOS2S has as a trigonometric expression as an

argument.

ACOS2S transforms the expression by replacing

arccos(x) with

− arcsin(x).

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Typing:

ACOS2S(ACOS(X) + ASIN(X))

gives, when simplified:

ASIN2C

Transform the arcsin into arccos

ASIN2C has as a trigonometric expression as an

argument.

ASIN2C transforms the expression by replacing arcsin(x)

with ----- − arccos(x).

Typing:

ASIN2C(ACOS(X) + ASIN(X))

gives, when simplified:

-----

ASIN2T

Transform the arccos into arctan

ASIN2T has a trigonometric expression as an argument.

ASIN2T transforms the expression by replacing arcsin(x)

⎛

⎜

⎝

⎞

⎟

⎠

-----------------

with arctan

1 – x²

Typing:

ASIN2T(ASIN(X))

gives:

⎛

⎜

⎝

⎞

⎟

⎠

-----------------

atan

1 – x²

ATAN2S

Transform the arctan into arcsin

ATAN2S has a trigonometric expression as an argument.

ATAN2S transforms the expression by replacing

⎛

⎜

⎝

⎞

⎟

⎠

------------------

arctan(x) with arcsin

1 + x²

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Typing:

ATAN2S(ATAN(X))

gives:

⎛

⎜

⎝

⎞

⎟

⎠

------------------

asin

x + 1

HALFTAN

Transform in terms of tan(x/2)

HALFTAN has a trigonometric expression as an

argument.

HALFTAN transforms sin(x), cos(x) and tan(x) in the

expression, rewriting them in terms of tan(x/2).

Typing:

HALFTAN(SIN(X) + COS(X) )

gives (SQ(X) = X ):

⎛ ⎞

⎛

⎛ ⎞⎞

⎛

⎜

⎝

⎞

⎟

⎠

⎛

⎜

⎝

⎞

⎟

⎠

2 ⋅ tan

1 – SQ tan

⎝ ⎠

⎝

⎝ ⎠⎠

---------------------------------------

⎛

SQ tan

⎝

⎛ ⎞⎞

⎛

SQ tan

⎝

⎛ ⎞⎞

+ 1

⎝ ⎠⎠

or, after simplification:

SINCOS

Transform the complex exponentials into sin and cos

SINCOS takes an expression containing complex

exponentials as an argument.

SINCOS then rewrites this expression in terms of sin(x)

and cos(x).

Typing:

SINCOS(EXP(i · X))

gives after turning on complex mode, if necessary:

cos(x) + i · sin(x)

TAN2CS2

Transform tan(x) with sin(2x) and cos(2x)

TAN2CS2 has a trigonometric expression as an

argument.

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TAN2CS2 transforms this expression by replacing tan(x)

1 – cos(2 ⋅ x)

sin(2 ⋅ x)

--------------------------------

with

Typing:

TAN2CS2(TAN(X))

gives:

1 – cos(2 ⋅ x)

--------------------------------

sin(2 ⋅ x)

TAN2SC

Replace tan(x) with sin(x)/cos(x)

TAN2SC has a trigonometric expression as an argument.

TAN2SC transforms this expression by replacing tan(x)

sin(x)

---------------

with

cos(x)

Typing:

TAN2SC(TAN(X))

gives:

sin(x)

---------------

cos(x)

TAN2SC2

Transform tan(x) with sin(2x) and cos(2x)

TAN2SC2 has a trigonometric expression as an

argument.

TAN2SC2 transforms this expression by replacing tan(x)

sin(2 ⋅ x)

--------------------------------

with

1 + cos(2 ⋅ x)

Typing:

TAN2SC2(TAN(X))

gives:

sin(2 ⋅ x)

--------------------------------

1 + cos(2 ⋅ x)

TCOLLECT

Reconstruct the sine and the cosine of the same angle

TCOLLECT has a trigonometric expression as an

argument.

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TCOLLECT linearizes this expression in terms of sin(n x)

and cos(n x), then (in Real mode) reconstructs the sine and

cosine of the same angle.

Typing:

TCOLLECT(SIN(X) + COS(X))

gives:

⎛

2 ⋅ cos x –

⎝

⎞

⎠

TEXPAND

Develop transcendental expressions

TEXPAND has as an argument a transcendental

expression (that is, an expression with trigonometric,

exponential or logarithmic functions). TEXPAND develops

this expression in terms of sin(x), cos(x), exp(x) or ln(x).

Example 1

Typing:

TEXPAND(EXP(X+Y))

gives:

exp(x)·exp(y)

Example 2

Typing:

TEXPAND(LN(X·Y))

gives:

ln(y) + ln(x)

Example 3

Typing:

TEXPAND(COS(X+Y))

gives:

cos(y)·cos(x)–sin(y)·sin(x)

Example 4

Typing:

TEXPAND(COS(3·X))

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gives:

4·cos(x)³–3·cos(x)

TLIN

Linearize a trigonometric expression

TLIN has as an argument a trigonometric expression.

TLIN linearizes this expression in terms of sin(n x) and

cos(n x).

Example 1

Typing:

TLIN(COS(X) · COS(Y))

gives:

⋅ cos(x – y) + ⋅ cos(x + y)

Example 2

Typing:

TLIN(COS(X) )

gives:

⋅ cos(3 ⋅ x) + ⋅ cos(x)

Example 3

Typing:

TLIN(4·COS(X) -2)

gives:

2 ⋅ cos(2 ⋅ x)

TRIG

Simplify using sin(x) + cos(x) = 1

TRIG has as an argument a trigonometric expression.

TRIG simplifies this expression using the identity

sin(x) + cos(x) = 1.

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Typing:

TRIG(SIN(X) + COS(X) + 1)

gives:

TRIGCOS

Simplify using the cosines

TRIGCOS has as an argument a trigonometric

expression.

TRIGCOS simplifies this expression, using the identity

sin(x) +cos(x) = 1 to rewrite it in terms of cosines.

Typing:

TRIGCOS(SIN(X) + COS(X) + 1)

gives:

cos(x)⁴– cos(x)²+ 2

TRIGSIN

Simplify using the sines

TRIGSIN has as an argument a trigonometric expression.

TRIGSIN simplifies this expression, using the identity

sin(x) + cos(x) = 1 to rewrite it in terms of sines.

Typing:

TRIGSIN(SIN(X) + COS(X) + 1)

gives:

sin(x)⁴– sin(x)²+ 2

TRIGTAN

Simplify using the tangents

TRIGTAN has as an argument a trigonometric expression.

TRIGTAN simplifies this expression, using the identity

sin(x) + cos(x) = 1 to rewrite it in terms of tangents.

Typing:

TRIGTAN(SIN(X) + COS(X) + 1)

gives:

2 ⋅ tan(x)⁴+ 3 ⋅ tan(x)²+ 2

------------------------------------------------------------------

tan(x)⁴+ 2 ⋅ tan(x)²+ 1

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CAS Functions on the MATH menu

When you are in the

Equation Writer and press

, a menu of

additional CAS functions

available to you is

displayed. Many of the

functions in this menu

match the functions available from the soft-key menus in

the Equation Writer; but there are other functions that are

only available from this menu. This section describes CAS

functions that are available when you press

Equation Writer (grouped by main menu name).

in the

Algebra menu

Complex menu

All the functions on this menu are also available on the

menu in the Equation Writer. See “ALGB menu”

on page 14-10 for a description of these functions.

Inserts i (= –1 ).

ABS

Determines the absolute value of the argument.

Example

Typing ABS(7 + 4i) yields 65 , as does ABS(7 – 4i).

ARG

See “ARG” on page 13-7.

See “CONJ” on page 13-7.

CONJ

DROITE

DROITE returns the equation of the line through the

Cartesian points, z₁, z₂. It takes two complex numbers, z₁

and z₂, as arguments.

Example

Typing:

DROITE((1, 2), (0, 1))

or:

DROITE(1 + 2·i, i)

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returns:

Y = X –1 + 2

Pressing

Y = X + 1

simplifies this to:

See “IM” on page 13-7.

–

Specifies the negation of the argument.

See “RE” on page 13-8.

SIGN

Determines the quotient of the argument divided by its

modulus.

Example

7 + 4i

-------------

Typing SIGN(7 + 4i) or SIGN(7,4) yields

Constant menu

e, i, π

See “Constants” on page 13-8.

Enters the sign for infinity.

∞

Diff & Int menu

All the functions on this menu are also available on the

menu in the Equation Writer. See “DIFF menu” on

page 14-16 for a description of these functions.

Hyperb menu

Integer menu

All the functions on this menu are described in

“Hyperbolic trigonometry” on page 13-9.

Note that many integer functions also work with Gaussian

integers (a + bi where a and b are integers).

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DIVIS

Gives the divisors of an integer.

Example

Typing:

DIVIS(12)

gives:

12 OR 6 OR 3 OR 4 OR 2 OR 1

Note: DIVIS(0) returns 0 OR 1.

EULER

Returns the Euler index of a whole number. The Euler

index of n is the number of whole numbers less than n that

are prime with n.

Example

Typing:

EULER(21)

gives:

Explanation: {2,4,5,7,8,10,11,13,15,16,17,19} is

the set of whole numbers less than 21 and prime with 21.

There are 12 members of the set, so the Euler index is12.

FACTOR

Decomposes an integer into its prime factors.

Example

Typing:

FACTOR(90)

gives:

2·3²·5

GCD

Returns the greatest common divisor of two integers.

Example

Typing:

GCD(18, 15)

gives:

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In step-by-step mode, there are a number of intermediate

results:

18 mod 15 = 3

15 mod 3 = 0

Result: 3

Pressing

Equation Writer.

then causes 3 to be written to the

Note that the last non-zero remainder in the sequence of

remainders shown in the intermediate steps is the GCD.

IDIV2

Returns the quotient and the remainder of the Euclidean

division between two integers.

Example

Typing:

IDIV2(148, 5)

gives:

29 AND 3

In step-by-step mode, the

calculator shows the

division process in

longhand.

IEGCD

Returns the value of Bézout’s Identity for two integers. For

example, IEGCD(A,B) returns U AND V = D, with U, V, D

such that AU+BV=D and D=GCD(A,B).

Example

Typing:

IEGCD(48, 30)

gives

2 AND –3 = 6

In other words: 2·48 + (–3)·30 = 6 and GCD(48,30) = 6.

In step-by-step mode, we get:

[z,u,v]:z=u*48+v*30

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[48,1,0]

[30,0,1]*–1

[18,1,–1]*–1

[12,–1,2]*–1

[6,2,–3]*–2

Result: [6,2,–3]

Pressing

then causes 2 AND –3 = 6 to be

written to the Equation Writer.

The intermediate steps shown are the combination of

lines. For example, to get line L(n + 2), take L(n) – q*L(n

+ 1) where q is the Euclidean quotient of the integers at

the beginning of the vector, these integers being the

sequence of remainders).

IQUOT

Returns the integer quotient of the Euclidean division of

two integers.

Example

Typing:

IQUOT(148, 5)

gives:

In step-by-step mode, the

division is carried out as if

in longhand

Pressing

then causes 29 to be

written to the Equation

Writer.

IREMAINDER

Returns the integer remainder from the Euclidean division

of two integers.

Example 1

Typing:

IREMAINDER(148, 5)

gives:

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IREMAINDER works with integers and with Gaussian

integers. This is what distinguishes it from MOD.

Example 2

Typing:

IREMAINDER(2 + 3·i, 1 + i)

gives:

ISPRIME?

Returns a value indicating whether an integer is a prime

number. ISPRIME?(n) returns 1 (TRUE) if n is a prime or

pseudo-prime, and 0 (FALSE) if n is not prime.

Definition: For numbers less than 10¹⁴, pseudo-prime

and prime mean the same thing. For numbers greater

than 10¹⁴, a pseudo-prime is a number with a large

probability of being prime.

Example 1

Typing:

ISPRIME?(13)

gives:

Example 2

Typing:

ISPRIME?(14)

gives:

LCM

Returns the least common multiple of two integers.

Example

Typing:

LCM(18, 15)

gives:

MOD

See “MOD” on page 13-15.

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NEXTPRIME

NEXTPRIME(n) returns the smallest prime or pseudo-prime

greater than n.

Example

Typing:

NEXTPRIME(75)

gives:

PREVPRIME

PREVPRIME(n) returns the greatest prime or pseudo-prime

less than n.

Example

Typing:

PREVPRIME(75)

gives:

Modular menu

All the examples of this section assume that p =13; that

is, you have entered MODSTO(13) or

STORE(13,MODULO), or have specified 13 for Modulo

in CAS MODESscreen (as explained on page 15-16).

ADDTMOD

Performs an addition in Z/pZ.

Example 1

Typing:

ADDTMOD(2, 18)

gives:

–6

ADDTMOD can also perform addition in Z/pZ[X].

Example 2

Typing:

ADDTMOD(11X + 5, 8X + 6)

gives:

6x – 2

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DIVMOD

Division in Z/pZ or Z/pZ[X].

Example 1

In Z/pZ, the arguments are two integers: A and B. When

B has an inverse in Z/pZ, the result is A/B simplified as

Z/pZ.

Typing:

DIVMOD(5, 3)

gives:

Example 2

In Z/pZ[X], the arguments are two polynomials: A[X] and

B[X]. The result is a rational fraction A[X]/B[X] simplified

as Z/pZ[X].

Typing:

DIVMOD(2X²+ 5, 5X²+ 2X –3)

gives:

4x + 5

3x + 3

--------------

–

EXPANDMOD

Expand and simplify expressions in Z/pZ or Z/pZ[X].

Example 1

In Z/pZ, the argument is an integer expression.

Typing:

EXPANDMOD(2 · 3 + 5 · 4)

gives:

Example 2

In Z/pZ[X], the argument is a polynomial.

Typing:

EXPANDMOD((2X²+ 12)·(5X – 4))

gives:

–(3 ⋅ x³– 5 ⋅ x²+ 5 ⋅ x – 4)

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FACTORMOD

Factors a polynomial in Z/pZ[X], providing that p ≤ 97,

p is prime and the order of the multiple factors is less than

the modulo.

Example

Typing:

FACTORMOD(–(3X³– 5X²+ 5X – 4))

gives:

–((3x – 5) ⋅ (x²+ 6))

GCDMOD

Calculates the GCD of the two polynomials in Z/pZ[X].

Example

Typing:

GCDMOD(2X²+ 5, 5X²+ 2X – 3)

gives:

–(6x – 1)

INVMOD

Calculates the inverse of an integer in Z/pZ.

Example

Typing:

INVMOD(5)

gives:

–5

since 5 · –5 = –25 = 1 (mod 13).

MODSTO

Sets the value of the MODULO variable p.

Example

Typing:

MODSTO(11)

sets the value of p to 11.

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MULTMOD

Performs a multiplication in Z/pZ or in Z/pZ[X].

Example 1

Typing:

MULTMOD(11, 8)

gives:

–3

Example 2

Typing:

MULTMOD(11X + 5, 8X + 6)

gives:

–(3x²– 2x – 4)

POWMOD

Calculates A to the power of N in Z/pZ[X], and A(X) to

the power of N in Z/pZ[X].

Example 1

If p = 13, typing:

POWMOD(11, 195)

gives:

In effect: 11¹²= 1 mod 13, so 11¹⁹⁵= 11^16×12+3= 5

mod 13.

Example 2

Typing:

POWMOD(2X + 1, 5)

gives:

6x⁵+ 2x⁴+ 2x³+ x²– 3x + 1

since 32 = 6 (mod 13), 80 = 2 (mod 13), 40 = 1 (mod

13), 10 = –3 (mod 13).

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SUBTMOD

Performs a subtraction in Z/pZ or Z/pZ[X].

Example 1

Typing:

SUBTMOD(29, 8)

gives:

–5

Example 2

Typing:

SUBTMOD(11X + 5, 8X + 6)

gives:

3x – 1

Polynomial menu

EGCD

Returns Bézout’s Identity, the Extended Greatest Common

Divisor (EGCD).

EGCD(A(X), B(X)) returns U(X) AND V(X) = D(X), with D,

U, V such that D(X) = U(X)·A(X) + V(X)·B(X).

Example 1

Typing:

EGCD(X²+ 2 · X + 1, X²– 1)

gives:

–1 AND –1 = 2x + 2

Example 2

Typing:

EGCD(X²+ 2 · X + 1, X³+ 1)

gives:

–(x – 2) AND 1 = 3x + 3

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FACTOR

Factors a polynomial.

Example 1

Typing:

FACTOR(X²– 2)

gives:

(x + 2) ⋅ (x – 2)

Example 2

Typing:

FACTOR(X²+ 2·X + 1)

gives:

(x + 1)²

GCD

Returns the GCD (Greatest Common Divisor) of two

polynomials.

Example

Typing:

GCD(X²+ 2·X + 1, X²– 1)

gives:

x + 1

HERMITE

Returns the Hermite polynomial of degree n (where n is a

whole number). This is a polynomial of the following type:

x²

²--^d------

x²

----

–

H_n(x) = (–1)ⁿ⋅ e

dxⁿ

Example

Typing:

HERMITE(6)

gives:

64x⁶– 480x⁴+ 720x²– 120

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LCM

Returns the LCM (Least Common Multiple) of two

polynomials.

Example

Typing:

LCM(X²+ 2·X + 1, X²– 1)

gives:

(x²+ 2x + 1) ⋅ (x – 1)

LEGENDRE

Returns the polynomial L_n, a non-null solution of the

differential equation:

(x²– 1) ⋅ y″ – 2 ⋅ x ⋅ y′ – n(n + 1) ⋅ y = 0

where n is a whole number.

Example

Typing:

LEGENDRE(4)

gives:

35 ⋅ x⁴– 30 ⋅ x²+ 3

----------------------------------------------

PARTFRAC

Returns the partial fraction decomposition of a rational

fraction.

Example

Typing:

X⁵– 2X³+ 1

⎛

-----------------------------------------------------------

ARTFRAC

⎜

⎝

X – 2X + 2X – 2X + 1

gives, in real and direct mode:

x – 3

2x²+ 2

–1

2x – 2

----------------- --------------

x + 2 +

and gives, in complex mode:

1 – 3 ⋅ i

–1

-----

1 + 3 ⋅ i

x – i

-----------------

----------------- ----------- ------------------

x + 2 +

x + i

x – 1

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PROPFRAC

PROPFRAC rewrites a rational fraction so as to bring out

its whole number part.

PROPFRAC(A(X)/ B(X)) writes the rational fraction A(X)/

B(X) in the form:

R(X)

B(X)

-----------

Q(X) +

where R”(X) = 0, or 0 ≤ deg (R(X) < deg (B(X).

Example

Typing:

(5X + 3) ⋅ (X – 1)

⎛

⎝

⎞

⎠

------------------------------------------

ROPFRAC

gives:

5x – 12 +

X + 2

-----------

x + 2

PTAYL

PTAYL rewrites a polynomial P(X) in order of its powers of

X – a.

Example

Typing:

PTAYL(X²+ 2·X + 1, 2)

produces the polynomial Q(X), namely:

x²+ 6x + 9

Note that P(X) = Q(X–2).

QUOT

QUOT returns the quotient of two polynomials, A(X) and

B(X), divided in decreasing order by exponent.

Example

Typing:

QUOT(X²+ 2·X + 1, X)

gives:

x + 2

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Note that in step-by-step mode, synthetic division is

shown, with each polynomial represented as the list of its

coefficients in descending order of power.

REMAINDER

Returns the remainder from the division of the two

polynomials, A(X) and B(X), divided in decreasing order

by exponent.

Example

Typing:

REMAINDER(X³– 1, X²– 1)

gives:

x – 1

Note that in step-by-step mode, synthetic division is

shown, with each polynomial represented as the list of its

coefficients in descending order of power.

TCHEBYCHEFF

For n > 0, TCHEBYCHEFF returns the polynomial T_nsuch

that:

Tn(x) = cos(n·arccos(x))

For n ≥ 0, we have:

[ ]

T_n(x) =

C²_n^k(x²– 1)^kx^{n – 2k}

∑

k = 0

For n ≥ 0 we also have:

′

″

(1 – x )T (x) – xT (x) + n T_n(x) = 0

For n ≥ 1, we have:

T_{n + 1}(x) = 2xT_n(x) – T_{n – 1}(x)

If n < 0, TCHEBYCHEFF returns the 2nd-species

Tchebycheff polynomial:

sin(n ⋅ arccos(x))

sin(arccos(x))

------------------------------------------

T_n(x) =

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Example 1

Typing:

TCHEBYCHEFF(4)

gives:

8x⁴– 8x²+ 1

Example 2

Typing:

TCHEBYCHEFF(–4)

gives:

8x³– 4x

Real menu

CEILING

FLOOR

FRAC

INT

See “CEILING” on page 13-14.

See “FLOOR” on page 13-14.

See “FRAC” on page 13-14.

See “INT” on page 13-15.

See “MAX” on page 13-15.

See “MIN” on page 13-15.

MAX

MIN

Rewrite menu

Solve menu

All the functions on this menu are also available on the

menu in the Equation Writer. See “REWRI menu”

on page 14-28 for a description of these functions.

All the functions on this menu are also available on the

menu in the Equation Writer. See “SOLV menu” on

page 14-33 for a description of these functions.

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Tests menu

ASSUME

Use this function to make a hypothesis about a specified

argument or variable.

Example

Typing:

ASSUME(X>Y)

sets an assumption that X is greater than Y. In fact, the

calculator works only with large not strict relations, and

thus ASSUME(X>Y) will actually set the assumption that X

≥ Y. (A message will indicate this when you enter an

ASSUME function.) Note that X ≥ Y will be stored in the

REALASSUME variable. To see the variable, press

, select REALASSUME and press

UNASSUME

Use this function to cancel all previously specified

assumptions about a particular argument or variable.

Example

Typing:

UNASSUME(X)

cancels any assumptions made about X. It returns X in the

Equation Writer. To see the assumptions, press

select REALASSUME and press

See “Test functions” on page 13-19.

See “AND” on page 13-19.

See “OR” on page 13-19.

>, ≥, <, ≤, ==, ≠

AND

NOT

IFTE

See “NOT” on page 13-19.

See “IFTE” on page 13-19.

Trig menu

All the functions on this menu are also available on the

menu in the Equation Writer. See “TRIG menu” on

page 14-38 for a description of these functions.

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CAS Functions on the CMDS menu

When you are in the

Equation Writer and press

, a menu of

the full set of CAS functions

available to you is

displayed. Many of the

functions in this menu

match the functions available from the soft-key menus in

the Equation Writer; but there are other functions that are

only available from this menu. This section describes the

additional CAS functions that are available when you

press

in the Equation Writer. (See the

previous section for other CAS commands.)

ABCUV

This command applies the Bézout identity like EGCD, but

the arguments are three polynomials A, B and C. (C must

be a multiple of GCD(A,B).)

ABCUV(A[X], B[X], C[X]) returns U[X] AND V[X], where U

and V satisfy:

C[X] = U[X] · A[X] + V[X] · B[X]

Example 1

Typing:

ABCUV(X²+ 2 · X + 1, X²– 1, X + 1)

gives:

AND –

CHINREM

Chinese Remainders: CHINREM has two sets of two

polynomials as arguments, each separated by AND.

CHINREM((A(X) AND R(X), B(X) AND Q(X)) returns an

AND with two polynomials as components: P(X) and S(X).

The polynomials P(X) and S(X) satisfy the following

relations when GCD(R(X),Q(X)) = 1:

S(X) = R(X) · Q(X),

P(X) = A(X) (modR(X)) and P(X) = B(X) (modQ(X)).

There is always a solution, P(X), if R(X) and Q(X) are

mutually primes and all solutions are congruent modulo

S(X) = R(X) · Q(X).

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Example

Find the solutions P(X) of:

P(X) = X (mod X²+ 1)

P(X) = X – 1 (mod X²– 1)

Typing:

CHINREM((X) AND (X²+ 1), (X – 1) AND (X²– 1))

gives:

x²– 2x + 1

x⁴– 1

--------------------------

-------------

–

AND

That is:

P[X] = –

x²– 2x + 1

x⁴– 1

⎛

⎝

⎞

⎠

--------------------------

-------------

mod–

CYCLOTOMIC

Returns the cyclotomic polynomial of order n. This is a

polynomial having the nth primitive roots of unity as

zeros.

CYCLOTOMIC has an integer n as its argument.

Example 1

When n = 4 the fourth roots of unity are {1, i, –1, –i}.

Among them, the primitive roots are: {i, –i}. Therefore, the

cyclotomic polynomial of order 4 is (X – i).(X + i) = X²+ 1.

Example 2

Typing:

CYCLOTOMIC(20)

gives:

x⁸– x⁶+ x⁴– x²+ 1

EXP2HYP

EXP2HYP has an expression enclosing exponentials as an

argument. It transforms that expression with the relation:

exp(a) = sinh(a) + cosh(a).

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Example 1

Typing:

EXP2HYP(EXP(A))

gives:

sinh(a) + cosh(a)

Example 2

Typing:

EXP2HYP(EXP(–A) + EXP(A))

gives:

2 · cosh(a)

GAMMA

Returns the values of the Γ function at a given point.

The Γ function is defined as:

Γ(x) = ^+∞e^–tt^{x – 1}dt

∫

We have:

Γ (1) = 1

Γ (x + 1) = x · Γ (x)

Example 1

Typing:

GAMMA(5)

gives:

Example 2

Typing:

GAMMA(1/2)

gives:

IABCUV

IABCUV(A,B,C) returns U AND V such that AU + BV = C

where A, B and C are whole numbers.

C must be a multiple of GCD(A,B) to obtain a solution.

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Example

Typing:

IABCUV(48, 30, 18)

gives:

6 AND –9

IBERNOULLI

Returns the nth Bernoulli’s number B(n) where:

+∞

B(n) ⁿ

-----------

------------

∑

e^t– 1

n = 0

Example

Typing:

IBERNOULLI(6)

gives:

-----------

ICHINREM

Chinese Remainders: ICHINREM(A AND P,B AND Q)

returns C AND R, where A, B, P and Q are whole

numbers.

The numbers X = C + k · R where k is an integer are such

that X = A mod P and X = B mod Q.

A solution X always exists when P and Q are mutually

prime, (GCD(P,Q) = 1) and in this case, all the solutions

are congruent modulo R = P · Q.

Example

Typing:

ICHINREM(7 AND 10, 12 AND 15)

gives:

–3 AND 30

ILAP

LAP is the Laplace transform of a given expression. The

expression is the value of a function of the variable stored

in VX.

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ILAP is the inverse Laplace transform of a given

expression. Again, the expression is the value of a

function of the variable stored in VX.

Laplace transform (LAP) and inverse Laplace transform

(ILAP) are useful in solving linear differential equations

with constant coefficients, for example:

y″ + p ⋅ y′ + q ⋅ y = f(x)

y(0) = a y′(0) = b

The following relations hold:

LAP(y)(x) = ^+∞e^{–x ⋅ t}y(t)dt

∫

2iπ

--------

ILAP(f)(x) =

⋅ e^zxf(z)dz

∫

where c is a closed contour enclosing the poles of f.

The following property is used:

LAP(y′)(x) = – y(0) + x ⋅ LAP(y)(x)

The solution, y, of:

y″ + p ⋅ y′ + q ⋅ y = f(x), y(0) = a, y′(0) = b

is then:

LAP(f(x)) + (x + p) ⋅ a + b

⎛

⎝

⎞

⎠

------------------------------------------------------------------

ILAP

x²+ px + q

Example

To solve:

y″–6 ⋅ y′ + 9 ⋅ y = x ⋅ e^3x, y(0) = a, y′(0) = b

type:

LAP(X · EXP(3 · X))

The result is:

--------------------------

x²– 6x + 9

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Typing:

----------------------------

⎛

⎜

⎝

+ (X–6) ⋅ a + b⎞

⎟

X – 6X + 9

------------------------------------------------------------------

ILAP

⎟

⎠

X²–6X + 9

gives:

x³

⎛

⎝

⎞

----

– (3a – b) ⋅ x + a ⋅ e

⎠

LAP

See ILAP above.

PA2B2

Decomposes a prime integer p congruent to 1 modulo 4,

as follows:

p = a²+ b².

The calculator gives the result as a + b · i.

Example 1

Typing:

PA2B2(17)

gives:

4 + i

that is, 17 = 4²+ 1²

Example 2

Typing:

PA2B2(29)

gives:

5 + 2 · i

that is, 29 = 5²+ 2²

PSI

Returns the value of the nth derivative of the Digamma

function at a.

The Digamma function is the derivative of ln(Γ(x)).

Example

Typing:

PSI(3, 1)

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gives:

-- --

⋅ π

–

Psi

Returns the value of the Digamma function at a.

The Digamma function is defined as the derivative of

ln(Γ(x)), so we have PSI(a,0) = Psi(a).

Example

Typing:

Psi(3)

and pressing

gives:

.922784335098

REORDER

Reorders the input expression following the order of

variables given in the second argument.

Example

Typing:

REORDER(X²+ 2 · X · A + A²+ Z²– X · Z, A AND X

AND Z)

gives:

A²+ 2 ⋅ X ⋅ A + X²– Z ⋅ X + Z²

SEVAL

SEVAL simplifies the given expression, operating on all

but the top-level operator of the expression.

Example

Typing:

SEVAL(SIN(3 · X -– X) + SIN(X + X))

gives:

sin(2 ⋅ x) + sin(2 ⋅ x)

SIGMA

Returns the discrete antiderivative of the input function,

that is, the function G, that satisfies the relation G(x + 1)

– G(x) = f(x). It has two arguments: the first is a function

f(x) of a variable x given as the second argument.

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Example

Typing:

SIGMA(X · X!, X)

gives:

because (X + 1)! – X! = X · X!.

SIGMAVX

Returns the discrete antiderivative of the input function,

that is a function, G, that satisfies the relation: G(x + 1) –

G(x) = f(x). SIGMAVX has as its argument a function f of

the current variable VX.

Example

Typing:

SIGMAVX(X²)

gives:

2x³– 3x²+ x

-------------------------------

because:

2(x + 1)³– 3(x + 1)²+ x + 1 – 2x³+ 3x²– x = 6x²

STURMAB

Returns the number of zeros of P in [a, b[ where P is a

polynomial and a and b are numbers.

Example 1

Typing:

STURMAB(X²· (X³+ 2), –2, 0)

gives:

Example 2

Typing:

STURMAB(X²· (X³+ 2), –2, 1)

gives:

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TSIMP

Simplifies a given expression by rewriting it as a function

of complex exponentials, and then reducing the number

of variables (enabling complex mode in the process).

Example

Typing:

SIN(3X) + SIN(7X)

⎛

⎝

⎞

⎠

---------------------------------------------------

TSIMP

gives:

SIN(5X)

EXP(i ⋅ x)⁴+ 1

-------------------------------------

EXP(i ⋅ x)²

VER

Returns the version number of your CAS.

Example

Typing:

VER

might give:

4.20050219

This particular result means that you have a version 4

CAS, dated 19 February 2005. Note that this is not the

same as VERSION (which returns the version of the

calculator’s ROM).

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Equation Writer

Using CAS in the Equation Writer

The Equation Writer enables you to type expressions that

you want to simplify, factor, differentiate, integrate, and

so on, and then work them through as if on paper.

The

key on the HOME

screen menu bar opens the

Equation Writer, and the

key closes it.

This chapter explains how to

write an expression in the Equation Writer using the

menus and the keyboard, how to select a subexpression,

how to apply CAS functions to an expression or

subexpression and how to store values in the Equation

Writer variables.

Chapter 14 explains all the symbolic calculation functions

contained in the various menus, and chapter 16 provides

numerous examples showing the use of the Equation

Writer.

The Equation Writer menu bar

The Equation Writer has a

number of soft menu keys.

TOOL menu

Unlike the other soft menu

keys, the

menu does

not give access to CAS

commands. Instead, it

provides access to a number

of utilities to help you work

with the Equation Writer. The following table explains

each of the utilities on the

menu.

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Cursor mode

Edit expr.

Enables you to go into cursor

mode, for quicker selection of

expressions and

subexpressions (see

page 15-10).

Enables you to edit the

highlighted expression on the

edit line, just as you do in the

HOME screen (see

page 15-11).

Change font

Cut

Enables you to choose to type

using large or small

characters (see page 15-10).

Copies the selection to the

clipboard and erases the

selection from Equation

Writer.

Copy

Copies the selection to the

clipboard.

Paste

Copies the contents of the

clipboard to the location of

the cursor. The clipboard

contents will be either

whatever Copyor Cut

selected the last time you

used these commands, or the

highlighted level when you

selected COPYin CAS

history.

ALGB menu

DIFF menu

The

menu contains

functions that enable you to

perform algebra, such as

factoring, expansion,

simplification, substitution,

and so on.

The

menu contains

functions that enable you to

perform differential calculus,

such as differentiation,

integration, series

expansion, limits, and so on.

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REWRI menu

SOLV menu

The

menu contains

functions that enable you to

rewrite an expression in

another form.

The

menu contains

functions that enable you to

solve equations, linear

systems, and differential

equations.

TRIG menu

The

menu contains

functions that enable you to

transform trigonometric

expressions.

N O T E

You can get online help about any CAS function by

pressing

2 and selecting that function (as

explained in “Online Help” on page 14-8).

Configuration menus

You can directly see, and change, CAS modes while

working with the Equation Writer. The first line in each of

the Equation Writer menus (except

current CAS mode settings.

) indicates the

In the example at the right,

the first line of the

menu reads:

CFG R= X S

CFG stands for

“configuration”, and the symbols to the right of it indicate

various mode settings.

•

The first symbol, R, indicates that you are in real

mode. If you were in complex mode, this symbol

would be C.

The second symbol, =, indicates that you are in exact

mode. If you were in approximate mode, this symbol

would be ~.

The third symbol, Xin the above example, indicates

the current independent variable.

Equation Writer

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•

The fourth symbol, S, in the above example, indicates

that you are in step-by-step mode. If you were not in

step-by-step mode, this symbol would be D(which

stands for Direct).

The first line of an Equation

Writer menu only indicates

some of the mode settings.

To see more settings,

highlight the first line and

press

. The configuration

menu appears. The header of the configuration menu has

additional symbols. In the example above, the upward-

pointing arrow indicates that polynomials are displayed

with increasing powers, and the 13indicates the modulo

value.

You can change CAS mode settings directly from the

configuration menu. Just press

until the setting you

want to choose is highlighted and then press

Note that the configuration menu includes only those

options that are not currently selected. For example, if

Rigorousis a current setting, its opposite, Sloppy, will

appear on the menu. If you choose Sloppy, then

Rigorousappears in its place.

To return your CAS modes to their default settings, select

Default cfgand press

To close the configuration menu, select Quit config

and press

N O T E

You can also change CAS mode settings from CAS

MODES screen. See “CAS modes” on page 14-5 for

information.

Online Help

language

One CAS setting that only

appears on the configuration

menu is the setting that

determines the language of

the online help. Two

languages are available:

English and French. To choose French, select Francais

and press

press

. To return to English, select Englishand

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Entering expressions and subexpressions

You type expressions in the Equation Writer is much the

same way as you type them in the HOME screen, using

the keys to directly enter numbers, letters and operators,

and menus to select various functions and commands.

When you type an expression in the Equation Writer, the

operator that you are typing always carries over to the

adjacent or selected expression. You don’t have to worry

about where the parentheses go: they are automatically

entered for you.

It will help you understand how the Equation Writer works

if you view a mathematical expression as a tree, with the

four arrow keys enabling you to move through the tree:

•

the

and

keys enable you to move from one

branch to another

the and keys enable you to move up and

down a particular tree

the and key combinations

enable you to make multiple selections.

How do I select?

There are two ways of going into selection mode:

•

Pressing

takes you into selection mode and

selects the element adjacent to the cursor. For

example:

1+2+3+4

selects 4. Pressing it again selects the entire tree:

1+2+3+4.

•

Pressing

takes you into selection mode and

selects the branch adjacent to the cursor. Pressing it

augments the selection, adding the next branch to the

right. For example:

1+2+3+4

selects 3+4. Pressing it again selects 2+3+4, and again

selects 1+2+3+4.

N O T E :

If you are typing a templated function with multiple

arguments (such as ∑ , ∫,SUBST, etc.), pressing

enables you to move from one argument to another. In

Equation Writer

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this case, you have to press

expression.

to select elements in the

The following illustration shows how an expression can

be viewed as a tree in the Equation Writer. It illustrates a

tree view of the expression:

(5x + 3) ⋅ (x – 1)

----------------------------------------

x + 3

–

Suppose that the cursor is positioned to the right of 3:

•

If you press

once, the 3 component is selected.

If you press

tree, with x + 3 now selected.

again, the selection moves up the

•

If you press

again, the selection moves up the

tree, and now the entire expression is selected.

If you had pressed

instead of

when the

cursor was positioned to the right of 3, the leaves of

the branch get selected (that is, x + 3).

•

If you press

again, the selection moves up the

tree, and now the entire expression is selected.

•

If you now press

, just the numerator is selected.

again, the top-most branch

selected (that is, (5x + 3).

•

Continue pressing

turn (5x and then 5).

to select each top-most leaf in

15-6

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•

Press

again and again to progressively select

more of the top-most branch, and then lower

branches (5x, 5x + 3, and then the entire numerator

and finally the entire expression).

More Examples

Example1

If you enter:

2 + X × 3– X

and press

the

entire expression is selected.

Pressing evaluates

what is selected (that is, the

entire expression) and

returns:

2X + 2

If you enter the same expression as earlier but press

after the first X, as in:

2 + X

× 3 – X

the 2 + X is selected and the

next operation,

multiplication, is applied to

to it. The expression

becomes:

(2 + X) × 3 – X

selects the

Pressing

entire expression, and

pressing

resulting in:

evaluates it,

2X + 6

Now enter the same expression, but press

3, as in:

after the

2 + X

× 3

– X

Note that

selects the

expression so far entered (2

+ X) thus making the next

operation apply to the entire

selection, not just the last

entered term. The

key

selects just the last entry (3) and makes the next operation

Equation Writer

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(– X) apply to it. As a result, the entered expression is

interpreted, and displayed, as (2 + X)(3 – X).

Select the entire expression

by pressing

and

evaluate it by pressing

. The result is:

–(X²–X–6)

Example2

To enter X²–3X+1, press:

– 3

If, instead, you had to enter

–x²–3X+1, you would need

to press:

(–)

– 3

Note that you press

twice to ensure that the exponent

applies to –X and not just to X.

Example 3

Suppose you want to enter:

-- -- -- --

Each fraction can be viewed

as a separate branch on the

equation tree. In the

Equation Writer type the first

branch:

1 ÷ 2

and then select this branch by pressing

Now type + and enter the second branch:

1 ÷ 3

Select the second branch by pressing

Now type + and enter the third branch:

1 ÷ 4

Likewise, select the third branch by pressing

and then the fourth branch:

, type +

1 ÷ 5

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Select the fifth branch by

pressing . At this point,

the desired expression is in

the Equation Writer, as

shown at the right.

Suppose that you want to select the second and third

-- --

branches, that is:

the second term.

. First press

. This selects

Now press

. This

key combination enables

you to select two contiguous

branches, the one already

selected and the one to the

right of it.

If you want, you can

evaluate the selected part by

pressing

. The result

is shown at the right.

Suppose now you want to

perform the partial calculation:

-- --

Because the two terms in this partial calculation are not

contiguous (that is, side by side), you must first perform a

permutation so that they are side by side.To do this,

press:

This exchanges the selected

element with its neighbour to

the left. The result is shown at

the right.

Now press:

to select just the branches

you are interested in:

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Pressing

produces

the result of the partial

calculation.

Summing up

Pressing

enables you to select the current

element and its neighbour to the right.

enables

you to exchange the selected element with its neighbour

to the left. The selected element remains selected after you

move it.

Cursor mode

In cursor mode you can select a large expression quickly.

To select cursor mode, press:

Cursor mode

As you press the arrow key,

various parts of the

expression are enclosed n in

a box.

When what you want to

select is enclosed, press

to select it.

Changing the

font

If you are entering a long expression, you may find it

useful to reduce the size of the font used in the Equation

Writer. Select Change fontfrom the

menu. This

enables you to view a large expression in its entirety

when you need to. Selecting Change fontagain

returns the font size to its previous setting.

You can also see the selected expression or

subexpression is a smaller or larger font size by pressing

and then

(to use the smaller font) or

(to use the larger font).

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How to modify an expression

If you’re typing an expression, the

key enables you

to erase what you’ve typed. If you’re selecting, you can:

•

Cancel the selection without deleting the expression

by pressing . The cursor moves to the end of the

deselected portion.

•

Replace the selection with an expression, just by

typing the desired expression.

Transform the selected expression by applying a CAS

function to it (which you can invoke from one of CAS

menus along the bottom of the screen).

•

Delete the selected expression by pressing:

Delete a selected unary operator at the top of the

expression tree by pressing:

For example, to replace SIN(expr) with COS(expr),

select SIN(expr), press

COS.

and then press

•

Delete a binary infix operator and one of its

arguments by selecting the argument you want delete

and pressing:

For example, if you have the expression 1+2 and

select 1, pressing

deletes 1+ and leaves

only 2. Similarly, to delete F(x)= in the expression F(x)

= x²– x +1, you select F(x) and then press

. This produces x = x²– x +1.

•

Delete a binary operator by selecting:

Edit expr.

from the

menu and then making the correction.

Copy an element from CAS history. You access CAS

history by pressing

details.

. See page 15-19 for

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Accessing CAS functions

While you are in the Equation Writer, you can access all

CAS functions, and you can access them in various ways.

General principle: When you have written an

expression in the Equation Writer, all you have to do is

press

to evaluate whatever you have selected (or

the entire expression, if nothing is selected).

How to type Σ and ∫

Press

to enter Σ and

to enter ∫.

These symbols and are treated as prefix functions with

multiple arguments. They are automatically placed before

the selected element, if there is one (hence the term prefix

functions).

You can move the cursor from argument to argument by

pressing

Enter the expressions according to the rules of selection

explained earlier, but you must first go into selection

mode by pressing

N O T E

Do not use the index i to define a summation, because i

designates the complex-number solution of x²+ 1 = 0.

Σ performs exact calculations if its argument has a

discrete primitive; otherwise it performs approximate

calculations, even in exact mode. For example, in both

approximate and exact mode:

---- = 2.70833333334

∑

k = 0

whereas in exact mode:

-----

---- ---- ---- ----

1 +

1! 2! 3! 4!

Note that Σ can symbolically calculate summations of

rational fractions and hypergeometric series that allow a

discrete primitive. For example, if you type:

--------------------------

∑

K ⋅ (K + 1)

K = 1

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select the entire expression and press

obtain:

, you

However, if you type:

∞

--------------------------

∑

K ⋅ (K + 1)

K = 1

select the entire expression and press

obtain 1.

, you

How to enter infix

functions

An infix function is one that is typed between its

arguments. For example, AND, |and MODare infix

functions.You can either:

•

type them in Alpha mode and then enter their

arguments, or

•

select them from a CAS menu or by pressing an

appropriate key, provided that you have already

written and selected the first argument.

You move from one argument to the other by pressing

and

. The comma enables you to write a

complex number: when you type (1,2), the

parentheses are automatically placed when you type

the comma. If you want to type (–1,2), you must

select –1 before you type the comma.

How to enter prefix

functions

A prefix function is one that is typed before its arguments.

To enter a prefix function, you can:

•

type the first argument, select it, then select the

function from a menu, or

you can select the function from a menu, or by

directly entering it in Alpha mode, and then type the

arguments.

The following example illustrates the various ways of

entering a prefix function. Suppose you want to factor the

expression x²– 4, then find its value for x = 4. FACTORis

the function for factoring, and it is found on the

menu. SUBSTis the function for substituting a value for a

variable in an expression, and it is also found in the

menu.

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First option: function first, then arguments

In the Equation Writer, press

, select FACTORand

then press

FACTOR()is displayed in

the Equation Writer, with the

cursor between the

parentheses (as shown at the right).

Enter your expression, using

the rules of selection

described earlier.

The entire expression is now selected.

Press

the result.

then produce

With a blank Equation

Writer screen, press

select SUBSTand then press

With the cursor between the

parentheses at the location of the first argument, type your

expression.

Note that SUBSThas two

arguments. When you have

finished entering the first

argument (the expression),

press

to move to the

second argument.

Now enter the second

argument, x=4.

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Press

to obtain the an

intermediate result (4²– 4)

and

again to

evaluate the intermediate

result. The final answer is 12.

Second option: arguments first, then function

Enter your expression, using

the rules of selection

described earlier.

The entire expression is now selected.

Now press and select

FACTOR. Notice that the

FACTORis applied to

whatever was selected

(which is automatically

placed in parentheses).

Press

to evaluate the

expression. The result is the

factors of the expression.

Because the result of an

evaluation is always

selected, you can immediately apply another command

to it.

To illustrate this, press

, select SUBSTand

then press

Note that SUBSTis applied

to whatever was selected

(which is automatically

placed in parentheses). Note too that the cursor is

automatically placed in the position of the second

argument.

Enter the second argument,

x=4.

Equation Writer

15-15

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Press

to obtain an

intermediate result, (4– 2)(4

+ 2), and

again to

evaluate the intermediate

result. The final answer, as

before, is 12.

N o t e

If you call a CAS function while you’re writing an

expression, whatever is currently selected is copied to the

function’s first or main argument. If nothing is selected,

the cursor is placed at the appropriate location for

completing the arguments.

Equation Writer variables

You can store objects in variables, then access an object

by using the name of its variable. However, you should

note the following:

•

Variables used in CAS cannot be used in HOME, and

vice versa.

In HOME or in the program editor, use

an object in a variable.

to store

In CAS, use the STORE command (on the

menu) to store a value in a variable.

•

The

available variables. Pressing

HOME displays the names of the variables defined in

HOME and in the Aplets. Pressing while you

key displays a menu that contains all the

while you are in

are in the Equation Writer displays the names of the

variables defined in CAS (as explained on

page 15-18).

Predefined CAS variables

• VXcontains the name of the current symbolic

variable. Generally, this is X, so you should not use X

as the name of a numeric variable. Nor should you

erase the contents of X with the UNASSIGNcommand

(on the

calculation.

menu) after having done a symbolic

• EPScontains the value of epsilon used in the EPSX0

command.

15-16

Equation Writer

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• MODULOcontains the value of p for performing

symbolic calculations in Z/pZ or in Z/pZ[X]. You can

change the value of p either with the MODSTO

command on the MODULARmenu, (by typing, for

example, MODSTO(n) to give p a value of n), or from

CAS MODES screen (see page 14-5).

• PERIODmust contain the period of a function before

you can find its Fourier coefficients.

• PRIMITcontains the primitive of the last integrated

function.

• REALASSUMEcontains a list of the names of the

symbolic variables that are considered reals. If you’ve

chosen the Cmplx varsoption on the CFG

configuration menu, the defaults are X, Y, t, S1 and

S2, as well as any integration variables that are in

use.

If you’ve chosen the Real varsoption on the CFG

configuration menu, all symbolic variables are

considered reals. You can also use an assumption to

define a variable such as X >1. In a case like this,

you use the ASSUME(X>1)command to make

REALASSUMEcontain X>1. The command

UNASSUME(X)cancels all the assumptions you have

previously made about X.

To see these variables, as well as those that you’ve

defined in CAS, press

in the Equation Editor

(see “CAS variables” on page 14-4).

The keyboard in the Equation Writer

The keys mentioned in this section have different functions

when pressed in the Equation Writer than when used

elsewhere.

MATH key

The

key, if pressed in

the Equation Writer, displays

just those functions used in

symbolic calculation. These

functions are contained in

the following menus:

•

The five function-containing Equation Writer menus

outlined in the previous section: Algebra( ),

Equation Writer

15-17

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Diff&Int (

), Rewrite(

), Solve

(

) and Trig(

•

The Complexmenu, providing functions specific to

manipulating with complex numbers.

•

The Constantmenu, containing e, i,∞ and π.

The Hyperb. menu, containing hyperbolic functions.

The Integermenu, containing functions that enable

you to perform integer arithmetic.

•

The Modularmenu, containing functions that enable

you to perform modular arithmetic (using the value

contained in the MODULOvariable).

•

The Polynom.menu, containing functions that enable

you to perform calculations with polynomials.

The Realmenu, containing functions specific to

common real-number calculations

The Testsmenu, containing logic functions for

working with hypotheses.

SHIFT MATH keys

The

key

combination opens an

alphabetical menu of all

CAS commands. You can

enter a command by

selecting it from this menu, so

that you don’t have to type it in ALPHA mode.

VARS key

Pressing

while you’re

in the Equation Writer

displays the names of the

variables defined in CAS.

Take special note of namVX,

which contains the name of

the current variable.

The menu options on the variables screen are:

Press to copy the name of the highlighted variable

to the position of the cursor in Equation Writer.

Press to see the contents of the highlighted

variable.

Press to change the contents of the highlighted

variable.

15-18

Equation Writer

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Press to clear the value of the highlighted variable.

Press to change the name of the highlighted

variable.

Press to define a new variable (which you do by

specifying an object and a name for the object.

SYMB key

Pressing the

key in

the Equation Writer gives

you access to CAS history.

As in the HOME screen

history, the calculations are

written on the left and the

results are written on the right. Using the arrow keys, you

can scroll through the history.

Press

clipboard in order to paste it in the Equation Writer. Press

or to replace the current selection in

to copy the highlighted entry in history to the

Equation Writer with the highlighted entry in CAS history.

Press

any way.

to leave CAS history without changing it in

SHIFT SYMB or

SHIFT HOME keys

While you are working in the

Equation Writer, pressing

opens CAS MODES

screen. The various CAS

modes are described in

“CAS modes” on page 14-5.

SHIFT , key

PLOT key

Pressing

followed by the comma key undoes (that

is, cancels) your last operation.

Pressing

in the

Equation Writer displays a

menu of plot types. You can

choose to graph a function,

a parametric curve, or a

polar curve.

Depending on what you

choose, the highlighted

expression is copied into the

appropriate aplet, to the

destination that you specify.

Equation Writer

15-19

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N O T E

This operation supposes that the current variable is also

the variable of the function or curve you want to graph.

When the expression is copied, it is evaluated, and the

current variable (contained in VX) is changed to X, T, or

θ, depending on the type of plot you chose.

If the function depends on a parameter, it is preferable to

give the parameter a value before pressing

. If,

however, you want the parameterized expression to be

copied with its parameter, then the name of the

parameter must consist of a single letter other than X, T,

or θ, so that there is no confusion. If the highlighted

expression has real values, the Function, Aplet or Polar

Aplet can be chosen, and the graph will be of Function or

Polar type. If the highlighted expression has complex

values, the Parametric Aplet must be chosen, and the

graph will be of Parametric type.

To summarize. If you choose:

•

the Function Aplet, the highlighted expression is

copied into the chosen function Fi, and the current

variable is changed to X.

•

the Parametric Aplet, the real part and the imaginary

part of the highlighted expression are copied into the

chosen functions Xi,Yi, and the current variable is

changed to T.

•

the Polar Aplet, the highlighted expression is copied

into the chosen function Ri and the current variable is

changed to θ.

NUM key

Pressing

in the Equation Writer causes the

highlighted expression to be replaced by a numeric

approximation.

approximate mode.

puts the calculator into

SHIFT NUM key

VIEWS key

Pressing

in the Equation Writer causes the

highlighted expression to be replaced by a rational

number.

mode.

puts the calculator into exact

Pressing

move the cursor with the

the entire highlighted expression. Press

the Equation Writer.

in the Equation Writer enables you to

and

arrow keys to see

to return in

15-20

Equation Writer

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Short-cut keys

In the Equation Writer, the following are short-cut keys to

the symbols indicated:

0 for ∞

1 for i

3 for π

5 for <

6 for >

8 for ≤

9 for ≥

Equation Writer

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Step-by-Step Examples

Introduction

This chapter illustrates the power of CAS, and the

Equation Writer, by working though a number of

examples. Some of these examples are variations on

questions from senior math examination papers.

The examples are given in order of increasing difficulty.

Example 1

If A is:

– 1

------------

+ 1

calculate the result of A in the form of an irreducible

fraction, showing each step of the calculation.

Solution: In the Equation

Writer, enter A by typing:

Now press

above).

to select the denominator (as shown

to simplify the

Press

denominator.

Now select the numerator

by pressing

Step-by-Step Examples

16-1

hp40g+.book Page 2 Friday, December 9, 2005 1:03 AM

Press

numerator.

to simplify the

Press

to select the

entire fraction.

Press

to simplify the

selected fraction, giving

the result shown at the

right.

Example 2

Given that C = 2 45 + 3 12 – 20 – 6 3

write C in the form d 5 , where d is a whole number.

Solution: In the Equation Writer, enter C by typing:

Press

select –6 3 .

Press

to select

– 20 and

select 20.

Now press

select FACTOR and

press

16-2

Step-by-Step Examples

hp40g+.book Page 3 Friday, December 9, 2005 1:03 AM

Press

to factor

20 into 2²⋅ 5 .

Press

to select

2²⋅ 5 and

simplify it.

Press

to select

–2 5 and

to exchange 3 12 with

–2 5 .

Press

to select

2 45 and

to select 45.

Press

, select

FACTOR and press

Press

to factor

45 into 3²⋅ 5 .

Press

to select

3²⋅ 5 and

simplify the selection.

Press

to select

2 ⋅ 3 5 , and

to select

2 ⋅ 3 5 – 2 5 .

Step-by-Step Examples

16-3

hp40g+.book Page 4 Friday, December 9, 2005 1:03 AM

Press

evaluate the selection.

It remains to transform

3 12 and combine it

with –6 3 . Follow the

same procedure as

undertaken a number of

times above. You will find that 3 12 is equal to

6 3 , and so the final two terms cancel each other

out.

Hence the result is

C = 4 5

Example 3

Given the expression D = (3x – 1)²– 81 :

•

expand and reduce D

factor D

•

solve the equation (3x – 10) ⋅ (3x + 8) = 0 and

evaluate D for x = 5.

Solution: First, enter D using the Equation Writer:

Press

to select

(3X – 1)²and

expandtheexpression. This

gives: 9x²– 6x + 1 – 81

16-4

Step-by-Step Examples

hp40g+.book Page 5 Friday, December 9, 2005 1:03 AM

Press

to select the entire

equation, then press

to reduce it to

9x²– 6x – 80 .

Press

FACTOR, press

then . The result is

as shown at the right.

, select

and

Now press

, select

and

SOLVEVX, press

press

shown at the right.

. The result is

Press

to display

CAS history, select D or a

version of it, and press

Press

press

, select SUBST,

and, then

complete the second

argument: x = –5

Press

to select

the entire expression and

then

to obtain the

intermediate result shown.

Press once more to

yield the result:175 .

Therefore, D = 175 when

x = –5 .

Example 4

A baker produces two assortments of biscuits and

macaroons. A packet of the first assortment contains 17

biscuits and 20 macaroons. A packet of the second

assortment contains 10 biscuits and 25 macaroons. Both

packets cost 90 cents.

Calculate the price of one biscuit, and the price of one

macaroon.

Solution: Let x be the price of one biscuit, and y the

price of one macaroon. The problem is to solve:

Step-by-Step Examples

16-5

hp40g+.book Page 6 Friday, December 9, 2005 1:03 AM

17x + 20y = 90

10x + 25y = 90

Press

, select

LINSOLVE andpress

Enter 17

If you are working in step

by step mode, pressing

produces the result

at the right.

Press

again to

produce the next step in the

solution:

Press

again to

produce the reduction

result:

Pressing

again

produces the final result:

-----

If you select

, and press

you get X = 2 and Y

= 2.8. In other words, the

price of one biscuit is 2

cents, and the price of one

macaroon is 2.8 cents.

Exercise 5

Suppose that A and B are points having the coordinates

(–1, 3) and (–3,–1) respectively, and where the unit of

measure is the centimetre.

16-6

Step-by-Step Examples

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1. Find the exact length of AB in centimetres.

2. Determine the equation of the line AB.

First method

Type:

STORE((-1,3),A)

and press

Accept the change to

Complexmode, if

necessary.

Note that pressing

returns the coordinates in

complex form: –1+3i.

Now type:

STORE((-3,-1),B)

and press

The coordinates this time are represented as –3+–1·i.

The vector AB has coordinates B – A.

Type:

(B - A)

Press

2 5 .

. The result is

Now apply the DROITE

command to determine the

equation of the line AB:

Complex

DROITE

Pressing

gives an

intermediate result.

Step-by-Step Examples

16-7

hp40g+.book Page 8 Friday, December 9, 2005 1:03 AM

Press

again to

simplify the result to

Y = 2X+5.

Second method

Type:

(-3,-1 )-(-1,3)

The answer is –(2+4i).

With the answer still

selected, apply the ABS

command by pressing

Pressing gives 2 5 , the same answer as with

method 1 above.

You can also determi1ne the equation of the line AB by

typing:

DROITE(( -1,3), (-3,-1))

Pressing

Y = –(2X+5).

then gives the result obtained before:

Exercise 6

In this exercise, we consider some examples of integer

arithmetic.

Part 1

For n, a strictly positive integer, we define:

a_n= 4 × 10ⁿ– 1, b_n= 2 × 10ⁿ– 1, c_n= 2 × 10ⁿ+ 1

1. Compute a₁, b₁, c₁, a₂, b₂, c₂, a₃, b₃and c₃.

2. Determine how many digits the decimal

representations of a_nand c_ncan have. Show that a_n

and c_nare divisible by 3.

3. Using a list of prime numbers less than 100, show

that b₃is a prime.

16-8

Step-by-Step Examples

hp40g+.book Page 9 Friday, December 9, 2005 1:03 AM

4. Show that for every integer n > 0, b_n× c_n= a_2n.

5. Deduce the prime factor decomposition of a₆.

6. Show that GCD(b_n,c_n) = GCD(c_n,2). Deduce that b_n

and c_nare prime together.

Solution: Begin by entering the three definitions. Type:

DEF(A(N) = 4 · 10^N–1)

DEF(B(N) = 2 · 10^N–1)

DEF(C(N) = 2 · 10^N+1)

Here are the keystrokes for entering the first definition:

First select the DEFcommand

by pressing

Now press

= 4

Finally press

Do likewise to define the

other two expressions.

You can now calculate various values of A(N), B(N) and

C(N) simply by typing the defined variable and a value

for N, and then pressing

. For example:

A(1)

A(2)

A(3)

yields 39

yields 399

yields 3999

B(1)

yields 19

B(2)

yields 199

yields 1999

B(3)

and so on.

In determining the number of digits the decimal

representations of a_nand c_ncan have, the calculator is

used only to try out different values of n.

Step-by-Step Examples

16-9

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Show that the whole numbers k such that:

10ⁿ≤ k < 10^{n + 1}have (n + 1) digits in decimal notation.

We have:

10ⁿ< 3 ⋅ 10ⁿ< a_n< 4 ⋅ 10ⁿ< 10^{n + 1}

10ⁿ< b_n< 2 ⋅ 10ⁿ< 10^{n + 1}

10ⁿ< 2 ⋅ 10ⁿ< c_n< 3 ⋅ 10ⁿ< 10^{n + 1}

so a_n,b_n,c_nhave (n + 1) digits in decimal notation.

Moreover, d_n= 10ⁿ– 1 is divisible by 9, since its

decimal notation can only end in 9.

We also have:

a_n= 3 ⋅ 10ⁿ+ d_n

and

c_n= 3 ⋅ 10ⁿ– d_n

so a_nand c_nare both divisible by 3.

Let’s consider whether B(3) is a prime number.

Type ISPRIME?(B(3))

and press

. The

result is 1, which means

true. In other words, B(3) is

a prime.

Note: ISPRIME? is not

available from a CAS soft menu, but you can select it from

from CAS FUNCTIONSmenu while you are in the

Equation Writer by pressing

, choosing the

INTEGERmenu, and scrolling to the ISPRIME?function.

To prove that b₃= 1999 is a prime number, it is

necessary to show that 1999 is not divisible by any of the

prime numbers less than or equal to 1999 . As

1999 < 2025 = 45², that means testing the divisibility of

1999 by n = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,

41. 1999 is not divisible by any of these numbers, so we

can conclude that 1999 is prime.

16-10

Step-by-Step Examples

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Now consider the product of two of the definitions

entered above: B(N) × C(N):

Press

to select EXP2POWand

press

Press

to evaluate

the expression, yielding the

result of B(N) × C(N).

Consider now the decomposition of A(6) into its prime

factors.

Press

to select FACTORand press

Now press

Finally, press

to get

the result. The factors are

listed, separated by a

medial period. In this case,

the factors are 3, 23, 29

and 1999.

Now let’s consider whether b_nand c_nare relatively prime.

Here, the calculator is useful only for trying out different

values of n.

To show that b_nand c_nare relatively prime, it is enough

to note that:

c_n= b_n+ 2

That means that the common divisors of b_nand c_nare the

common divisors of b_nand 2, as well as the common

divisors of c_nand 2. b_nand 2 are relatively prime

because b_nis a prime number other than 2. So:

Step-by-Step Examples

16-11

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GCD(c_n,b_n) = GCD(c_n,2) = GCD(b_n,2) = 1

Part 2

Given the equation:

b₃⋅ x + c₃⋅ y = 1

[1]

where the integers x and y are unknown and b₃and c₃

are defined as in part 1 above:

1. Show that [1] has at least one solution.

2. Apply Euclid’s algorithm to b₃and c₃and find a

solution to [1].

3. Find all solutions of [1].

Solution: Equation [1] must have at least one solution,

as it is actually a form of Bézout’s Identity.

In effect, Bézout’s Theorem states that if a and b are

relatively prime, there exists an x and y such that:

a ⋅ x + b ⋅ y = 1

Therefore, the equation b₃⋅ x + c₃⋅ y = 1 has at least

one solution.

Now enter IEGCD(B(3),

C(3)).

Note that the IEGCD

function can be found on

the INTEGERsubmenu of

the MATHmenu.

Pressing

a number

of times returns the result

shown at the right:

In other words:

b₃× 1000 + c₃× (–999) = 1

Therefore, we have a particular solution:

x = 1000, y = –999.

The rest can be done on paper:

c₃= b₃+ 2 , b₃= 999 × 2 + 1

16-12

Step-by-Step Examples

hp40g+.book Page 13 Friday, December 9, 2005 1:03 AM

so, b₃= 999 × (c₃– b₃) + 1 , or

b₃× 1000 + c₃× (–999) = 1

The calculator is not needed for finding the general

solution to equation [1].

We started with b₃⋅ x + c₃⋅ y = 1

and have established that b₃× 1000 + c₃× (–999) = 1 .

So, by subtraction we have:

b₃⋅ (x – 1000) + c₃⋅ (y + 999) = 0

or b₃⋅ (x – 1000) = –c₃⋅ (y + 999)

According to Gauss’s Theorem, c₃is prime with b₃, so

c₃is a divisor of (x – 1000) .

Hence there exists k ∈ Z such that:

(x – 1000) = k × c₃

and

–(y + 999) = k × b₃

Solving for x and y, we get:

x = 1000 + k × c₃

and

y = – 999 – k × b₃

for k ∈ Z .

This gives us:

b₃⋅ x + c₃⋅ y = b₃× 1000 + c₃× (–999) = 1

The general solution for all k ∈ Z is therefore:

x = 1000 + k × c₃

y = – 999 – k × b₃

Exercise 7

Let m be a point on the circle C of center O and radius 1.

Consider the image M of m defined on their affixes by the

transformation F : z – > ⋅ z – Z . When m moves on

Step-by-Step Examples

16-13

hp40g+.book Page 14 Friday, December 9, 2005 1:03 AM

the circle C, M will move on a curve Γ. In this exercise we

will study and plot Γ.

1. Let t ∉ [–π,π] and m be the point on C of affix

z = e^{i ⋅ t}. Find the coordinates of M in terms of t.

2. Compare x(–t) with x(t) and y(–t) with y(t).

3. Compute x′(t) and find the variations of x over [0, π].

4. Repeat step 3 for y.

5. Show the variations of x and y in the same table.

6. Put the points of Γ corresponding to t = 0, π/3,

2π/3 and π, and draw the tangent to Γ at these

points.

Part 1

First go to CAS MODES

screen and make t the VX

variable. To do this, press

to open the Equation

Writer, and then press

. This opens

CAS MODESscreen. Press

and delete the current

T and press

variable. Type

Now enter the expression

⋅ z – z and press

to select it.

Now invoke the SUBST

command from the

menu. Because the

expression was

highlighted, the SUBST

command is automatically

applied to it.

Note that the cursor is

positioned in the second

parameter. Since we know

that z = e^{i ⋅ t}, we can

enter this as the second

parameter.

16-14

Step-by-Step Examples

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Selecting the entire

expression and pressing

gives the result at

the right:

Now linearize the result by

applying the LIN

command (which can be

found on the

menu).

The result, after accepting

the switch to complex

mode, is shown at the right:

Now store the result in

variable M. Note that

STOREis on the

menu.

To calculate the real part of

the expression, apply the

REcommand (available on

the COMPLEXsubmenu of

the MATHmenu).

Pressing

yields the

result at the right:

We are now going to

define this result as x(t).

To do this, enter =X(t),

highlight the X(t) by

pressing

and press

to swap the two

parts of the expression, as

shown at the right:

Now select the entire

expression and apply the

Step-by-Step Examples

16-15

hp40g+.book Page 16 Friday, December 9, 2005 1:03 AM

DEFcommand to it. Press

to complete the

definition.

To calculate the real part of

the expression, apply the

IMcommand (available on

the COMPLEXsubmenu of

the MATHmenu) to the

stored variable M.

Press

to get the

result at the right:

Finally, define the result as

Y(t) in the same way that

you defined X(t): by firstly

adding Y(t) = to the

expression (as shown at the

right) and then applying the DEFcommand.

We have now found the coordinates of M in terms of t.

Part 2

To find an axis of symmetry for Γ, calculate x(–t) and

y(–t) by typing:

Press

to highlight the

expression.

Then press

produce the result at the

right:

In other words,

x(–t) = x(t)

Now type

Press

to highlight the

expression.

16-16

Step-by-Step Examples

hp40g+.book Page 17 Friday, December 9, 2005 1:03 AM

Then press

produce the result at the

right:

In other words,

y(–t) = –y(t) .

If M₁(x(t),y(t)) is part of Γ , then M_x(x(–t),y(–t)) is also

part of Γ .

Since M₁and M₂are symmetrical with respect to the x-

axis, we can deduce that the x-axis is an axis of symmetry

for Γ .

Part 3

Calculate x′(t) by typing:

DERVX

t. Press

to highlight the

expression.

Pressing

returns the

result at the right:

Press

result:

to simplify the

You can now define the

function x′(t) by invoking

DEF.

Note: You will first need to type =X1(t)then exchange

X1(t)with the previous expression.

To do this, highlight X1(t)

and type

Now select the entire

expression and apply the

DEFcommand to it:

Finally press

finish the definition.

Step-by-Step Examples

16-17

hp40g+.book Page 18 Friday, December 9, 2005 1:03 AM

Part 4

To calculate y′(t) , begin

by typing: DERVX(Y(t)).

Pressing

returns:

Press

again to

simplify the result:

Select FACTORand press

You can now define the

function y′(t) (in the same

way that you defined

x′(t) ).

Part 5

To show the variations of x(t) and y(t) , we will trace

x(t) and y(t) on the same graph.

The independent variable must be t which it should be as

a result of the previous calculations. (You can check this

by pressing

Type X(t)in the Equation

Writer and press

The corresponding

expression is displayed.

Now press

Function, press

, select

select F1as the destination and press

Now do the same thing with Y(t), making F2the

destination.

To graph the functions, quit

CAS (by pressing

choose the Function

aplet, and check F1and

F2.

16-18

Step-by-Step Examples

hp40g+.book Page 19 Friday, December 9, 2005 1:03 AM

Now press

the graphs.

to see

π 2 ⋅ π

-- ----------

, π

Part 6

To find the values of x(t) and y(t) for t = 0,

return to CAS, type each function in turn and press

. (You may need to press

simplification).

twice for further

For example, pressing

gives the result at the right:

Likewise, pressing

gives this

answer at the right:

The other results are:

2π

⎛

⎝

⎞

⎠

------

X(π)=

Y(0)= 0

– 3

---------

⎛ ⎞

⎝ ⎠

2π

–3 ⋅

⎛

⎝

⎞

⎠

------

-----------------

Y(π)= 0

y'(t)

x'(t)

----------

The slope of the tangents is m =

y'(t)

x'(t)

π 2 ⋅ π

----------

-- ----------

, π by

We can find the values of

using the limcommand.

for t = 0,

Step-by-Step Examples

16-19

hp40g+.book Page 20 Friday, December 9, 2005 1:03 AM

The example at the right

shows the case for t = 0.

Select the entire expression

and press

answer:

to get the

The example at the right

shows the case for t = π/3.

Selecting the entire

expression and pressing

displays the

message shown at the right.

Accept YESand press

Press

again to get

the result:

∞

The next example is for t =

2π/3. Selecting the entire

expression and pressing

displays the result:

The final example is for the

case where t = π. Press

, accept YESto the

messageUNSIGNEDINF.

SOLVE?, press

and

to get the

press

result:

∞

Here, then, are the variations of x(t) and y(t) :

16-20

Step-by-Step Examples

hp40g+.book Page 21 Friday, December 9, 2005 1:03 AM

2π

------

–

x'(t)

↓

↑

–1

x(t)

–3

-----

↓

↑

y(t)

–

–3

---------

-------------

–

–1

–

y'(t)

∞

Now we will graph Γ, which is a parametric curve.

In the Equation Writer, type

X(t) + i × Y(t).

Select the entire expression

and press

Now press

, select

Parametricand press

. Select X1,Y1as the

destination and press

To make the graph of Γ, quit CAS and choose the

Parametricaplet. Check X1(T)and Y1(T).

Now press

the graph.

to see

Step-by-Step Examples

16-21

hp40g+.book Page 22 Friday, December 9, 2005 1:03 AM

Exercise 8

For this exercise, make sure that the calculator is in exact

real mode with X as the current variable.

Part 1

For an integer, n, define the following:

²2x + 3

--------------

e dx

u_n

∫

x + 2

Define g over [0,2] where:

2x + 3

x + 2

--------------

g(x) =

1. Find the variations of g over [0,2]. Show that for

every real x in [0,2]:

≤ g(x) ≤

2. Show that for every real x in [0,2]:

e ≤ g(x)e ≤

3. After integration, show that:

⎛

⎜

⎝

⎞

⎛

⎞

ne – n ≤ u ≤ neⁿ– n

⎟

⎠

⎜

⎝

⎟

⎠

4. Using:

e^x– 1

lim ------------- = 1

x → 0

show that if u_nhas a limit L as n approaches infinity,

then:

3 ≤ L ≤

16-22

Step-by-Step Examples

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Solution 1

Start by defining G(X):

DEF

= 2

Now press

Press

and

to select

the numerator and

denominator, and then

press

. This

leaves G(X) displayed:

Finally, apply the TABVAR

function:

TABVAR

and press

number of times until

the variation table appears (shown above).

The first line of the variation table gives the sign of

g′(x) according to x, and the second line the variations

of g (x). Note that for TABVARthe function is always

called F.

We can deduce, then, that g(x) increases over [0, 2].

If you had been in step-by-step mode, you would have

obtained:

2 ⋅ X + 3

X + 2

-------------------

F =

Press to get the

result at the right.

Step-by-Step Examples

16-23

hp40g+.book Page 24 Friday, December 9, 2005 1:03 AM

Now press

and scroll down the screen to:

------------------

→

(x + 2)²

Now press

to obtain the table of variations.

If you are not in step-by -step mode, you can also get the

calculation of the derivative by typing:

DERVX(G(X))

which produces the preceding result.

To prove the stated inequality, first calculate g(0) by

typing G(0) and pressing

. The answer is:

Now calculate g(2) by typing G(2) and pressing

The answer is

The two results prove that:

≤ g(x) ≤ for x ∈ [0,2]

Solution 2

The calculator is not needed here. Simply stating that:

eⁿ≥ 0 for x ∈ [0,2]

is sufficient to show that, for x ∈ [0,2] , we have:

e ≤ g(x)e ≤

Solution 3

To integrate the preceding

inequality, type the

expression at the right:

Pressing

produces

the result at the right:

16-24

Step-by-Step Examples

hp40g+.book Page 25 Friday, December 9, 2005 1:03 AM

We can now see that:

⎛

⎜

⎝

⎞

⎛

⎞

ne – n ≤ u ≤ neⁿ– n

⎟

⎠

⎜

⎝

⎟

⎠

To justify the preceding calculation, we must assume that

n ⋅ eⁿis a primitive of eⁿ.

If you are not sure, you can

use the INTVXfunction as

illustrated at the right:

Note that the INTVX

command is on the

menu.

The simplified result, got by

pressing

twice, is

shown at the right:

Solution 4

⎛

⎞

To find the limit of neⁿ– n when n → +∞ , enter the

⎜

⎟

⎠

⎝

expression at the right:

Note that the lim

command is on the

menu. The infinity sign can

be selected from the

character map, opened by

pressing

Pressing

once after selecting the infinity sign adds

a “+” character to the infinity sign.

Select the entire expression

ans press

result, which is:

to get the

Step-by-Step Examples

16-25

hp40g+.book Page 26 Friday, December 9, 2005 1:03 AM

NOTE: The variable VXis now set to N. Reset it to Xby

pressing

(to display CAS MODESscreen)

and change the INDEP VARsetting.

To check the result, we can say that:

e^x– 1

lim ------------- = 1

x → 0

and that therefore:

eⁿ– 1

lim ------------- = 1

n → +∞

or, simplifying:

⎛

⎞

lim eⁿ– 1 ⋅ n = 2

⎜

⎝

⎟

⎠

n → +∞

If the limit L of u_nexists as n approaches + ∞ in the

inequalities in solution 2 above, we get:

⋅ 2 ≤ L ≤ ⋅ 2

Part 2

1. Show that for every x in [0,2]:

2x + 3

x + 2

--------------

-----------

= 2 –

x + 2

2. Find the value of:

²2x + 3

--------------

I =

∫

x + 2

3. Show that for every x in [0,2]:

1 ≤ eⁿ≤ eⁿ

4. Deduce that:

1 ≤ u_n≤ eⁿ⋅ I

5. Show that u_nis convergent and find its limit, L.

16-26

Step-by-Step Examples

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Solution 1

Start by defining the

following: g(x) = 2 –

-----------

x + 2

Now type

PROPFRAC(G(X)). Note

that PROPFRACcan be

found on the POLYNOMIAL

submenu of the MATH

menu.

Pressing

yields the

result shown at the right.

Solution 2

Enter the integral:

I = g(x)dx .

∫

Pressing

yields the

result shown at the right:

Pressing

yields:

again

Working by hand:

2x + 3 = 2(x + 2) – 1 , so: g(x) = 2 –

-----------

x + 2

Then, integrating term by term between 0 and 2

produces:

x = 2

g(x)dx = [2x – ln(x + 2)]

∫

x = 0

that is, since ln4 = 2ln2 :

g(x)dx = 4 – ln2

∫

Step-by-Step Examples

16-27

hp40g+.book Page 28 Friday, December 9, 2005 1:03 AM

Solution 3

The calculator is not needed here. Simply stating that

eⁿ^--increases for x ∈ [0,2] is sufficient to yield the

inequality:

1 ≤ eⁿ≤ eⁿ

Solution 4

Since g(x) is positive over [0, 2], through multiplication

we get:

g(x) ≤ g(x)eⁿ≤ g(x)eⁿ

and then, integrating:

I ≤ u_n≤ eⁿI

Solution 5

First find the limit of eⁿ

when n → +∞ .

Note: pressing

after you have selected the

infinity sign from the

character map places a “+”

character in front of the infinity sign.

Selecting the entire

expression and pressing

yields:

In effect, tends to 0 as n

eⁿ

tends to +∞ , so

tends to e⁰= 1 as n tends to +∞ .

As n tends to +∞ , u_nis the portion between I and a

quantity that tends to I .

Hence, u_nconverges, and its limit is I .

We have therefore shown that: L = I = 4 – ln2

16-28

Step-by-Step Examples

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Step-by-Step Examples

16-29

hp40g+.book Page 30 Friday, December 9, 2005 1:03 AM

hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Variables and memory management

Introduction

The HP 40gs has approximately 200K of user memory.

The calculator uses this memory to store variables,

perform computations, and store history.

A variable is an object that you create in memory to hold

data. The HP 40gs has two types of variables, home

variables and aplet variables.

•

Home variables are available in all aplets. For

example, you can store real numbers in variables A

to Z and complex numbers in variables Z0 to Z9.

These can be numbers you have entered, or the

results of calculations. These variables are available

within all aplets and within any programs.

•

Aplet variables apply only to a single aplet. Aplets

have specific variables allocated to them which vary

from aplet to aplet.

You use the calculator’s memory to store the following

objects:

•

copies of aplets with specific configurations

new aplets that you download

aplet variables

home variables

variables created through a catalog or editor, for

example a matrix or a text note

•

programs that you create.

You can use the Memory Manager (

MEMORY) to

view the amount of memory available. The catalog views,

which are accessible via the Memory Manager, can be

used to transfer variables such as lists or matrices

between calculators.

Variables and memory management

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Storing and recalling variables

You can store numbers or expressions from a previous

input or result into variables.

Numeric Precision

A number stored in a variable is always stored as a 12-

digit mantissa with a 3-digit exponent. Numeric precision

in the display, however, depends on the display mode

(Standard, Fixed, Scientific, Engineering, or Fraction). A

displayed number has only the precision that is

displayed. If you copy it from the HOME view display

history, you obtain only the precision displayed, not the

full internal precision. On the other hand, the variable

Ans always contains the most recent result to full

precision.

To store a value

1. On the command line,

enter the value or the

calculation for the result

you wish to store.

2. Press

3. Enter a name for the

variable.

4. Press

To store the results

of a calculation

If the value you want to store is in the HOME view display

history, for example the results of a previous calculation,

you need to copy it to the command line, then store it.

1. Perform the calculation for the result you want to store.

2. Press

3. Press

4. Press

to highlight to the result you wish to store.

to copy the result to the command line.

17-2

Variables and memory management

hp40g+.book Page 3 Friday, December 9, 2005 1:03 AM

5. Enter a name for the variable.

6. Press

the result.

to store

The results of a calculation can also be stored directly to

a variable. For example:

To recall a value

To recall a variable’s value, type the name of the variable

and press

To use variables in

calculations

You can use variables in calculations. The calculator

substitutes the variable’s value in the calculation:

To clear a variable

You can use the CLRVAR

command to clear a

specified variable. For

example, if you have

stored{1,2,3,4}invariable

L1, entering CLRVAR L1

will clear L1. (You can find the ^CLRVARcommand

by pressing

category of commands.)

and choosing the PROMPT

Variables and memory management

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The VARS menu

You use the VARS menu to access all variables in the

calculator. The VARS menu is organised by category. For

each variable category in the left column, there is a list of

variables in the right column. You select a variable

category and then select a variable in the category.

1. Open the VARS menu.

2. Use the arrow keys or press the alpha key of the first

letter in the category to select a variable category.

For example, to select

the Matrix category,

press

Note: In this instance,

there is no need to

press the ALPHA key.

3. Move the highlight to the variables column.

4. Use the arrow keys to select the variable that you

want. For example, to select the M2 variable, press

17-4

Variables and memory management

hp40g+.book Page 5 Friday, December 9, 2005 1:03 AM

5. Choose whether to place the variable name or the

variable value on the command line.

–

Press

to indicate that you want the

variable’s contents to appear on the command

line.

–

Press

to indicate that you want the

variable’s name to appear on the command line.

6. Press to place the value or name on the

command line. The selected object appears on the

command line.

Note: The VARS menu can also be used to enter the

names or values of variables into programs.

Example

This example demonstrates how to use the VARS menu to

add the contents of two list variables, and to store the

result in another list variable.

1. Display the List Catalog.

LIST

to select L1

2. Enter the data for L1.

3. Return to the List Catalog to create L2.

LIST

to select L2

Variables and memory management

17-5

hp40g+.book Page 6 Friday, December 9, 2005 1:03 AM

4. Enter data for L2.

5. Press

to access HOME.

6. Open the variable menu and select L1.

7. Copy it to the command line. Note: Because the

option is highlighted, the variable’s name,

rather than its contents, is copied to the command

line.

8. Insert the + operator and select the L2 variable from

the List variables.

9. Store the answer in the List catalog L3 variable.

Note: You can also

type list names directly

from the keyboard.

17-6

Variables and memory management

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Home variables

It is not possible to store data of one type in a variable of

another type. For example, you use the Matrix catalog to

create matrices. You can create up to ten matrices, and

you can store these in variables M0 to M9. You cannot

store matrices in variables other than M0 to M9.

Cate-

gory

Available names

Complex Z0 to Z9

For example, (1,2)

Z0 or 2+3i

Z1. You can enter a complex

number by typing (r,i), where r represents

the real part, and i represents the

imaginary part.

Graphic

Library

G0 to G9

See“Graphic commands” on page 21-21

for more information on storing graphic

objects via programming commands. See

“To store into a graphics variable” on

page 20-5 for more information on

storing graphic object via the sketch view.

Aplet library variables can store aplets

that you have created, either by saving a

copy of a standard aplet, or downloading

an aplet from another source.

List

L0 to L9

For example, {1,2,3}

M0 to M9 can store matrices or vectors.

For example, [[1,2],[3,4]] M0.

L1.

Matrix

Modes

Modes variables store the modes settings

that you can configure using

MODES.

Notepad Notepad variables store notes.

Program

Real

Program variables store programs.

A to Z and θ.

For example, 7.45

Symbolic E0…9, S1…S5, s1…s5 and n1…n5.

Variables and memory management

17-7

hp40g+.book Page 8 Friday, December 9, 2005 1:03 AM

Aplet variables

Most aplet variables store values that are unique to a

particular aplet. These include symbolic expressions and

equations (see below), settings for the Plot and Numeric

views, and the results of some calculations such as roots

and intersections.

See the Reference Information chapter for more

information about aplet variables.

Category Available names

Function

F0 to F9 (Symbolic view). See “Function

aplet variables” on page R-7.

Parametric

X0, Y0 to X9, Y9 (Symbolic view). See

“Parametric aplet variables” on page

R-8.

Polar

R0 to R9 (Symbolic view). See “Polar

aplet variables” on page R-9.

Sequence

U0 to U9 (Symbolic view). See

“Sequence aplet variables” on page

R-10.

Solve

E0 to E9 (Symbolic view). See “Solve

aplet variables” on page R-11.

Statistics

C0 to C9 (Numeric view). See

“Statistics aplet variables” on page

R-12.

To access an aplet

variable

1. Open the aplet that contains the variable you want to

recall.

2. Press

to display the VARS menu.

3. Use the arrow keys to select a variable category in

the left column, then press

in the right column.

to access the variables

4. Use the arrow keys to select a variable in the right

column.

5. To copy the name of the variable onto the edit line,

press

. (

is the default setting.)

6. To copy the value of the variable into the edit line,

press

and press

17-8

Variables and memory management

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Memory Manager

You can use the Memory Manager to determine the

amount of available memory on the calculator. You can

also use Memory Manager to organize memory. For

example, if the available memory is low, you can use the

Memory Manager to determine which aplets or variables

consume large amounts of memory. You can make

deletions to free up memory.

Example

1. Start the Memory Manager. A list of variable

categories is displayed.

MEMORY

Free memory is

displayed in the top

right corner and the

body of the screen lists

each category, the memory it uses, and the

percentage of the total memory it uses.

2. Select the category with which you want to work and

press

. Memory Manager displays memory

details of variables within the category.

3. To delete variables in a

category:

–

Press

to delete the selected variable.

CLEAR to delete all variables in the

selected category.

Variables and memory management

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Matrices

Introduction

You can perform matrix calculations in HOME and in

programs. The matrix and each row of a matrix appear

in brackets, and the elements and rows are separated by

commas. For example, the following matrix:

1 2 3

4 5 6

is displayed in the history as:

[[1,2,3],[4,5,6]]

(If the Decimal Mark mode is set to Comma, then separate

each element and each row with a period.)

You can enter matrices directly in the command line, or

create them in the matrix editor.

Vectors

Vectors are one-dimensional arrays. They are composed

of just one row. A vector is represented with single

brackets; for example, [1,2,3]. A vector can be a real

number vector or a complex number vector, for example

[(1,2), (7,3)].

Matrices

Matrices are two-dimensional arrays. They are composed

of more than one row and more than one column.

Two-dimensional matrices are represented with nested

brackets; for example, [[1,2,3],[4,5,6]]. You can create

complex matrices, for example, [[(1,2), (3,4)], [(4,5),

(6,7)]].

Matrix Variables

There are ten matrix variables available, named M0 to

M9. You can use them in calculations in HOME or in a

program. You can retrieve the matrix names from the

VARS menu, or just type their names from the keyboard.

Matrices

18-1

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Creating and storing matrices

You can create, edit,

delete, send, and receive

matrices in the Matrix

catalog.

To open the Matrix

catalog, press

MATRIX.

You can also create and store matrices—named or

unnamed—-in HOME. For example, the command:

POLYROOT([1,0,–1,0])XM1

stores the root of the complex vector of length 3 into the

M1 variable. M1 now contains the three roots of

x³– x = 0

Matrix Catalog

keys

The table below lists the operations of the menu keys in

the Matrix Catalog, as well as the use of Delete (

)

and Clear (

CLEAR).

Key

Meaning

Opens the highlighted matrix for

editing.

Prompts for a matrix type, then

opens an empty matrix with the

highlighted name.

Transmits the highlighted matrix to

another HP 40gs or a disk drive.

See.

Receives a matrix from another

HP 40gs or a disk drive. See .

Clears the highlighted matrix.

Clears all matrices.

CLEAR

Moves to the end or the beginning

of the catalog.

To create a matrix

in the Matrix

Catalog

1. Press

MATRIX to open the Matrix Catalog. The

Matrix catalog lists the 10 available matrix variables,

M0 to M9.

18-2

Matrices

hp40g+.book Page 3 Friday, December 9, 2005 1:03 AM

2. Highlight the matrix variable name you want to use

and press

3. Select the type of matrix to create.

– For a vector (one-dimensional array),

select Realvectoror Complex vector.

Certain operations (+, –, CROSS) do not

recognize a one-dimensional matrix as a vector,

so this selection is important.

– For a matrix (two-dimensional array),

select Realmatrix or Complexmatrix.

4. For each element in the matrix, type a number or an

expression, and press

. (The expression may

not contain symbolic variable names.)

For complex numbers, enter each number in

complex form; that is, (a, b), where a is the real part

and b is the imaginary part. You must include the

parentheses and the comma.

5. Use the cursor keys to move to a different row or

column. You can change the direction of the highlight

bar by pressing

. The

menu key toggles

between the following three options:

–

specifies that the cursor moves to the cell

below the current cell when you press

specifies that the cursor moves to the cell to

the right of the current cell when you press

–

specifies that the cursor stays in the current

cell when you press

6. When done, press

catalog, or press

MATRIX to see the Matrix

to return to HOME. The

matrix entries are automatically stored.

A matrix is listed with two dimensions, even if it is 3×1. A

vector is listed with the number of elements, such as 3.

Matrices

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To transmit a

matrix

You can send matrices between calculators just as you

can send aplets, programs, lists, and notes.

1. Connect the calculators using an appropriate cable.

2. Open the Matrix catalogs on both calculators.

3. Highlight the matrix to send.

4. Press

5. Press

and choose the method of sending.

on the receiving calculator and choose

the method of receiving.

For more information on sending and receiving files, see

“Sending and receiving aplets” on page 22-4.

Working with matrices

To edit a matrix

In the Matrix catalog, highlight the name of the matrix

you want to edit and press

Matrix edit keys

The following table lists the matrix edit key operations.

Key

Meaning

Copies the highlighted element to

the edit line.

Inserts a row of zeros above, or a

column of zeros to the left, of the

highlighted cell. (You are prompted

to choose row or column.)

A three-way toggle for cursor

advancement in the Matrix editor.

advances to the right,

advances downward, and

does not advance at all.

Switches between larger and

smaller font sizes.

Deletes the highlighted cells, row,

or column (you are prompted to

make a choice).

CLEAR

Clears all elements from the matrix.

18-4

Matrices

hp40g+.book Page 5 Friday, December 9, 2005 1:03 AM

Key

Meaning (Continued)

Moves to the first row, last row, first

column, or last column respectively.

To display a matrix

•

In the Matrix catalog (

matrix name and press

In HOME, enter the name of the matrix variable and

press

MATRIX), highlight the

To display one

element

In HOME, enter matrixname(row,column). For example,

if M2is [[3,4],[5,6]], then M2(1,2)

returns

To create a matrix

in HOME

1. Enter the matrix in the edit line. Start and end the

matrix and each row with square brackets (the shifted

and

keys).

2. Separate each element and each row with a comma.

Example: [[1,2],[3,4]].

3. Press

to enter and display the matrix.

The left screen below shows the matrix

[[2.5,729],[16,2]]being stored into M5. The

screen on the right shows the vector [66,33,11]being

stored into M6. Note that you can enter an expression

(like 5/2) for an element of the matrix, and it will be

evaluated.

Matrices

18-5

hp40g+.book Page 6 Friday, December 9, 2005 1:03 AM

To store one

element

In HOME, enter, value

matrixname(row,column).

For example, to change the element in the first row and

second column of M5 to 728, then display the resulting

matrix:

728

An attempt to store an element to a row or column beyond

the size of the matrix results in an error message.

Matrix arithmetic

You can use the arithmetic functions (+, –, ×, / and

powers) with matrix arguments. Division left-multiplies by

the inverse of the divisor. You can enter the matrices

themselves or enter the names of stored matrix variables.

The matrices can be real or complex.

For the next examples, store [[1,2],[3,4]] into M1 and

[[5,6],[7,8]] into M2.

Example

1. Create the first matrix.

MATRIX

2. Create the second

matrix.

MATRIX

3. Add the matrices that

you created.

18-6

Matrices

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To multiply and

divide by a scalar

For division by a scalar, enter the matrix first, then the

operator, then the scalar. For multiplication, the order of

the operands does not matter.

The matrix and the scalar can be real or complex. For

example, to divide the result of the previous example by

2, press the following keys:

To multiply two

matrices

To multiply the two matrices M1 and M2 that you created

for the previous example, press the following keys:

To multiply a matrix by a

vector, enter the matrix

first, then the vector. The

number of elements in the vector must equal the number

of columns in the matrix.

To raise a matrix to

a power

You can raise a matrix to any power as long as the power

is an integer. The following example shows the result of

raising matrix M1, created earlier, to the power of 5.

Note: You can also raise a

matrix to a power without

first storing it as a variable.

Matrices can be raised to negative powers. In this case,

the result is equivalent to 1/[matrix]^ABS(power). In the

following example, M1 is raised to the power of –2.

Matrices

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To divide by a

square matrix

For division of a matrix or a vector by a square matrix,

the number of rows of the dividend (or the number of

elements, if it is a vector) must equal the number of rows

in the divisor.

This operation is not a mathematical division: it is a left-

multiplication by the inverse of the divisor. M1/M2 is

–1

equivalent to M2 * M1.

To divide the two matrices M1 and M2 that you created

for the previous example, press the following keys:

To invert a matrix

You can invert a square matrix in HOME by typing the

–1

matrix (or its variable name) and pressing

. Or you can use the matrix INVERSE command.

Enter INVERSE(matrixname) in HOME and press

To negate each

element

You can change the sign of each element in a matrix by

pressing

before the matrix name.

Solving systems of linear equations

Example

Solve the following linear system:

2x + 3y + 4z = 5

x + y – z = 7

4x – y + 2z = 1

1. Open the Matrix

catalog and create a

vector.

MATRIX

2. Create the vector of the

constants in the linear

system.

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3. Return to the Matrix

Catalog.

MATRIX

In this example, the

vector you created is

listed as M1.

4. Create a new matrix.

Select Real matrix

5. Enter the equation

coefficients.

In this example, the matrix you created is listed as

M2.

6. Return to HOME and enter the calculation to

left-multiply the constants vector by the inverse of the

coefficients matrix.

–1

The result is a vector of the

solutions x = 2, y = 3 and z = –2.

An alternative method, is to use the RREF function. See

“RREF” on page 18-12.

Matrices

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Matrix functions and commands

About functions

•

Functions can be used in any aplet or in HOME. They

are listed in the MATH menu under the Matrix

category. They can be used in mathematical

expressions—primarily in HOME—as well as in

programs.

•

Functions always produce and display a result. They

do not change any stored variables, such as a matrix

variable.

Functions have arguments that are enclosed in

parentheses and separated by commas; for example,

CROSS(vector1,vector2). The matrix input can be

either a matrix variable name (such as M1) or the

actual matrix data inside brackets. For example,

CROSS(M1,[1,2]).

About commands

Matrix commands are listed in the CMDS menu (

CMDS), in the matrix category.

See “Matrix commands” on page 21-24 for details of the

matrix commands available for use in programming.

Functions differ from commands in that a function can be

used in an expression. Commands cannot be used in an

expression.

Argument conventions

•

For row# or column#, supply the number of the row

(counting from the top, starting with 1) or the number

of the column (counting from the left, starting with 1).

•

The argument matrix can refer to either a vector or a

matrix.

Matrix functions

COLNORM

Column Norm. Finds the maximum value (over all

columns) of the sums of the absolute values of all elements

in a column.

COLNORM(matrix)

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COND

Condition Number. Finds the 1-norm (column norm) of a

square matrix.

COND(matrix)

CROSS

DET

Cross Product of vector1 with vector2.

CROSS(vector1, vector2)

Determinant of a square matrix.

DET(matrix)

DOT

Dot Product of two arrays, matrix1 matrix2.

DOT(matrix1, matrix2)

EIGENVAL

EIGENVV

Displays the eigenvalues in vector form for matrix.

EIGENVAL(matrix)

Eigenvectors and Eigenvalues for a square matrix.

Displays a list of two arrays. The first contains the

eigenvectors and the second contains the eigenvalues.

EIGENVV(matrix)

IDENMAT

Identity matrix. Creates a square matrix of dimension

size × size whose diagonal elements are 1 and off-

diagonal elements are zero.

IDENMAT(size)

INVERSE

Inverts a square matrix (real or complex).

INVERSE(matrix)

LQ Factorization. Factors an m × n matrix into three

matrices:

{[[ m × n lowertrapezoidal]],[[ n × n orthogonal]],

[[ m × m permutation]]}.

LQ(matrix)

LSQ

Least Squares. Displays the minimum norm least squares

matrix (or vector).

LSQ(matrix1, matrix2)

Matrices

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LU Decomposition. Factors a square matrix into three

matrices:

{[[lowertriangular]],[[uppertriangular]],[[permutation]]}

The uppertriangular has ones on its diagonal.

LU(matrix)

MAKEMAT

Make Matrix. Creates a matrix of dimension rows ×

columns, using expression to calculate each element. If

expression contains the variables I and J, then the

calculation for each element substitutes the current row

number for I and the current column number for J.

MAKEMAT(expression, rows, columns)

Example

MAKEMAT(0,3,3)returns a 3×3 zero matrix,

[[0,0,0],[0,0,0],[0,0,0]].

QR Factorization. Factors an m×n matrix into three

matrices: {[[m×m orthogonal]],[[m×n

uppertrapezoidal]],[[n×n permutation]]}.

QR(matrix)

RANK

Rank of a rectangular matrix.

RANK(matrix)

ROWNORM

Row Norm. Finds the maximum value (over all rows) for

the sums of the absolute values of all elements in a row.

ROWNORM(matrix)

RREF

Reduced-Row Echelon Form. Changes a rectangular

matrix to its reduced row-echelon form.

RREF(matrix)

SCHUR

Schur Decomposition. Factors a square matrix into two

matrices. If matrix is real, then the result is

{[[orthogonal]],[[upper-quasi triangular]]}.

If matrix is complex, then the result is

{[[unitary]],[[upper-triangular]]}.

SCHUR(matrix)

SIZE

Dimensions of matrix. Returned as a list: {rows,columns}.

SIZE(matrix)

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SPECNORM

SPECRAD

SVD

Spectral Norm of matrix.

SPECNORM(matrix)

Spectral Radius of a square matrix.

SPECRAD(matrix)

Singular Value Decomposition. Factors an m × n matrix

into two matrices and a vector:

{[[m × m square orthogonal]],[[n × n square orthogonal]],

[real]}.

SVD(matrix)

SVL

Singular Values. Returns a vector containing the singular

values of matrix.

SVL(matrix)

TRACE

Finds the trace of a square matrix. The trace is equal to

the sum of the diagonal elements. (It is also equal to the

sum of the eigenvalues.)

TRACE(matrix)

TRN

Transposes matrix. For a complex matrix, TRN finds the

conjugate transpose.

TRN(matrix)

Examples

Identity Matrix

You can create an identity matrix with the IDENMAT

function. For example, IDENMAT(2) creates the 2×2

identity matrix [[1,0],[0,1]].

You can also create an identity matrix using the

MAKEMAT (make matrix) function. For example, entering

MAKEMAT(I¼J,4,4) creates a 4 × 4 matrix showing the

numeral 1 for all elements except zeros on the diagonal.

The logical operator ¼ returns 0 when I (the row number)

and J (the column number) are equal, and returns 1 when

they are not equal.

Transposing a

Matrix

The TRN function swaps the row-column and column-row

elements of a matrix. For instance, element 1,2 (row 1,

Matrices

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column 2) is swapped with element 2,1; element 2,3 is

swapped with element 3,2; and so on.

For example, TRN([[1,2],[3,4]])creates the matrix

[[1,3],[2,4]].

Reduced-Row

Echelon Form

The following set of equations x – 2y + 3z = 14

2x + y – z = – 3

4x – 2y + 2z = 14

can be written as the augmented matrix

1 –2 3 14

2 1 –1 –3

4 –2 2 14

which can then stored as a

3 × 4 real matrix in any

matrix variable. M1 is used

in this example.

You can use the RREF

function to change this to

reduced row echelon form,

storing it in any matrix

variable. M2 is used in this

example.

The reduced row echelon

matrix gives the solution to

the linear equation in the

fourth column.

An advantage of using the

RREF function is that it will also work with inconsistent

matrices resulting from systems of equations which have

no solution or infinite solutions.

For example, the following set of equations has an infinite

number of solutions:

x + y – z = 5

2x – y = 7

x – 2y + z = 2

18-14

Matrices

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The final row of zeros in the

reduced-row echelon form

of the augmented matrix

indicates an inconsistent

system with infinite

solutions.

Matrices

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Lists

You can do list operations in HOME and in programs. A

list consists of comma-separated real or complex

numbers, expressions, or matrices, all enclosed in braces.

A list may, for example, contain a sequence of real

numbers such as {1,2,3}. (If the Decimal Mark mode is

set to Comma, then the separators are periods.) Lists

represent a convenient way to group related objects.

There are ten list variables available, named L0 to L9. You

can use them in calculations or expressions in HOME or

in a program. Retrieve the list names from the VARS

menu, or just type their names from the keyboard.

You can create, edit, delete, send, and receive named

lists in the List catalog (

LIST). You can also create

and store lists—named or unnnamed—in HOME lists

List variables are identical in behaviour to the columns

C1.C0 in the Statistics aplet. You can store a statistics

column to a list (or vice versa) and use any of the list

functions on the statistics columns, or the statistics

functions, on the list variables.

Create a list in

the List Catalog

1. Open the List catalog.

LIST.

2. Highlight the list name

you want to assign to

the new list (L1, etc.)

and press

display the List editor.

Lists

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3. Enter the values you want in the list, pressing

after each one.

Values can be real or

complex numbers (or

an expression). If you

enter a calculation, it is

evaluated and the

result is inserted in the

list.

4. When done, press

LIST to see the List catalog,

or press

to return to HOME.

List catalog keys

The list catalog keys are:

Key

Meaning

Opens the highlighted list for

editing.

Transmits the highlighted list to

another HP 40gs or a PC. See

“Sending and receiving aplets” on

page 22-4 for further information.

Receives a list from another

HP 40gs or a PC. See “Sending

and receiving aplets” on page 22-4

for further information.

Clears the highlighted list.

Clears all lists.

CLEAR

or Moves to the end or the beginning

of the catalog.

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Lists

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List edit keys

When you press

to create or change a list, the

following keys are available to you:

Key

Meaning

Copies the highlighted list item into

the edit line.

Inserts a new value before the

highlighted item.

Deletes the highlighted item from

the list.

CLEAR

Clears all elements from the list.

or Moves to the end or the beginning

of the list.

Create a list in

HOME

1. Enter the list on the edit line. Start and end the list

with braces (the shifted and keys) and

separate each element with a comma.

2. Press

to evaluate and display the list.

Immediately after typing in the list, you can store it in

a variable by pressing

listname

. The

list variable names are L0 through L9.

This example stores the

list {25,147,8} in L1.

Note: You can omit the

final brace when

entering a list.

Lists

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Displaying and editing lists

To display a list

•

In the List catalog, highlight the list name and press

•

In HOME, enter the name of the list and press

To display one

element

In HOME, enter listname(element#). For example, if L2 is

{3,4,5,6}, then L2(2)

returns 4.

To edit a list

1. Open the List catalog.

LIST.

2. Press

to highlight the name of the list you

want to edit (L1, etc.) and press

list contents.

to display the

3. Press

to highlight the element you want to

edit. In this example, edit the third element so that it

has a value of 5.

4. Press

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Lists

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To insertan element

in a list

1. Open the List catalog.

LIST.

2. Press

highlight the name of

the list you want to edit

(L1, etc.) and press

to display the list

contents.

New elements are inserted above the highlighted

position. In this example, an element, with the value

of 9, is inserted between the first and second

elements in the list.

3. Press

to the

insertion position, then

press

, and press

4. Press

To store one

element

In HOME, enter value

example, to store 148 as the second element in L1, type

148 L1(2)

listname(element). For

Lists

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Deleting lists

To delete a list

In the List catalog, highlight the list name and press

You are prompted to confirm that you want to delete the

contents of the highlighted list variable. Press

delete the contents.

To delete all lists

In the List catalog, press

CLEAR.

Transmitting lists

You can send lists to calculators or PCs just as you can

aplets, programs, matrices, and notes.

1. Connect the calculators using an appropriate cable).

2. Open the List catalogs on both calculators.

3. Highlight the list to send.

4. Press

5. Press

and choose the method of sending.

on the receiving calculator and choose

the method of receiving.

For more information on sending and receiving files, see

“Sending and receiving aplets” on page 22-4.

List functions

List functions are found in the MATH menu. You can use

them in HOME, as well as in programs.

You can type in the name

of the function, or you can

copy the name of the

function from the List

category of the MATH

menu. Press

(the

alpha L character key). This

highlights the List category in the left column. Press

move the cursor to the right column which contain the List

functions, select a function, and press

List functions have the following syntax:

•

Functions have arguments that are enclosed in

parentheses and separated by commas. Example:

CONCAT(L1,L2). An argument can be either a list

19-6

Lists

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variable name (such as L1) or the actual list. For

example, REVERSE({1,2,3}).

•

If Decimal Mark in Modes is set to Comma, use

periods to separate arguments. For example,

CONCAT(L1.L2).

Common operators like +, –, ×, and / can take lists as

arguments. If there are two arguments and both are lists,

then the lists must have the same length, since the

calculation pairs the elements. If there are two arguments

and one is a real number, then the calculation pairs the

number with each element of the list.

Example

5*{1,2,3} returns {5,10,15}.

Besides the common operators that can take numbers,

matrices, or lists as arguments, there are commands that

can only operate on lists.

CONCAT

^ΔLIST

Concatenates two lists into a new list.

CONCAT(list1,list2)

Example

CONCAT({1,2,3},{4})returns {1,2,3,4}.

Creates a new list composed of the first differences, that

is, the differences between the sequential elements in

list1. The new list has one fewer elements than list1. The

first differences for {x x ... x } are {x –x ... x –x }.

n–1

ΔLIST(list1)

Example

In HOME, store {3,5,8,12,17,23} in L5 and find the first

differences for the list.

{3,5,8,12,17,23

}

L 5

Select ΔLIST

Lists

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MAKELIST

Calculates a sequence of elements for a new list.

Evaluates expression with variable from begin to end

values, taken at increment steps.

MAKELIST(expression,variable,begin,end,

increment)

The MAKELIST function generates a series by

automatically producing a list from the repeated

evaluation of an expression.

Example

In HOME, generate a series of squares from 23 to 27.

Select

MAKELIST

ΠLIST

Calculates the product of all elements in list.

ΠLIST(list)

Example

ΠLIST({2,3,4})returns 24.

POS

Returns the position of an element within a list. The

element can be a value, a variable, or an expression. If

there is more than one instance of the element, the

position of the first occurrence is returned. A value of 0 is

returned if there is no occurrence of the specified element.

POS(list, element)

Example

POS ({3, 7, 12, 19},12)returns 3

REVERSE

Creates a list by reversing the order of the elements in a

list.

REVERSE(list)

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Lists

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SIZE

Calculates the number of elements in a list.

SIZE(list)

Also works with matrices.

ΣLIST

Calculates the sum of all elements in list.

ΣLIST(list)

Example

ΣLIST({2,3,4})returns 9.

SORT

Sorts elements in ascending order.

SORT(list)

Finding statistical values for list elements

To find values such as the mean, median, maximum, and

minimum values of the elements in a list, use the Statistics

aplet.

Example

In this example, use the Statistics aplet to find the mean,

median, maximum, and minimum values of the elements

in the list, L1.

1. Create L1 with values 88, 90, 89, 65, 70, and 89.

{ 88

}

2. In HOME, store L1 into

C1. You will then be

able to see the list data in the Numeric view of the

Statistics aplet.

Lists

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3. Start the Statistics aplet, and select 1-variable mode

(press , if necessary, to display ).

Select

Statistics

Note: Your list values are now in column 1 (C1).

4. In the Symbolic view, define H1 (for example) as C1

(sample) and 1 (frequency).

5. Go to the Numeric view to display calculated

statistics.

See “One-variable” on page 10-14 for the meaning

of each computed statistic.

19-10

Lists

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Notes and sketches

Introduction

The HP 40gs has text and picture editors for entering

notes and sketches.

•

Each aplet has its own independent Note view and

Sketch view. Notes and sketches that you create in

these views are associated with the aplet. When you

save the aplet, or send it to another calculator, the

notes and sketches are saved or sent as well.

•

The Notepad is a collection of notes independent of

all aplets. These notes can also be sent to another

calculator via the Notepad Catalog.

Aplet note view

You can attach text to an aplet in its Note view.

To write a note in

Note view

1. In an aplet, press NOTE for the Note view.

2. Use the note editing keys shown in the table in the

following section.

3. Set Alpha lock (

) for quick entry of letters. For

lowercase Alpha lock, press

4. While Alpha lock is on:

–

To type a single letter of the opposite case, press

letter.

–

To type a single non-alpha character (such as 5

or [ ), press

first. (This turns off Alpha

lock for one character.)

Your work is automatically saved. Press any view key

) or to exit

(

the Notes view.

Notes and sketches

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Note edit keys

Key

Meaning

Space key for text entry.

Displays next page of a multi-page

note.

Alpha-lock for letter entry.

Lower-case alpha-lock for letter

entry.

Backspaces cursor and deletes

character.

Deletes current character.

Starts a new line.

CLEAR

Erases the entire note.

Menu for entering variable names,

and contents of variables.

Menu for entering math

operations, and constants.

Menu for entering program

commands.

CMDS

Displays special characters. To

type one, highlight it and press

. To copy a character without

closing the CHARS screen, press

CHARS

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Notes and sketches

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Aplet sketch view

You can attach pictures to an aplet in its Sketch view

SKETCH). Your work is automatically saved with the

(

aplet. Press any other view key or

Sketch view

to exit the

Sketch keys

Key

Meaning

Stores the specified portion of the

current sketch to a graphics

variable (G1 through G0).

Adds a new, blank page to the

current sketch set.

Displays next sketch in the sketch

set. Animates if held down.

Opens the edit line to type a text

label.

Displays the menu-key labels for

drawing.

Deletes the current sketch.

Erases the entire sketch set.

CLEAR

Toggles menu key labels on and

off. If menu key labels are hidden,

or any menu key, redisplays

the menu key labels.

To draw a line

1. In an aplet, press

2. In Sketch view, press

where you want to start the line

3. Press . This turns on line-drawing.

4. Move the cursor in any direction to the end point of

the line by pressing the keys.

5. Press to finish the line.

SKETCH for the Sketch view.

and move the cursor to

Notes and sketches

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To draw a box

1. In Sketch view, press

and move the cursor to

where you want any corner of the box to be.

2. Press

3. Move the cursor to mark the opposite corner for the

box. You can adjust the size of the box by moving the

cursor.

4. Press

to finish the box.

To draw a circle

DRAW keys

1. In Sketch view, press

and move the cursor to

where you want the center of the circle to be.

2. Press

. This turns on circle drawing.

3. Move the cursor the distance of the radius.

4. Press

to draw the circle.

Key

Meaning

Dot on. Turns pixels on as the cursor

moves.

Dot off. Turns pixels off as the cursor

moves.

Draws a line from the cursor’s starting

position to the cursor’s current position.

Press

when you have finished. You

can draw a line at any angle.

Draws a box from the cursor’s starting

position to the cursor’s current position.

Press

when you have finished.

Draws a circle with the cursor’s starting

position as the center. The radius is the

distance between the cursor’s starting

and ending position. Press

the circle.

to draw

20-4

Notes and sketches

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To label parts of a

sketch

1. Press

the Alpha shift on, press

(for lowercase).

To make the label a smaller character size, turn off

before pressing . ( is a toggle

and type the text on the edit line. To lock

(for uppercase) or

between small and large font size). The smaller

character size cannot display lowercase letters.

2. Press

3. Position the label where you want it by pressing the

4. Press

5. Press

keys.

again to affix the label.

to continue

drawing, or press

to exit the

Sketch view.

To create a set of

sketches

You can create a set of up to ten sketches. This allows for

simple animation.

•

After making a sketch, press

to add a new,

blank page. You can now make a new sketch, which

becomes part of the current set of sketches.

•

To view the next sketch in an existing set, press

. Hold

down for animation.

To remove the current page in the current sketch

series, press

To store into a

graphics variable

You can define a portion of a sketch inside a box, and

then store that graphic into a graphics variable.

1. In the Sketch view, display the sketch you want to

copy (store into a variable).

2. Press

3. Highlight the variable name you want to use and

press

4. Draw a box around the portion you want to copy:

move the cursor to one corner, press , then move

the cursor to the opposite corner, and press

Notes and sketches

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To import a

graphics variable

You can copy the contents of a graphics variable into the

Sketch view of an aplet.

1. Open the Sketch view of the aplet (

The graphic will be copied here.

SKETCH).

2. Press

3. Highlight Graphic, then press

and highlight the

name of the variable (G1, etc.).

4. Press

variable.

to recall the contents of the graphics

5. Move the box to where you would like to copy the

graphic, then press

The notepad

Subject to available memory, you can store as many

notes as you want in the Notepad (

NOTEPAD).

These notes are independent of any aplet. The Notepad

catalog lists the existing entries by name. It does not

include notes that were created in aplets’ Note views, but

these can be imported. See “To import a note” on

page 20-8.

To create a note in

the Notepad

1. Display the Notepad

catalog.

NOTEPAD

2. Create a new note.

3. Enter a name for your

note.

MYNOTE

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Notes and sketches

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4. Write your note.

See “Note edit keys”

on page 20-2 for more

information on the

entry and editing of

notes.

5. When you are finished, press

or an aplet key

to exit Notepad. Your work is automatically saved.

Notepad Catalog keys

Key

Meaning

Opens the selected note for

editing.

Begins a new note, and asks

for a name.

Transmits the selected note to

another HP 40gs or PC.

Receives a note being

transmitted from another HP

40gs or PC.

Deletes the selected note.

CLEAR

Deletes all notes in the

catalog.

Notes and sketches

20-7

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To import a note

You can import a note from the Notepad into an aplet’s

Note view, and vice versa. Suppose you want to copy a

note named “Assignments” from the Notepad into the

Function Note view:

1. In the Function aplet, display the Note view

(

NOTE).

2. Press

, highlight Notepadin the left

column, then highlight the name “Assignments” in the

right column.

3. Press

to copy the contents of

“Assignments” to the Function Note view.

Note: To recall the name instead of the contents,

press

instead of

Suppose you want to copy the Note view from the current

aplet into the note, Assignments, in the Notepad.

1. In the Notepad (

“Assignments”.

NOTEPAD), open the note,

2. Press

, highlight Notein the left

column, then press

the right column.

and highlight NoteTextin

3. Press

to recall the contents of the Note

view into the note “Assignments”.

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Programming

Introduction

This chapter describes how to program using the HP

40gs. In this chapter you’ll learn about:

•

using the Program catalog to create and edit

programs

•

programming commands

storing and retrieving variables in programs

programming variables.

H I N T

More information on programming, including examples

and special tools, can be found at HP’s calculators web

site:

http://www.hp.com/calculators

The Contents of a

Program

An HP 40gs program contains a sequence of numbers,

mathematical expressions, and commands that execute

automatically to perform a task.

These items are separated by a colon ( : ). Commands

that take multiple arguments have those arguments

separated by a semicolon ( ; ). For example,

PIXON xposition;yposition:

Structured

Programming

Inside a program you can use branching structures to

control the execution flow. You can take advantage of

structured programming by creating building-block

programs. Each building-block program stands

alone—and it can be called from other programs. Note:

If a program has a space in its name then you have to put

quotes around it when you want to run it.

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Example

RUN GETVALUE: RUN CALCULATE: RUN

"SHOW ANSWER":

This program is separated into three main tasks, each an

individual program. Within each program, the task can

be simple—or it can be divided further into other

programs that perform smaller tasks.

Program catalog

The Program catalog is where you create, edit, delete,

send, receive, or run programs. This section describes

how to

•

open the Program catalog

create a new program

•

enter commands from the program commands menu

enter functions from the MATH menu

edit a program

run and debug a program

stop a program

copy a program

send and receive a program

delete a program or its contents

customize an aplet.

Open Program

Catalog

1. Press

PROGRM.

The Program Catalog displays a list of program

names. The Program Catalog contains a built-in entry

called Editline.

Editlinecontains the last expression that you

entered from the edit line in HOME, or the last data

you entered in an input form. (If you press

from HOME without entering any data, the HP 40gs

runs the contents of Editline.)

Before starting to work with programs, you should

take a few minutes to become familiar with the

Program catalog menu keys. You can use any of the

following keys (both menu and keyboard), to perform

tasks in the Program catalog.

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Program catalog keys

The program catalog keys are:

Key

Meaning

Opens the highlighted program

for editing.

Prompts for a new program

name, then opens an empty

program.

Transmits the highlighted

program to another HP 40gs or to

a disk drive.

Receives the highlighted program

from another HP 40gs or from a

disk drive.

Runs the highlighted program.

Moves to the beginning or end of

the Program catalog.

Deletes the highlighted program.

CLEAR

Deletes all programs in the

program catalog.

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Creating and editing programs

Create a new

program

1. Press

2. Press

PROGRM to open the Program catalog.

The HP 40gs prompts

you for a name.

A program name can contain special characters,

such as a space. However, if you use special

characters and then run the program by typing it in

HOME, you must enclose the program name in

double quotes (" "). Don't use the " symbol within your

program name.

3. Type your program

name, then press

When you press

the Program Editor

opens.

4. Enter your program. When done, start any other

activity. Your work is saved automatically.

Entercommands

Until you become familiar with the HP 40gs commands,

the easiest way to enter commands is to select them from

the Commands menu from the Program editor. You can

also type in commands using alpha characters.

1. From the Program editor, press

the Program Commands menu.

CMDS to open

CMDS

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2. On the left, use

category, then press

to highlight a command

to access the commands in

the category. Select the command that you want.

3. Press

to paste the command into the program

editor.

Edit a program

1. Press

PROGRM to

open the Program

catalog.

2. Use the arrow keys to highlight the program you want

to edit, and press . The HP 40gs opens the

Program Editor. The name of your program appears

in the title bar of the display. You can use the

following keys to edit your program.

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Editing keys

The editing keys are:

Key Meaning

Inserts the

editing point.

character at the

Inserts space into text.

Displays previous page of the

program.

Displays next page of the program.

Moves up or down one line.

Moves right or left one character.

Alpha-lock for letter entry. Press

A...Z to lock lower case.

Backspaces cursor and deletes

character.

Deletes current character.

Starts a new line.

CLEAR

Erases the entire program.

Displays menus for selecting variable

names, contents of variables, math

functions, and program constants.

CMDS

Displays menus for selecting program

conmmands.

CHARS

Displays all characters. To type one,

highlight it and press

To enter several characters in a row,

use the

CHARS menu.

menu key while in the

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Using programs

Run a program

From HOME, type RUN program_name.

From the Program catalog, highlight the program you

want to run and press

Regardless of where you start the program, all programs

run in HOME. What you see will differ slightly depending

on where you started the program. If you start the

program from HOME, the HP 40gs displays the contents

of Ans (Home variable containing the last result), when

the program has finished. If you start the program from

the Program catalog, the HP 40gs returns you to the

Program catalog when the program ends.

Debug a

program

If you run a program that contains errors, the program

will stop and you will see an error message.

To debug the program:

1. Press

to edit the program.

The insert cursor appears in the program at the point

where the error occurred.

2. Edit the program to fix the error.

3. Run the program.

4. Repeat the process until you correct all errors.

Stop a program

You can stop the running of a program at any time by

pressing CANCEL (the

key). Note: You may have to

press it a couple of times.

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Copy a program You can use the following procedure if you want to make

a copy of your work before editing—or if you want to use

one program as a template for another.

1. Press

2. Press

PROGRM to open the Program catalog.

3. Type a new file name, then choose

The Program Editor opens with a new program.

4. Press

to open the variables menu.

to quickly scroll to Program.

5. Press

6. Press

copy.

, then highlight the program you want to

7. Press

, then press

The contents of the highlighted program are copied

into the current program at the cursor location.

H I N T

If you use a programming routine often, save the routine

under a different program name, then use the above

method to copy it into your programs.

Transmit a

program

You can send programs to, and receive programs from,

other calculators just as you can send and receive aplets,

matrices, lists, and notes.

After connecting the calculators with an appropriate

cable, open the Program catalogs on both calculators.

Highlight the program to send, then press

on the

sending calculator and on the receiving calculator.

You can also send programs to, and receive programs

from, a remote storage device (aplet disk drive or

computer). This takes place via a cable connection and

requires an aplet disk drive or specialized software

running on a PC (such as a connectivity kit).

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Delete a

program

To delete a program:

1. Press

PROGRM to open the Program catalog.

2. Highlight a program to delete, then press

Delete all

programs

You can delete all programs at once.

1. In the Program catalog, press

2. Press

CLEAR.

Delete the

contents of a

program

You can clear the contents of a program without deleting

the program name.

1. Press

2. Highlight a program, then press

3. Press CLEAR, then press

PROGRM to open the Program catalog.

4. The contents of the program are deleted, but the

program name remains.

Customizing an aplet

You can customize an aplet and develop a set of

programs to work with the aplet.

Use the SETVIEWS command to create a custom VIEWS

menu which links specially written programs to the new

aplet.

A useful method for customizing an aplet is illustrated

below:

1. Decide on the built-in aplet that you want to

customize. For example you could customize the

Function aplet or the Statistics aplet. The customized

aplet inherits all the properties of the built-in aplet.

Save the customized aplet with a unique name.

2. Customize the new aplet if you need to, for example

by presetting axes or angle measures.

3. Develop the programs to work with your customized

aplet. When you develop the aplet’s programs, use

the standard aplet naming convention. This allows

you to keep track of the programs in the Program

catalog that belong to each aplet. See “Aplet naming

convention” on page 21-10.

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4. Develop a program that uses the SETVIEWS

command to modify the aplet’s VIEWS menu. The

menu options provide links to associated programs.

You can specify any other programs that you want

transferred with the aplet. See “SETVIEWS” on page

21-14 for information on the command.

5. Ensure that the customized aplet is selected, then run

the menu configuration program to configure the

aplet’s VIEWS menu.

6. Test the customized aplet and debug the associated

programs. (Refer to “Debug a program” on page

16-7).

Aplet naming convention

To assist users in keeping track of aplets and associated

programs, use the following naming convention when

setting up an aplet’s programs:

•

Start all program names with an abbreviation of the

aplet name. We will use APL in this example.

•

Name programs called by menu entries in the VIEWS

menu number, after the entry, for example:

–

APL.ME1 for the program called by menu option

–

APL.ME2 for the program called by menu option

•

Name the program that configures the new VIEWS

menu option APL.SV where SV stands for SETVIEWS.

For example, a customized aplet called “Differentiation”

might call programs called DIFF.ME1, DIFF.ME2, and

DIFF.SV.

Example

This example aplet is designed to demonstrate the

process of customizing an aplet. The new aplet is based

on the Function aplet. Note: This aplet is not intended to

serve a serious use, merely to illustrate the process.

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Save the aplet

1. Open the Function aplet and save it as

“EXPERIMENT”. The new aplet appears in the Aplet

library.

Select

Function

EXPERIMENT

2. Create a program

called EXP.ME1 with

contents as shown. This

program configures the

plot ranges, then runs a

program that allows

you to set the angle format.

3. Create a program

called EXP.ME2 with

contents as shown. This

program sets the

numeric view options

for the aplet, and runs

the program that you can use to configure the angle

mode.

4. Create a program

called EXP.ANG which

the previous two

programs call.

5. Create a program

called EXP.S which runs

when you start the

aplet, as shown. This

program sets the angle

mode to degrees, and

sets up the initial function that the aplet plots.

Configuring the

Setviews menu

option

In this section we will begin by configuring the

VIEWS menu by using the SETVIEWS command. We

will then create the “helper” programs called by the

VIEWS menu which will do the actual work.

programs

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6. Open the Program catalog and create a program

named “EXP.SV”. Include the following code in the

program.

Each entry line after the command SETVIEWS is a trio

that consists of a VIEWS menu text line (a space

indicates none), a program name, and a number that

defines the view to go to after the program has run its

course. All programs listed here will transfer with an

aplet when the aplet is transferred.

SETVIEWS ’’’’;’’’’;18;

Sets the first menu option to be “Auto

scale”. This is the fourth standard Function

aplet view menu option and the 18 “Auto

scale”, specifies that it is to be included in

the new menu. The empty quotes will

ensure that the old name of “Auto scale”

appears on the new menu. See

“SETVIEWS” on page 21-14.

’’My Entry1’’;’’EXP.ME1’’;1;

Sets the second menu option. This option

runs program EXP.ME1, then returns to

view 1, Plot view.

’’My Entry2’’;’’EXP.ME2’’;3;

Sets the third menu option. This option

runs the program EXP.ME2, then returns to

view 3, the NUM view.

’’’’;’’EXP.SV’’;0;

This line specifies that the program to set

the View menu (this program) is

transferred with the aplet. The space

character between the first set of quotes in

the trio specifies that no menu option

appears for the entry. You do not need to

transfer this program with the aplet, but it

allows users to modify the aplet’s menu if

they want to.

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’’’’;’’EXP.ANG’’;0;

The program EXP.ANG is a small routine

that is called by other programs that the

aplet uses. This entry specifies that the

program EXP.ANGis transferred when the

aplet is transferred, but the space in the

first quotes ensures that no entry appears

on the menu.

’’Start’’;’’EXP.S’’;7:

This specifies the Start menu option. The

program that is associated with this entry,

EXP.S, runs automatically when you

start the aplet. Because this menu option

specifies view 7, the VIEWS menu opens

when you start the aplet.

You only need to run this program once to configure

your aplet’s VIEWS menu. Once the aplet’s VIEWS

menu is configured, it remains that way until you run

SETVIEWS again.

You do not need to include this program for your

aplet to work, but it is useful to specify that the

program is attached to the aplet, and transmitted

when the aplet is transmitted.

7. Return to the program

catalog. The programs

that you created should

appear as follows:

8. You must now

the

program EXP.SV to execute the SETVIEWS command

and create the modified VIEWS menu. Check that the

name of the new aplet is highlighted in the Aplet

view.

9. You can now return to the Aplet library and press

to run your new aplet.

Programming commands

This section describes the commands for programming

with HP 40gs. You can enter these commands in your

program by typing them or by accessing them from the

Commands menu.

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Aplet commands

CHECK

Checks (selects) the corresponding function in the current

aplet. For example, Check 3 would check F3 if the current

aplet is Function. Then a checkmark would appear next

to F3 in Symbolic view, F3 would be plotted in Plot view,

and evaluated in Numeric view.

CHECKn:

SELECT

Selects the named aplet and makes it the current aplet.

Note: Quotes are needed if the name contains spaces or

other special characters.

SELECTapletname:

SETVIEWS

The SETVIEWS command is used to define entries in the

VIEWS menu for aplets that you customize. See

“Customizing an aplet” on page 21-9 for an example of

using the SETVIEWS command.

When you use the SETVIEWS command, the aplet’s

standard VIEWS menu is deleted and the customized

menu is used in its place. You only need to apply the

command to an aplet once. The VIEWS menu changes

remain unless you apply the command again.

Typically, you develop a program that uses the

SETVIEWS command only. The command contains a trio

of arguments for each menu option to create, or program

to attach. Keep the following points in mind when using

this command:

•

The SETVIEWS command deletes an aplet’s standard

Views menu options. If you want to use any of the

standard options on your reconfigured VIEWS menu,

you must include them in the configuration.

•

When you invoke the SETVIEWS command, the

changes to an aplet’s VIEWS menu remain with the

aplet. You need to invoke the command on the aplet

again to change the VIEWS menu.

•

All the programs that are called from the VIEWS

menu are transferred when the aplet is transferred, for

example to another calculator or to a PC.

As part of the VIEWS menu configuration, you can

specify programs that you want transferred with the

aplet, but are not called as menu options. For

example, these can be sub-programs that menu

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options use, or the program that defines the aplet’s

VIEWS menu.

•

You can include a “Start” option in the VIEWS menu

to specify a program that you want to run

automatically when the aplet starts. This program

typically sets up the aplet’s initial configuration. The

START option on the menu is also useful for resetting

the aplet.

Command syntax

The syntax for the command is as follows:

SETVIEWS

"Prompt1";"ProgramName1";ViewNumber1;

"Prompt2";"ProgramName2";ViewNumber2:

(You can repeat as many Prompt/ProgramName/

ViewNumber trios of arguments as you like.)

Within each Prompt/ProgramName/ViewNumber trio,

you separate each item with a semi-colon.

Prompt

Prompt is the text that is displayed for the corresponding

entry in the Views menu. Enclose the prompt text in

double quotes.

Associating programs with your aplet

If Prompt consists of a single space, then no entry appears

in the view menu. The program specified in the

ProgramName item is associated with the aplet and

transferred whenever the aplet is transmitted. Typically,

you do this if you want to transfer the Setviews program

with the aplet, or you want to transfer a sub-program that

other menu programs use.

Auto-run programs

If the Prompt item is “Start”, then the ProgramName

program runs whenever you start the aplet. This is useful

for setting up a program to configure the aplet. Users can

select the Start item from the VIEWS menu to reset the

aplet if they change configurations.

You can also define a menu item called “Reset” which is

auto-run if the user chooses the

view.

button in the APLET

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ProgramName

ProgramName is the name of the program that runs when

the corresponding menu entry is selected. All programs

that are identified in the aplet’s SETVIEWS command are

transferred when the aplet is transmitted.

ViewNumber

ViewNumber is the number of a view to start after the

program finishes running. For example, if you want the

menu option to display the Plot view when the associated

program finishes, you would specify 1 as the

ViewNumber value.

Including standard menu options

To include one of an aplet’s standard VIEWS menu

options in your customized menu, set up the arguments

trio as follows:

•

The first argument specifies the menu item name:

–

Leave the argument empty to use the standard

Views menu name for the item, or

–

Enter a menu item name to replace the standard

name.

The second argument specifies the program to run:

–

Leave the argument empty to run the standard

menu option.

–

Insert a program name to run the program before

the standard menu option is executed.

The third argument specifies the view and the menu

number for the item. Determine the menu number

from the View numbers table below.

Note: SETVIEWS with no arguments resets the views

to default of the base aplet.

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View numbers

The Function aplet views are numbered as follows:

HOME

List Catalog

Matrix Catalog

Notepad Catalog

Program Catalog

Plot-Detail

Plot

Symbolic

Numeric

Plot-Setup

Symbolic-Setup

Numeric-Setup

Views

Plot-Table

Overlay Plot

Auto scale

Decimal

Note

Sketch view

Aplet Catalog

Integer

Trig

View numbers from 15 on will vary according to the

parent aplet. The list shown above is for the Function

aplet. Whatever the normal VIEWS menu for the parent

aplet, the first entry will become number 15, the second

number 16 and so on.

UNCHECK

Unchecks (unselects) the corresponding function in the

current aplet. For example, Uncheck 3 would uncheck F3

if the current aplet is Function.

UNCHECKn:

Branch commands

Branch commands let a program make a decision based

on the result of one or more tests. Unlike the other

programming commands, the branch commands work in

logical groups. Therefore, the commands are described

together rather than each independently.

IF...THEN...END

Executes a sequence of commands in the true-clause only

if the test-clause evaluates to true. Its syntax is:

IFtest-clause

THENtrue-clause END

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Example

1XA :

IF A==1

THEN MSGBOX " A EQUALS 1" :

END:

IF... THEN... ELSE...

END

Executes the true-clause sequence of commands if the test-

clause is true, or the false-clause sequence of commands

if the test-clause is false.

IF test-clause

THEN true-clause ELSE false-clause END

Example

1XA :

IF A==1 THEN

MSGBOX "A EQUALS 1" :

ELSE

MSGBOX "A IS NOT EQUAL TO 1" :

A+1XA :

END:

CASE...END

Executes a series of test-clause commands that execute

the appropriate true-clause sequence of commands. Its

syntax is:

CASE

IF test-clause THEN true-clause END

END:

When CASE is executed, test-clause is evaluated. If the

test is true, true-clause is executed, and execution skips

to END. If test-clause if false, execution proceeds to test-

clause . Execution with the CASE structure continues until

a true-clause is executed (or until all the test-clauses

evaluate to false).

IFERR...

THEN...

ELSE…

END...

Many conditions are automatically recognized by the HP

40gs as error conditions and are automatically treated as

errors in programs.

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IFERR...THEN...ELSE...END allows a program to intercept

error conditions that otherwise would cause the program

to abort. Its syntax is:

IFERR trap-clause

THEN clause_1

ELSE clause_2

END:

Example

IFERR

60/X X Y:

THEN

MSGBOX "Error: X is zero.":

ELSE

MSGBOX "Value is "Y:

END:

RUN

Runs the named program. If your program name contains

special characters, such as a space, then you must

enclose the file name in double quotes (" ").

RUN"program name":or RUNprogramname:

STOP

Stops the current program.

STOP:

Drawing commands

The drawing commands act on the display. The scale of

the display depends on the current aplet's Xmin, Xmax,

Ymin, and Ymax values. The following examples assume

the HP 40gs default settings with the Function aplet as the

current aplet.

ARC

Draws a circular arc, of given radius, whose centre is at

(x,y) The arc is drawn from start_angle_measurement to

end_angle_measurement.

ARCx;y;radius;start_angle_measurement;

end_angle_measurement:

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Example

ARC0;0;2;0;2π:

FREEZE:

Draws a circle centered

at (0,0) of radius 2. The

FREEZE command

causes the circle to

remain displayed on the screen until you press a key.

BOX

Draws a box with diagonally opposite corners (x1,y1) and

(x2,y2).

BOXx1;y1;x2;y2:

Example

BOX -1;-1;1;1:

FREEZE:

Draws a box, lower

corner at (–1,–1),

upper corner at (1,1)

ERASE

Clears the display

ERASE:

FREEZE

LINE

Halts the program, freezing the current display.

Execution resumes when any key is pressed.

Draws a line from (x1, y1) to (x2, y2).

LINEx1;y1;x2;y2:

PIXOFF

PIXON

TLINE

Turns off the pixel at the specified coordinates (x,y).

PIXOFFx;y:

Turns on the pixel at the specified coordinates (x,y).

PIXONx;y:

Toggles the pixels along the line from (x1, y1) to (x2, y2)

on and off. Any pixel that was turned off, is turned on;

any pixel that was turned on, is turned off. TLINE can be

used to erase a line.

TLINEx1;y1;x2;y2:

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Example

TLINE 0;0;3;3:

Erases previously drawn 45 degree line from (0,0) to

(3,3), or draws that line if it doesn’t already exist.

Graphic commands

The graphic commands use the graphics variables G0

through G9—or the Page variable from Sketch—as

graphicname arguments. The position argument takes the

form (x,y). Position coordinates depend on the current

aplet’s scale, which is specified by Xmin, Xmax, Ymin,

and Ymax. The upper left corner of the target graphic

(graphic2) is at (Xmin,Ymax).

You can capture the current display and store it in G0 by

simultaneously pressing

DISPLAY→

Stores the current display in graphicname.

DISPLAY→ graphicname:

→DISPLAY

Displays graphic from graphicname in the display.

→DISPLAY graphicname:

→GROB

Creates a graphic from expression, using font_size, and

stores the resulting graphic in graphicname. Font sizes

are 1, 2, or 3. If the fontsize argument is 0, the HP 40gs

creates a graphic display like that created by the SHOW

operation.

→GROB graphicname;expression;fontsize:

GROBNOT

GROBOR

Replaces graphic in graphicname with bitwise-inverted

graphic.

GROBNOT graphicname:

Using the logical OR, superimposes graphicname2 onto

graphicname1. The upper left corner of graphicname2 is

placed at position.

GROBORgraphicname1;(position);graphicname2:

where position is expressed in terms of the current axes

settings, not in terms of pixel postion.

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GROBXOR

Using the logical XOR, superimposes graphicname2 onto

graphicname1. The upper left corner of graphicname2 is

placed at position.

GROBXOR

graphicname1;(position);graphicname2:

MAKEGROB

Creates graphic with given width, height, and

hexadecimal data, and stores it in graphicname.

MAKEGROB graphicname;width;height;hexdata:

PLOT→

Stores the Plot view display as a graphic in graphicname.

PLOT→ graphicname:

PLOT→ and DISPLAY→ can be used to transfer a copy

of the current PLOT view into the sketch view of the aplet

for later use and editing.

Example

XPageNum:

PLOT→ Page:

→DISPLAYPage:

FREEZE:

This program stores the current PLOT view to the first page

in the sketch view of the current aplet and then displays

the sketch as a graphic object until any key is pressed.

→PLOT

Puts graph from graphicname into the Plot view display.

→PLOTgraphicname:

REPLACE

Replaces portion of graphic in graphicname1 with

graphicname2,starting at position.REPLACEalso

works for lists and matrices.

REPLACE

graphicname1;(position);graphicname2:

SUB

Extracts a portion of the named graphic (or list or matrix),

and stores it in a new variable, name. The portion is

specified by position and positions.

SUBname;graphicname;(position);(positions):

ZEROGROB

Creates a blank graphic with given width and height,

and stores it in graphicname.

ZEROGROB graphicname;width;height:

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Loop commands

Loop hp allow a program to execute a routine repeatedly.

The HP 40gs has three loop structures. The example

programs below illustrate each of these structures

incrementing the variable A from 1 to 12.

DO…UNTIL …END Do... Until... Endis a loop command that executes the

loop-clause repeatedly until test-clause returns a true

(nonzero) result. Because the test is executed after the

loop-clause, the loop-clause is always executed at least

once. Its syntax is:

DO loop-clause UNTIL test-clause END

X A:

A + 1 X A:

DISP 3;A:

UNTIL A == 12 END:

WHILE…

REPEAT…

END

While... Repeat... Endis a loop command that

repeatedly evaluates test-clause and executes loop-clause

sequence if the test is true. Because the test-clause is

executed before the loop-clause, the loop-clause is not

executed if the test is initially false. Its syntax is:

WHILEtest-clause REPEAT loop-clause END

1X A:

WHILE A < 12 REPEAT

A+1 X A:

DISP 3;A:

END:

FOR…TO…STEP

...END

FOR name=start-expression TO end-expression

[STEP increment]; loop-clause END

FOR A=1 TO 12 STEP 1;

DISP 3;A:

END:

Note that the STEP parameter is optional. If it is omitted,

a step value of 1 is assumed.

BREAK

Terminates loop.

BREAK:

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Matrix commands

The matrix commands take variables M0–M9 as

arguments.

ADDCOL

Add Column. Inserts values into a column before

column_number in the specified matrix. You enter the

values as a vector. The values must be separated by

commas and the number of values must be the same as

the number of rows in the matrix name.

ADDCOL

name;[value ,...,value ];column_number:

ADDROW

Add Row. Inserts values into a row before row_number in

the specified matrix. You enter the values as a vector. The

values must be separated by commas and the number of

values must be the same as the number of columns in the

matrix name.

ADDROW name;[value ,..., value ];row_number:

DELCOL

Delete Column. Deletes the specified column from the

specified matrix.

DELCOL name;column_number:

DELROW

EDITMAT

Delete Row. Deletes the specified row from the specified

matrix.

DELROWname;row_number:

Starts the Matrix Editor and displays the specified matrix.

If used in programming, returns to the program when user

presses

EDITMATname:

RANDMAT

Creates random matrix with a specified number of rows

and columns and stores the result in name

(name must be M0...M9). The entries will be integers

ranging from –9 to 9.

RANDMATname;rows;columns:

REDIM

Redimensions the specified matrix or vector to size. For a

matrix, size is a list of two integers {n1,n2}. For a vector,

size is a list containing one integer {n}.

REDIMname;size:

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REPLACE

Replaces portion of a matrix or vector stored in name with

an object starting at position start. start for a matrix is a

list containing two numbers; for a vector, it is a single

number. Replace also works with lists and graphics.

REPLACEname;start;object:

SCALE

Multiplies the specified row_number of the specified

matrix by value.

SCALEname;value;rownumber:

SCALEADD

SUB

Multiplies the row of the matrix name by value, then adds

this result to the second specified row.

SCALEADDname;value;row1;row2:

Extracts a sub-object—a portion of a list, matrix, or

graphic from object—and stores it into name. start and

end are each specified using a list with two numbers for

a matrix, a number for vector or lists, or an ordered pair,

(X,Y), for graphics.

SUBname;object;start;end:

SWAPCOL

Swaps Columns. Exchanges column1 and column2 of the

specified matrix.

SWAPCOL name;column1;column2:

SWAPROW

Swap Rows. Exchanges row1 and row2 in the specified

matrix.

SWAPROWname;row1;row2:

Print commands

These commands print to an HP infrared printer, for

example the HP 82240B printer.

PRDISPLAY

PRHISTORY

Prints the contents of the display.

PRDISPLAY:

Prints all objects in the history.

PRHISTORY:

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PRVAR

Prints name and contents of variablename.

PRVARvariablename:

You can also use the PRVAR command to print the

contents of a program or a note.

PRVARprogramname;PROG:

PRVARnotename;NOTE:

Prompt commands

BEEP

Beeps at the frequency and for the time you specify.

BEEPfrequency;seconds:

CHOOSE

Creates a choose box, which is a box containing a list of

options from which the user chooses one. Each option is

numbered, 1 through n. The result of the choose

command is to store the number of the option chosen in a

variable. The syntax is:

CHOOSEvariable_name;title;option ;option ;

...option :

where variable_name is the name of a variable for

storing a default option number, title is the text displayed

in the title bar of the choose box, and option ...option

are the options listed in the choose box.

By pre-storing a value into variable_name you can

specify the default option number, as shown in the

example below.

Example

X A:CHOOSE A;

"COMIC STRIPS";

"DILBERT";

"CALVIN&HOBBES";

"BLONDIE":

CLRVAR

Clears the specified variable. The syntax is:

CLRVAR variable :

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Example

If you have stored

{1,2,3,4} in variable L1,

entering CLVAR L1

will clear L1.

DISP

Displays textitem in a row of the display at the

line_number. A text item consists of any number of

expressions and quoted strings of text. The expressions

are evaluated and turned into strings. Lines are numbered

from the top of the screen, 1 being the top and 7 being

the bottom.

DISP line_number;textitem:

Example

DISP 3;"A is" 2+2

Result: A is 4

(displayed on line 3)

DISPXY

Displays object at position (x_pos, y_pos) in size font. The

syntax is:

DISPXY x_pos;y_pos;font;object:

The value of object can be a text string, a variable, or a

combination of both. x_pos and y_pos are relative to the

current settings of Xmin, Xmax, Ymin and Ymax (which

you set in the PLOT SETUP view). The value of font is either

1 (small) or 2 (large).

Example

DISPXY

–3.5;1.5;2;"HELLO

WORLD":

DISPTIME

Displays the current date and time.

DISPTIME

To set the date and time, simply store the correct settings

in the date and time variables. Use the following formats:

M.DDYYYY for the date and H.MMSSfor the time.

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Examples

5.152000X DATE(sets the date to May 15, 2000).

10.1500X TIME(sets the time to 10:15 am).

EDITMAT

Matrix Editor. Opens the Matrix editor for the specified

matrix. Returns to the program when user presses

EDITMAT matrixname:

The EDITMAT command can also be used to create

matrices.

1. Press

2. Press

CMDS

M 1, and then press

The Matrix catalog opens with M1 available for

editing.

EDITMAT matrixname is an alternative to opening the

matrix editor with matrixname. It can be used in a

program to enter a matrix.

FREEZE

This command prevents the display from being updated

after the program runs. This allows you to view the

graphics created by the program. Cancel FREEZEby

pressing any key.

FREEZE:

GETKEY

Waits for a key, then stores the keycode rc.p in name,

where r is row number, c is column number, and p is key-

plane number. The key-planes numbers are: 1 for

unshifted; 2 for shifted; 4 for alpha-shifted; and 5 for both

alpha-shifted and shifted.

GETKEYname:

INPUT

Creates an input form with a title bar and one field. The

field has a label and a default value. There is text help at

the bottom of the form. The user enters a value and

presses the

menu key. The value that the user enters

is stored in the variable name. The title, label, and help

items are text strings and need to be enclosed in double

quotes.

Use

CHARS to type the quote marks " ".

INPUTname;title,label;help;default:

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Example

INPUT R; "Circular Area";

"Radius";

"Enter Number";1:

MSGBOX

Displays a message box containing textitem. A text item

consists of any number of expressions and quoted strings

of text. The expressions are evaluated and turned into

strings of text.

For example, "AREA IS:"2+2becomes AREA IS:4.

Use

CHARS to type the quote marks " ".

MSGBOXtextitem:

Example

1X A:

MSGBOX "AREA IS: "π*A^2:

You can also use the NoteText variable to provide text

arguments. This can be used to insert line breaks. For

example, press

NOTE and type AREAIS

The position line

MSGBOXNoteText " " π*A^2:

will display the same message box as the previous

example.

PROMPT

WAIT

Displays an input box with name as the title, and prompts

for a value for name. name can be a variable such as

A…Z, θ, L1…L9, C1…C9 or Z1…Z9..

PROMPTname:

Halts program execution for the specified number of

seconds.

WAITseconds:

Stat-One and Stat-Two commands

The following commands are used for analyzing one-

variable and two-variable statistical data.

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Stat-One commands

DO1VSTATS

Calculates STATS using datasetname and stores the

results in the corresponding variables: NΣ, TotΣ, MeanΣ,

PVarΣ, SVarΣ, PSDev, SSDev, MinΣ, Q1, Median, Q3,

and MaxΣ. Datasetname can be H1, H2, ..., or H5.

Datasetname must include at least two data points.

DO1VSTATSdatasetname:

SETFREQ

Sets datasetname frequency according to column or

value. Datasetname can be H1, H2,..., or H5, column

can be C0–C9 and value can be any positive integer.

SETFREQdatasetname;column:

SETFREQdefinition;value:

SETSAMPLE

Sets datasetname sample according to column.

Datasetname can be H1–H5, and column can be

CO–C9.

SETSAMPLEdatasetname;column:

Stat-Two commands

DO2VSTATS

Calculates STATS using datasetname and stores the

results in corresponding variables: MeanX, ΣX, ΣX2,

MeanY, ΣY, ΣY2, ΣXY, Corr, PCov, SCov, and RELERR.

Datasetname can be SI, S2,..., or S5. Datasetname must

include at least two pairs of data points.

DO2VSTATSdatasetname:

SETDEPEND

SETINDEP

Sets datasetname dependent column. Datasetname can

be S1, S2, …, or S5 and column can be C0–C9.

SETDEPENDdatasetname;column:

Sets datasetname independent column. Datasetname can

be S1, S2,…, or S5 and column can be C0–C9.

SETINDEPdatasetname;column:

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Storing and retrieving variables in programs

The HP 40gs has both Home variables and Aplet

variables. Home variables are used for real numbers,

complex numbers, graphics, lists, and matrices. Home

variables keep the same values in HOME and in aplets.

Aplet variables are those whose values depend on the

current aplet. The aplet variables are used in

programming to emulate the definitions and settings you

make when working with aplets interactively.

You use the Variable menu (

) to retrieve either

Home variables or aplet variables. See “The VARS menu”

on page 17-4. Not all variables are available in every

aplet. S1fit–S5fit, for example, are only available in the

Statistics aplet. Under each variable name is a list of the

aplets where the variable can be used.

Plot-view variables

Area

Function

Contains the last value found by the Area function in Plot-

FCN menu.

Axes

All Aplets

Turns axes on or off.

From Plot Setup, check (or uncheck) AXES.

In a program, type:

X Axes—to turn axes on (default).

X Axes—to turn axes off.

Connect

Function

Parametric

Polar

Draws lines between successively plotted points.

From Plot Setup, check (or uncheck) CONNECT.

Solve

In a program, type

Statistics

X Connect—to connect plotted points (default,

except in Statistics where the default is off).

X Connect—not to connect plotted points.

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Coord

Turns the coordinate-display mode in Plot view on or off.

Function

Parametric

Polar

Sequence

Solve

From Plot view, use the Menu mean key to toggle

coordinate display on an off.

In a program, type

X Coord—to turn coordinate display on (default).

X Coord—to turn coordinate display off.

Statistics

Extremum

Function

Contains the last value found by the Extremum operation

in the Plot-FCN menu.

FastRes

Function

Solve

Toggles resolution between plotting in every other column

(faster), or plotting in every column (more detail).

From Plot Setup, choose Faster or More Detail.

In a program, type

X FastRes—for faster.

X FastRes—for more detail (default).

Grid

All Aplets

Turns the background grid in Plot view on or off. From Plot

setup, check (or uncheck) GRID.

In a program, type

X Gridto turn the grid on.

X Gridto turn the grid off (default).

Hmin/Hmax

Statistics

Defines minimum and maximum values for histogram

bars.

From Plot Setup for one-variable statistics, set values for

HRNG.

In a program, type

n₁X Hmin

n₂X Hmax

where n₂> n₁

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Hwidth

Statistics

Sets the width of histogram bars.

From Plot Setup in 1VAR stats set a value for Hwidth

In a program, type

nX Hwidth

Indep

All Aplets

Defines the value of the independent variable used in

tracing mode.

In a program, type

nX Indep

InvCross

All Aplets

Toggles between solid crosshairs or inverted crosshairs.

(Inverted is useful if the background is solid).

From Plot Setup, check (or uncheck) InvCross

In a program, type:

X InvCross—to invert the crosshairs.

X InvCross—for solid crosshairs (default).

Isect

Function

Contains the last value found by the Intersection function

in the Plot-FCN menu.

Labels

All Aplets

Draws labels in Plot view showing X and Y ranges.

From Plot Setup, check (or uncheck) Labels

In a program, type

XLabels—to turn labels on.

XLabels—to turn labels off (default).

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Nmin / Nmax

Sequence

Defines the minimum and maximum independent variable

values. Appears as the NRNGfields in the Plot Setup input

form.

From Plot Setup, enter values for NRNG.

In a program, type

n₁XNmin

n₂XNmax

where n₂> n₁

Recenter

All Aplets

Recenters at the crosshairs locations when zooming.

From Plot-Zoom-Set Factors, check (or uncheck)

Recenter

In a program, type

X Recenter— to turn recenter on (default).

X Recenter—to turn recenter off.

Root

Function

Contains the last value found by the Root function in the

Plot-FCN menu.

S1mark–S5mark

Statistics

Sets the mark to use for scatter plots.

From Plot Setup for two-variable statistics, S1mark-

S5mark, then choose a mark.

In a program, type

X S1mark

where n is 1,2,3,...5

SeqPlot

Sequence

Enables you to choose types of sequence plot: Stairstep

or Cobweb.

From Plot Setup, select SeqPlot, then choose

Stairstepor Cobweb.

In a program, type

X SeqPlot—for Stairstep.

2X SeqPlot—for Cobweb.

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Simult

Enables you to choose between simultaneous and

sequential graphing of all selected expressions.

Function

Parametric

Polar

From Plot Setup, check (or uncheck) _SIMULT

Sequence

In a program, type

X Simult—for simultaneous graphing (default).

X Simult—for sequential graphing.

Slope

Function

Contains the last value found by the Slope function in the

Plot-FCN menu.

StatPlot

Statistics

Enables you to choose types of 1-variable statistics plot

between Histogram or Box-and-Whisker.

From Plot Setup, select StatPlot, then choose

Histogramor BoxWhisker.

In a program, type

1X StatPlot—for Histogram.

2X StatPlot—for Box-and-Whisker.

Umin/Umax

Polar

Sets the minimum and maximum independent values.

Appears as the URNGfield in the Plot Setup input form.

From the Plot Setup input form, enter values for URNG.

In a program, type

n₁X Umin

n₂X Umax

where n₂> n₁

Ustep

Polar

Sets the step size for an independent variable.

From the Plot Setup input form, enter values for USTEP.

In a program, type

n ^XUstep

where n > 0

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Tmin / Tmax

Parametric

Sets the minimum and maximum independent variable

values. Appears as the TRNGfield in the Plot Setup input

form.

From Plot Setup, enter values for TRNG.

In a program, type

n₁X Tmin

n₂X Tmax

where n₂> n₁

Tracing

Turns the tracing mode on or off in Plot view.

In a program, type

All Aplets

X Tracing—to turn Tracing mode on (default).

X Tracing—to turn Tracing mode off.

Tstep

Sets the step size for the independent variable.

Parametric

From the Plot Setup input form, enter values for TSTEP.

In a program, type

n ^XTstep

where n > 0

Xcross

Sets the horizontal coordinate of the crosshairs. Only

works with TRACEoff.

All Aplets

In a program, type

n ^XXcross

Ycross

Sets the vertical coordinate of the crosshairs. Only works

with TRACEoff.

All Aplets

In a program, type

n X Ycross

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Xtick

Sets the distance between tick marks for the horizontal

axis.

AAll Aplets

From the Plot Setup input form, enter a value for Xtick.

In a program, type

n ^XXtick where n > 0

Ytick

All Aplets

Sets the distance between tick marks for the vertical axis.

From the Plot Setup input form, enter a value for Ytick.

In a program, type

n ^XYtick where n > 0

Xmin / Xmax

All Aplets

Sets the minimum and maximum horizontal values of the

plot screen. Appears as the XRNGfields (horizontal

range) in the Plot Setup input form.

From Plot Setup, enter values for XRNG.

In a program, type

n₁X Xmin

n₂X Xmax

where n₂> n₁

Ymin / Ymax

All Aplets

Sets the minimum and maximum vertical values of the plot

screen. Appears as the YRNGfields (vertical range) in the

Plot Setup input form.

From Plot Setup, enter the values for YRNG.

In a program, type

n₁X Ymin

n₂X Ymax

where n₂> n₁

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Xzoom

All Aplets

Sets the horizontal zoom factor.

From Plot-ZOOM-Set Factors, enter the value for XZOOM.

In a program, type

X XZOOM

where n > 0

The default value is 4.

Yzoom

All Aplets

Sets the vertical zoom factor.

From Plot-ZOOM-Set Factors, enter the value for YZOOM.

In a program, type

X YZOOM

The default value is 4.

Symbolic-view variables

Angle

All Aplets

Sets the angle mode.

From Symbolic Setup, choose Degrees, Radians, or

Gradsfor angle measure.

In a program, type

1X Angle—for Degrees.

2X Angle—for Radians.

3X Angle—for Grads.

F1...F9, F0

Function

Can contain any expression. Independent variable is X.

Example

'SIN(X)' X F1(X)

You must put single quotes around an expression to keep

it from being evaluated before it is stored. Use

CHARS to type the single quote mark.

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X1, Y1...X9,Y9

X0,Y0

Parametric

Can contain any expression. Independent variable is T.

Example

'SIN(4*T)' X Y1(T):'2*SIN(6*T)'

X1(T)

R1...R9, R0

Polar

Can contain any expression. Independent variable is θ.

Example

'2*SIN(2*θ)' X R1(θ)

U1...U9, U0

Sequence

Can contain any expression. Independent variable is N.

Example

RECURSE (U,U(N-1)*N,1,2) X U1(N)

E1...E9, E0

Solve

Can contain any equation or expression. Independent

variable is selected by highlighting it in Numeric View.

Example

'X+Y*X-2=Y' X E1

S1fit...S5fit

Statistics

Sets the type of fit to be used by the FIT operation in

drawing the regression line.

From Symbolic Setup view, specify the fit in the field for

S1FIT, S2FIT, etc.

In a program, store one of the following constant numbers

or names into a variable S1fit, S2fit, etc.

1 Linear

2 LogFit

3 ExpFit

4 Power

5 QuadFit

6 Cubic

7 Logist

8 ExptFit

9 TrigFit

10 User

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Example

Cubic X S2fit

X S2fit

Numeric-view variables

The following aplet variables control the Numeric view.

The value of the variable applies to the current aplet only.

C1...C9, C0

Statistics

C0through C9, for columns of data. Can contain lists.

Enter data in the Numeric view

In a program, type

LISTXCn

where n = 0, 1, 2, 3 ... 9

Digits

All Aplets

Number of decimal places to use for Number format in

the HOME view and for labeling axes in the Plot view.

From the Modes view, enter a value in the second field of

NumberFormat.

In a program, type

n ^XDigits

where 0<n <11

Format

All Aplets

Defines the number display format to use for numeric

format on the HOME view and for labeling axes in the

Plot view.

From the Modes view, choose Standard, Fixed,

Scientific, Engineering, Fraction or Mixed

Fraction in the Number Formatfield.

In a program, store the constant number (or its name) into

the variable Format.

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1 Standard

2 Fixed

3 Sci

4 Eng

5 Fraction

6 MixFraction

Note: if Fractionor Mixed Fractionis chosen, the

setting will be disregarded when labeling axes in the Plot

view. A setting of Scientificwill be used instead.

Example

ScientificX Format

X Format

NumCol

All Aplets except

Statistics aplet

Sets the column to be highlighted in Numeric view.

In a program, type

n ^XNumCol

where n can be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

NumFont

Function

Parametric

Polar

Sequence

Statistics

Enables you to choose the font size in Numeric view.

Does not appear in the Num Setup input form.

Corresponds to the

key in Numeric view.

In a program, type

X NumFontfor small (default).

X NumFontfor big.

NumIndep

Function

Parametric

Polar

Specifies the list of independent values to be used by

Build Your Own Table.

In a program, type

LISTX NumIndep

Sequence

NumRow

All Aplets except

Statistics aplet

Sets the row to be highlighted in Numeric view.

In a program, type

n ^XNumRow

where n >0

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NumStart

Function

Parametric

Polar

Sets the starting value for a table in Numeric view.

From Num Setup, enter a value for NUMSTART.

Sequence

In a program, type

n ^XNumStart

NumStep

Function

Parametric

Polar

Sets the step size (increment value) for an independent

variable in Numeric view.

From Num Setup, enter a value for NUMSTEP.

Sequence

In a program, type

n ^XNumStep

where n > 0

NumType

Function

Parametric

Polar

Sets the table format.

From Num Setup, choose Automaticor BuildYour

Own.

Sequence

In a program, type

X NumTypefor Build Your Own.

X NumTypefor Automatic (default).

NumZoom

Function

Parametric

Polar

Sets the zoom factor in the Numeric view.

From Num Setup, type in a value for NUMZOOM.

Sequence

In a program, type

n ^XNumZoom

where n > 0

StatMode

Statistics

Enables you to choose between 1-variable and 2-variable

statistics in the Statistics aplet. Does not appear in the Plot

Setup input form. Corresponds to the

menu keys in Numeric View.

and

In a program, store the constant name (or its number) into

the variable StatMode. 1VAR=1, 2VAR=2.

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Example

1VAR X StatMode

X StatMode

Note variables

The following aplet variable is available in Note view.

NoteText

All Aplets

Use NoteTextto recall text previously entered in Note

view.

Sketch variables

The following aplet variables are available in Sketch

view.

Page

All Aplets

Sets a page in a sketch set. The graphics can be viewed

one at a time using the

and

keys.

The Page variable refers to the currently displayed page

of a sketch set.

In a program, type

graphicname ^XPage

PageNum

All Aplets

Sets a number for referring to a particular page of the

sketch set (in Sketch view).

In a program, type the page that is shown when

SKETCH is pressed.

X PageNum

Programming

21-43

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hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Extending aplets

Aplets are the application environments where you

explore different classes of mathematical operations.

You can extend the capability of the HP 40gs in the

following ways:

•

Create new aplets, based on existing aplets, with

specific configurations such as angle measure,

graphical or tabular settings, and annotations.

•

Transmit aplets between HP 40gs calculators via a

serial or USB cable.

Download e-lessons (teaching aplets) from

Hewlett-Packard’s Calculator web site.

Program new aplets. See chapter 21,

“Programming”, for further details.

Creating new aplets based on existing aplets

You can create a new aplet based on an existing aplet.

To create a new aplet, save an existing aplet under a new

name, then modify the aplet to add the configurations

and the functionality that you want.

Information that defines an aplet is saved automatically

as it is entered into the calculator.

To keep as much memory available for storage as

possible, delete any aplets you no longer need.

Example

This example demonstrates how to create a new aplet by

saving a copy of the built-in Solve aplet. The new aplet is

saved under the name “TRIANGLES” and contains the

formulas commonly used in calculations involving

right-angled triangles.

Extending aplets

22-1

hp40g+.book Page 2 Friday, December 9, 2005 1:03 AM

1. Open the Solve aplet and save it under the new

name.

Solve

T R I A N G L E S

2. Enter the four formulas:

3. Decide whether you want the aplet to operate in

Degrees, Radians, or Grads.

MODES

Degrees

4. View the Aplet Library. The “TRIANGLES” aplet is

listed in the Aplet Library.

The Solve aplet can now

be reset and used for

Category

Complex

Graphic

Library

Available name

Z1...Z9, Z0

G1...G9, G0

Function

Parametric

Polar

Sequence

Solve

Statistics

User-named

List

L1...L9, L0

Matrix

Modes

M1...M9, M0

Ans

Date

HAngle

HDigits

HFormat

Ierr

Time

Notepad

Program

User-named

Editline

User-named

Real

A...Z, θ

R-6

hp40g+.book Page 7 Friday, December 9, 2005 1:03 AM

Function aplet variables

The function aplet variables are:

Category

Available name

Plot

Axes

Xcross

Ycross

Xtick

Ytick

Xmin

Connect

Coord

FastRes

Grid

Indep

InvCross

Labels

Recenter

Simult

Tracing

Xmax

Ymin

Ymax

Xzoom

Yxoom

Plot-FCN

Symbolic

Area

Root

Extremum

Isect

Slope

Angle

Numeric

Digits

NumRow

Format

NumStart

NumStep

NumType

NumZoom

NumCol

NumFont

NumIndep

Note

NoteText

Page

Sketch

PageNum

R-7

hp40g+.book Page 8 Friday, December 9, 2005 1:03 AM

Parametric aplet variables

The parametric aplet variables are:

Category

Available name

Plot

Axes

Tracing

Tstep

Xcross

Ycross

Xtick

Ytick

Xmin

Connect

Coord

Grid

Indep

InvCross

Labels

Recenter

Simult

Tmin

Xmax

Ymin

Ymax

Xzoom

Yzoom

Tmax

Symbolic

Angle

Numeric

Digits

NumRow

Format

NumStart

NumStep

NumType

NumZoom

NumCol

NumFont

NumIndep

Note

NoteText

Page

Sketch

PageNum

R-8

hp40g+.book Page 9 Friday, December 9, 2005 1:03 AM

Polar aplet variables

The polar aplet variables are:

Category

Available names

Plot

Axes

Connect

Coord

Xcross

Ycross

Xtick

Ytick

Xmin

Grid

Indep

InvCross

Labels

Recenter

Simult

Umin

Xmax

Ymin

Ymax

Xzoom

Yxoom

Umax

θstep

Tracing

Symbolic

Numeric

Angle

Digits

NumRow

Format

NumStart

NumStep

NumType

NumZoom

NumCol

NumFont

NumIndep

Note

NoteText

Page

Sketch

PageNum

R-9

hp40g+.book Page 10 Friday, December 9, 2005 1:03 AM

Sequence aplet variables

The sequence aplet variables are:

Category

Available name

Plot

Axes

Coord

Grid

Indep

InvCross

Labels

Nmin

Tracing

Xcross

Ycross

Xtick

Ytick

Xmin

Xmax

Nmax

Ymin

Recenter

SeqPlot

Simult

Ymax

Xzoom

Yzoom

Symbolic

Numeric

Angle

Digits

Format

NumCol

NumFont

NumIndep

NumRow

NumStart

NumStep

NumType

NumZoom

Note

NoteText

Page

Sketch

PageNum

R-10

hp40g+.book Page 11 Friday, December 9, 2005 1:03 AM

Solve aplet variables

The solve aplet variables are:

Category

Available name

Plot

Axes

Connect

Coord

FastRes

Grid

Xcross

Ycross

Xtick

Ytick

Xmin

Indep

Xmax

InvCross

Labels

Recenter

Tracing

Ymin

Ymax

Xzoom

Yxoom

Symbolic

Numeric

Angle

Digits

Format

NumCol

NumRow

Note

NoteText

Page

Sketch

PageNum

R-11

hp40g+.book Page 12 Friday, December 9, 2005 1:03 AM

Statistics aplet variables

The statistics aplet variables are:

Category

Available name

Plot

Axes

S4mark

S5mark

StatPlot

Tracing

Xcross

Ycross

Xtick

Ytick

Xmin

Connect

Coord

Grid

Hmin

Hmax

Hwidth

Indep

InvCross

Labels

Recenter

S1mark

S2mark

S3mark

Xmax

Ymin

Ymax

Xzoom

Yxoom

Symbolic

Numeric

Angle

S1fit

S2fit

S3fit

S4fit

S5fit

C0,...C9

Digits

Format

NumCol

NumFont

NumRow

StatMode

Stat-One

Stat-Two

MaxΣ

MeanΣ

Median

MinΣ

NΣ

PSDev

SSDev

PVarΣ

SVarΣ

TotΣ

Corr

Cov

Fit

MeanX

MeanY

RelErr

ΣX

ΣX2

ΣXY

ΣY

ΣY2

Note

NoteText

Page

Sketch

PageNum

R-12

hp40g+.book Page 13 Friday, December 9, 2005 1:03 AM

MATH menu categories

Math functions

The math functions are:

Category

Available name

Calculus

∂

∫

TAYLOR

Complex

Constant

ARG

CONJ

MAXREAL

MINREAL

Hyperb.

ACOSH

ASINH

ATANH

COSH

TANH

ALOG

EXP

EXPM1

LNP1

SINH

List

CONCAT

ΔLIST

MAKELIST

πLIST

POS

REVERSE

SIZE

ΣLIST

SORT

Loop

ITERATE

RECURSE

R-13

hp40g+.book Page 14 Friday, December 9, 2005 1:03 AM

Category

Available name (Continued)

Matrix

COLNORM

COND

RANK

CROSS

DET

ROWNORM

RREF

DOT

SCHUR

SIZE

SPECNORM

SPECRAD

SVD

EIGENVAL

EIGENVV

IDENMAT

INVERSE

SVL

LSQ

TRACE

TRN

MAKEMAT

Polynom.

Prob.

POLYCOEF

POLYEVAL

POLYFORM

POLYROOT

COMB

PERM

RANDOM

UTPC

UTPF

UTPN

UTPT

Real

CEILING

DEG→RAD

FLOOR

FNROOT

FRAC

HMS→

→HMS

INT

MIN

MOD

%CHANGE

%TOTAL

RAD→DEG

ROUND

SIGN

MANT

MAX

TRUNCATE

XPON

Stat-Two

Symbolic

PREDX

PREDY

QUAD

QUOTE

ISOLATE

LINEAR?

R-14

hp40g+.book Page 15 Friday, December 9, 2005 1:03 AM

Category

Available name (Continued)

Tests

AND

≤

= =

≠

IFTE

NOT

XOR

≥

Trig

ACOT

ACSC

ASEC

COT

CSC

SEC

Program constants

The program constants are:

Category

Available name

Angle

Degrees

Grads

Radians

Format

Standard

Fixed

Sci

Eng

Fraction

SeqPlot

S1...5fit

Cobweb

Stairstep

Linear

LogFit

ExpFit

Power

QuadFit

Cubic

Logist

User

Trigonometric Exponent

StatMode

StatPlot

Stat1Var

Stat2Var

Hist

BoxW

R-15

hp40g+.book Page 16 Friday, December 9, 2005 1:03 AM

Physical Constants

The physical constants are:

Category

Available Name

Chemist

• Avogadro(Avagadro’s Number,

NA)

• Boltz. (Boltmann, k)

• mol. vo... (molar volume, Vm)

• univ gas(universal gas, R)

• std temp(standard temperature,

St dT)

• std pres(standard pressure,

St dP)

Phyics

• StefBolt(Stefan-Boltzmann, σ)

•

light s...(speed of light, c)

• permitti(permittivity, ε0)

• permeab(permeability, μ0)

• acce gr... (acceleration of

gravity, g)

• gravita...(gravitation, G)

Quantum

• Plank’s(Plank’s constant, h)

• Dirac’s(Dirac’s, hbar)

• e charge(electronic charge, q)

• e mass(electron mass, me)

• q/me ra...(q/me ratio, qme)

• proton m(proton mass, mp)

• mp/me r...(mp/me ratio,

mpme)

• fine str(fine structure, α)

• mag flux(magnetic flux, φ)

• Faraday(Faraday, F)

• Rydberg(Rydberg, R∞)

• Bohr rad (Bohr radius, a0)

• Bohr mag(Bohr magneton, μB)

• nuc. mag (nuclear magneton,

μN)

• photon...(photon wavelength,

λ)

• photon...(photon frequency,

f0)

• Compt w...(Compton

wavelength, λc)

R-16

hp40g+.book Page 17 Friday, December 9, 2005 1:03 AM

CAS functions

CAS functions are:

Category

Function

Algebra

COLLECT

DEF

STORE

EXPAND

FACTOR

PARTFRAC

QUOTE

SUBST

TEXPAND

UNASSIGN

Complex

ABS

–

ARG

CONJ

DROITE

SIGN

Constant

Diff & Int

∞

DERIV

DERVX

DIVPC

FOURIER

IBP

PREVAL

RISCH

SERIES

TABVAR

TAYLOR0

TRUNC

INTVX

lim

Hyperb.

Integer

ACOSH

ASINH

ATANH

COSH

SINH

TANH

DIVIS

EULER

FACTOR

GCD

IREMAINDER

ISPRIME?

LCM

MOD

IDIV2

IEGCD

IQUOT

NEXTPRIME

PREVPRIME

Modular

ADDTMOD

DIVMOD

EXPANDMOD

FACTORMOD

GCDMOD

INVMOD

MODSTO

MULTMOD

POWMOD

SUBTMOD

R-17

hp40g+.book Page 18 Friday, December 9, 2005 1:03 AM

Category

Function (Continued)

Polynom.

EGCD

PARTFRAC

PROPFRAC

PTAYL

FACTOR

GCD

HERMITE

LCM

LEGENDRE

QUOT

REMAINDER

TCHEBYCHEFF

Real

CEILING

FLOOR

FRAC

INT

MAX

MIN

Rewrite

DISTRIB

EPSX0

POWEXPAND

SINCOS

SIMPLIFY

XNUM

EXPLN

EXP2POW

FDISTRIB

LIN

LNCOLLECT

Solve

Tests

DESOLVE

ISOLATE

LDEC

LINSOLVE

SOLVE

SOLVEVX

ASSUME

UNASSUME

≥

≤

= =

≠

AND

NOT

IFTE

Trig

ACOS2S

ASIN2C

ASIN2T

ATAN2S

HALFTAN

SINCOS

TAN2CS2

TAN2SC

TAN2SC2

TCOLLECT

TEXPAMD

TLIN

TRIG

TRIGCOS

TRIGSIN

TRIGTAN

R-18

hp40g+.book Page 19 Friday, December 9, 2005 1:03 AM

Program commands

The program commands are:

Category

Command

Aplet

CHECK

SELECT

SETVIEWS

UNCHECK

Branch

CASE

IFERR

RUN

THEN

ELSE

END

STOP

Drawing

Graphic

ARC

LINE

BOX

PIXOFF

PIXON

TLINE

ERASE

FREEZE

DISPLAY→

→DISPLAY

→GROB

MAKEGROB

PLOT→

→PLOT

GROBNOT

GROBOR

REPLACE

SUB

GROBXOR

ZEROGROB

Loop

FOR

UNTIL

END

WHILE

REPEAT

END

STEP

END

BREAK

Matrix

ADDCOL

ADDROW

DELCOL

DELROW

EDITMAT

RANDMAT

REDIM

REPLACE

SCALE

SCALEADD

SUB

SWAPCOL

SWAPROW

PRDISPLAY

PRHISTORY

PRVAR

Prompt

BEEP

FREEZE

GETKEY

INPUT

MSGBOX

PROMPT

WAIT

CHOOSE

CLRVAR

DISP

DISPXY

DISPTIME

EDITMAT

Stat-One

DO1VSTATS

RANDSEED

SETFREQ

SETSAMPLE

R-19

hp40g+.book Page 20 Friday, December 9, 2005 1:03 AM

Category

Command (Continued)

Stat-Two

DO2VSTATS

SETDEPEND

SETINDEP

Status messages

Message

Meaning

Bad Argument

Type

Incorrect input for this

operation.

Bad Argument

Value

The value is out of range for this

operation.

Infinite Result

Math exception, such as 1/0.

Insufficient

Memory

You must recover some memory

to continue operation. Delete

one or more matrices, lists,

notes, or programs (using

catalogs), or custom (not built-

in) aplets (using

MEMORY).

Insufficient

Statistics Data

Not enough data points for the

calculation. For two-variable

statistics there must be two

columns of data, and each

column must have at least four

numbers.

Invalid Dimension

Array argument had wrong

dimensions.

Invalid Statistics

Data

Need two columns with equal

numbers of data values.

hp40g+.book Page 21 Friday, December 9, 2005 1:03 AM

Message

Meaning (Continued)

Invalid Syntax

The function or command you

entered does not include the

proper arguments or order of

arguments. The delimiters

(parentheses, commas,

periods, and semi-colons) must

also be correct. Look up the

function name in the index to

find its proper syntax.

Name Conflict

The | (where) function

attempted to assign a value to

the variable of integration or

summation index.

No Equations

Checked

You must enter and check an

equation (Symbolic view)

before evaluating this function.

(OFF SCREEN)

Receive Error

Function value, root, extremum,

or intersection is not visible in

the current screen.

Problem with data reception

from another calculator. Re-

send the data.

Too Few

Arguments

The command requires more

arguments than you supplied.

Undefined Name

Undefined Result

The global variable named

does not exist.

The calculation has a

mathematicallyundefinedresult

(such as 0/0).

Out of Memory

You must recover a lot of

memory to continue operation.

Delete one or more matrices,

lists, notes, or programs (using

catalogs), or custom (not built-

in) aplets (using

MEMORY).

R-21

hp40g+.book Page 22 Friday, December 9, 2005 1:03 AM

hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Limited Warranty

HP 40gs Graphing Calculator; Warranty period: 12

months

1. HP warrants to you, the end-user customer, that HP

hardware, accessories and supplies will be free from

defects in materials and workmanship after the date

of purchase, for the period specified above. If HP

receives notice of such defects during the warranty

period, HP will, at its option, either repair or replace

products which prove to be defective. Replacement

products may be either new or like-new.

2. HP warrants to you that HP software will not fail to

execute its programming instructions after the date of

purchase, for the period specified above, due to

defects in material and workmanship when properly

installed and used. If HP receives notice of such

defects during the warranty period, HP will replace

software media which does not execute its

programming instructions due to such defects.

3. HP does not warrant that the operation of HP

products will be uninterrupted or error free. If HP is

unable, within a reasonable time, to repair or replace

any product to a condition as warranted, you will be

entitled to a refund of the purchase price upon

prompt return of the product with proof of purchase.

4. HP products may contain remanufactured parts

equivalent to new in performance or may have been

subject to incidental use.

5. Warranty does not apply to defects resulting from (a)

improper or inadequate maintenance or calibration,

(b) software, interfacing, parts or supplies not

supplied by HP, (c) unauthorized modification or

misuse, (d) operation outside of the published

environmental specifications for the product, or (e)

improper site preparation or maintenance.

W-1

hp40g+.book Page 2 Friday, December 9, 2005 1:03 AM

6. HP MAKES NO OTHER EXPRESS WARRANTY OR

CONDITION WHETHER WRITTEN OR ORAL. TO

THE EXTENT ALLOWED BY LOCAL LAW, ANY

IMPLIED WARRANTY OR CONDITION OF

MERCHANTABILITY, SATISFACTORY QUALITY, OR

FITNESS FOR A PARTICULAR PURPOSE IS LIMITED

TO THE DURATION OF THE EXPRESS WARRANTY

SET FORTH ABOVE. Some countries, states or

provinces do not allow limitations on the duration of

an implied warranty, so the above limitation or

exclusion might not apply to you. This warranty gives

you specific legal rights and you might also have

other rights that vary from country to country, state to

state, or province to province.

7. TO THE EXTENT ALLOWED BY LOCAL LAW, THE

REMEDIES IN THIS WARRANTY STATEMENT ARE

YOUR SOLE AND EXCLUSIVE REMEDIES. EXCEPT AS

INDICATED ABOVE, IN NO EVENT WILL HP OR ITS

SUPPLIERS BE LIABLE FOR LOSS OF DATA OR FOR

DIRECT, SPECIAL, INCIDENTAL, CONSEQUENTIAL

(INCLUDING LOST PROFIT OR DATA), OR OTHER

DAMAGE, WHETHER BASED IN CONTRACT, TORT,

OR OTHERWISE. Some countries, States or provinces

do not allow the exclusion or limitation of incidental

or consequential damages, so the above limitation or

exclusion may not apply to you.

8. The only warranties for HP products and services are

set forth in the express warranty statements

accompanying such products and services . HP shall

not be liable for technical or editorial errors or

omissions contained herein.

FOR CONSUMER TRANSACTIONS IN AUSTRALIA AND

NEW ZEALAND: THE WARRANTY TERMS CONTAINED

IN THIS STATEMENT, EXCEPT TO THE EXTENT

LAWFULLY PERMITTED, DO NOT EXCLUDE, RESTRICT

OR MODIFY AND ARE IN ADDITION TO THE

MANDATORY STATUTORY RIGHTS APPLICABLE TO THE

SALE OF THIS PRODUCT TO YOU.

W-2

hp40g+.book Page 3 Friday, December 9, 2005 1:03 AM

Service

Europe

Country :

Telephone numbers

+43-1-3602771203

+32-2-7126219

Austria

Belgium

Denmark

+45-8-2332844

Eastern Europe +420-5-41422523

countries

Finland

France

+35-89640009

+33-1-49939006

+49-69-95307103

+420-5-41422523

+31-2-06545301

+39-02-75419782

+47-63849309

Germany

Greece

Holland

Italy

Norway

Portugal

Spain

+351-229570200

+34-915-642095

+46-851992065

Sweden

Switzerland

+41-1-4395358

(German)

+41-22-8278780

(French)

+39-02-75419782

(Italian)

Turkey

+420-5-41422523

+44-207-4580161

Czech Republic +420-5-41422523

South Africa

Luxembourg

+27-11-2376200

+32-2-7126219

Other European +420-5-41422523

countries

Asia Pacific Country :

Australia

Telephone numbers

+61-3-9841-5211

Singapore

W-3

hp40g+.book Page 4 Friday, December 9, 2005 1:03 AM

L.America Country:

Argentina

Telephone numbers

0-810-555-5520

Brazil

Sao Paulo 3747-7799;

ROTC 0-800-157751

Mexico

Mx City 5258-9922;

ROTC 01-800-472-6684

Venezuela

Chile

0800-4746-8368

800-360999

Columbia

Peru

9-800-114726

0-800-10111

1-800-711-2884

Central

America &

Caribbean

Guatemala

Puerto Rico

Costa Rica

1-800-999-5105

1-877-232-0589

0-800-011-0524

N.America Country :

Telephone numbers

U.S.

1800-HP INVENT

Canada

(905) 206-4663 or

800- HP INVENT

ROTC = Rest of the country

Please logon to http://www.hp.com for the latest service

and support information.h

W-4

hp40g+.book Page 5 Friday, December 9, 2005 1:03 AM

Regulatory Notices

Federal Commu- This equipment has been tested and found to comply with

the limits for a Class B digital device, pursuant to Part 15

nications

of the FCC Rules. These limits are designed to provide

Commission

reasonable protection against harmful interference in a

Notice

residential installation. This equipment generates, uses,

and can radiate radio frequency energy and, if not

installed and used in accordance with the instructions,

may cause harmful interference to radio communications.

However, there is no guarantee that interference will not

occur in a particular installation. If this equipment does

cause harmful interference to radio or television

reception, which can be determined by turning the

equipment off and on, the user is encouraged to try to

correct the interference by one or more of the following

measures:

•

Reorient or relocate the receiving antenna.

Increase the separation between the equipment and

the receiver.

•

Connect the equipment into an outlet on a circuit

different from that to which the receiver is connected.

Consult the dealer or an experienced radio or

television technician for help.

Modifications

Cables

The FCC requires the user to be notified that any changes

or modifications made to this device that are not

expressly approved by Hewlett-Packard Company may

void the user's authority to operate the equipment.

Connections to this device must be made with shielded

cables with metallic RFI/EMI connector hoods to maintain

compliance with FCC rules and regulations.

Declaration of

Conformity for

Products

Marked with

FCC Logo,

This device complies with Part 15 of the FCC Rules.

Operation is subject to the following two conditions: (1)

this device may not cause harmful interference, and (2)

this device must accept any interference received,

including interference that may cause undesired

operation.

United States

Only

For questions regarding your product, contact:

W-5

hp40g+.book Page 6 Friday, December 9, 2005 1:03 AM

Hewlett-Packard Company

P. O. Box 692000, Mail Stop 530113

Houston, Texas 77269-2000

Or, call

1-800-474-6836

For questions regarding this FCC declaration, contact:

Hewlett-Packard Company

P. O. Box 692000, Mail Stop 510101

Houston, Texas 77269-2000

Or, call

1-281-514-3333

To identify this product, refer to the part, series, or model

number found on the product.

Canadian

Notice

This Class B digital apparatus meets all requirements of

the Canadian Interference-Causing Equipment

Regulations.

Avis Canadien

Cet appareil numérique de la classe B respecte toutes les

exigences du Règlement sur le matériel brouilleur du

Canada.

European Union

Regulatory

Notice

This product complies with the following EU Directives:

•

Low Voltage Directive 73/23/EEC

EMC Directive 89/336/EEC

Compliance with these directives implies conformity to

applicable harmonized European standards (European

Norms) which are listed on the EU Declaration of

Conformity issued by Hewlett-Packard for this product or

product family.

This compliance is indicated by the following conformity

marking placed on the product:

xxxx*

*Notified body number (used only if applicable - refer to the product label)

This marking is valid for non-Telecom products

This marking is valid for EU non-harmonized Telecom products .

and EU harmonized Telecom products (e.g. Bluetooth).

W-6

hp40g+.book Page 7 Friday, December 9, 2005 1:03 AM

Japanese Notice

Korean Notice

この装置は、情報処理装置等電波障害自主規制協議会

（VCCI）の基準に基づくクラス B 情報技術装置です。この装

置は、家庭環境で使用することを目的としていますが、この

装置がラジオやテレビジョン受信機に近接して使用されると、

受信障害を引き起こすことがあります。

取り扱い説明書に従って正しい取り扱いをしてください。

Disposal of Waste

Equipment by Users

in Private

Household in the

European Union

This symbol on the product or on its

packaging indicates that this product

must not be disposed of with your other

household waste. Instead, it is your

responsibility to dispose of your waste

equipment by handing it over to a

designated collection point for the

recycling of waste electrical and

electronic equipment. The separate collection and

recycling of your waste equipment at the time of disposal

will help to conserve natural resources and ensure that it

is recycled in a manner that protects human health and

the environment. For more information about where you

can drop off your waste equipment for recycling, please

contact your local city office, your household waste

disposal service or the shop where you purchased the

product.

W-7

hp40g+.book Page 8 Friday, December 9, 2005 1:03 AM

hp40g+.book Page 1 Friday, December 9, 2005 1:03 AM

Index

CHECK 21-14

SELECT 21-14

SETVIEWS 21-17

UNCHECK 21-17

aplet variables

definition 17-1, 17-8

in Plot view 21-31

new 17-1

ABCUV 14-62

ABS 14-45

absolute value 13-6

ACOS2S 14-38

add 13-4

ADDTMOD 14-51

ALGB menu 14-10

algebraic entry 1-19

alpha characters

typing 1-6

alphabetical sorting 22-6

angle measure 1-10

in statistics 10-12

setting 1-11

animation 20-5

creating 20-5

aplet views

canceling operations in 1-1

changing 1-19

note 1-18

Numeric view 1-17

Plot view 1-16

sketch 1-18

split-screen 1-17

Symbolic view 1-16

approximation 14-32

arc cosecant 13-20

arc cosine 13-5

arc cotangent 13-20

arc secant 13-20

arc sine 13-4

annunciators 1-3

Ans (last answer) 1-24

antiderivative 14-68, 14-69

antilogarithm 13-4, 13-10

aplet

arc tangent 13-5

area

attaching notes 22-4

clearing 22-3

graphical 3-10

interactive 3-10

variable 21-31

ARG 13-7

copying 22-4

definition of R-1

deleting 22-6

Function 13-21

Inference 11-1

key 1-4

arguments

with matrices 18-10

ASIN2C 14-39

ASIN2T 14-39

ASSUME 14-61

ATAN2S 14-39

attaching

library 22-6

Linear Solver 8-1

opening 1-16

Parametric 4-1

Polar 5-1

receiving 22-5

resetting 22-3

a note to an aplet 20-1

a sketch to an aplet 20-3

auto scale 2-14

axes

sending 22-4, 22-5

Sketch view 20-1

Solve 7-1

plotting 2-7

variable 21-31

sorting 22-6

statistics 10-1

transmitting 22-5

Triangle Solver 9-1

aplet commands

bad argument R-20

I-1

hp40g+.book Page 2 Friday, December 9, 2005 1:03 AM

bad guesses error message 7-7

batteries R-4

graphic 21-21

loop 21-23

print 21-25

Bernoulli’s number 14-65

box-and-whisker plot 10-16

branch commands

program 21-4, R-19

stat-one 21-29

stat-two 21-30

with matrices 18-10

CASE...END 21-18

IF...THEN...ELSE...END 21-18

complex number functions 13-6,

IFERR...THEN...ELSE 21-18

branch structures 21-17

13-17

conjugate 13-7

build your own table 2-19

imaginary part 13-7

real part 13-8

complex numbers 1-29

entering 1-29

calculus

operations 13-7

CAS 14-1, 15-1

configuration 15-3

help 15-4

math functions 13-7

storing 1-29

computer algebra system See CAS

confidence intervals 11-15

CONJ 13-7

history 14-8

in HOME 14-7

list of functions 14-9, R-17

modes 14-5, 15-3

online help 14-8

variables 14-4

catalogs 1-30

conjugate 13-7

connecting

data points 10-19

variable 21-31

via serial cable 22-5

via USB cable 22-5

connectivity kit 22-4

constant? error message 7-7

constants

CFG 15-3

Chinese remainders 14-62, 14-65

CHINREM 14-62

chronological sorting 22-6

circle drawing 20-4

clearing

e 13-8

i 13-8

maximum real number 13-8

minimum real number 13-8

physical 1-8, 13-25, R-16

program R-15, R-16

contrast

aplet 22-3

characters 1-22

display 1-22

display history 1-25

edit line 1-22

decreasing display 1-2

increasing display 1-2

conversions 13-8

coordinate display 2-9

copying

lists 19-6

plot 2-7

cobweb graph 6-1

coefficients

polynomial 13-11

COLLECT 14-10

columns

display 1-22

graphics 20-6

notes 20-8

changing position 21-25

combinations 13-12

commands

programs 21-8

correlation

coefficient 10-17

aplet 21-14

CORR 10-17

branch 21-17

statistical 10-15

cosecant 13-20

definition of R-1

drawing 21-19

I-2

hp40g+.book Page 3 Friday, December 9, 2005 1:03 AM

cosine 13-4

differentiation 13-6, 14-33

digamma function 14-67, 14-68

display 21-21

inverse hyperbolic 13-9

cotangent 13-20

covariance

adjusting contrast 1-2

annunciator line 1-2

capture 21-21

statistical 10-15

creating

clearing 1-2

aplet 22-1

date and time 21-27

element 18-5

lists 19-1

matrices 18-2

elements 19-4

notes in Notepad 20-6

programs 21-4

sketches 20-3

engineering 1-10

fixed 1-10

fraction 1-10

critical value(s) displayed 11-4

cross product

history 1-22

line 1-23

vector 18-11

curve fitting 10-12, 10-17

CYCLOTOMIC 14-63

matrices 18-5

parts of 1-2

printing contents 21-25

rescaling 2-13

scientific 1-10

data set definition 10-8

date, setting 21-27

debugging programs 21-7

decimal

scrolling through history 1-25

soft key labels 1-2

standard 1-10

DISTRIB 14-28

distributivity 14-12, 14-28, 14-30

divide 13-4

changing format 1-10

scaling 2-14, 2-15

decreasing display contrast 1-2

DEF 14-10

DIVIS 14-47

DIVMOD 14-52

DIVPC 14-17

definite integral 13-6

deleting

drawing

aplet 22-6

circles 20-4

keys 20-4

lists 19-6

matrices 18-4

lines and boxes 20-3

drawing commands

ARC 21-19

programs 21-9

statistical data 10-11

delimiters, programming 21-1

DERIV 14-16

BOX 21-20

ERASE 21-20

derivative 14-16

derivatives

FREEZE 21-20

LINE 21-20

PIXOFF 21-20

definition of 13-6

in Function aplet 13-22

in Home 13-21

PIXON 21-20

TLINE 21-20

DROITE 14-45

DERVX 14-16

DESOLVE 14-33

determinant

square matrix 18-11

DIFF menu 14-16

differential equations 14-33, 14-35,

14-57

e 13-8

edit line 1-2

editing

matrices 18-4

I-3

hp40g+.book Page 4 Friday, December 9, 2005 1:03 AM

notes 20-2

extremum 3-10

programs 21-5

Editline

Program catalog 21-2

FACTOR 14-12, 14-47, 14-56

factorial 13-13

editors 1-30

EGCD 14-55

factorization 14-12

FACTORMOD 14-53

FastRes variable 21-32

FDISTRIB 14-30

fit

a curve to 2VAR data 10-17

choosing 10-12

defining your own 10-13

fixed number format 1-10

font size

change 3-8, 15-2, 20-5

forecasting 10-20

FOURIER 14-17

fraction number format 1-11

full-precision display 1-10

function

eigenvalues 18-11

eigenvectors 18-11

element

storing 18-6

E-lessons 1-12

engineering number format 1-11

EPSX0 14-29

equals

for equations 13-17

logical test 13-19

Equation Writer 14-2, 15-1, 16-1

selecting terms 15-5

equations

solving 7-1

erasing a line in Sketch view 21-20

error messages

bad guesses 7-7

constant? 7-7

Euclidean division 14-48, 14-49

EULER 14-47

analyze graph with FCN tools 3-4

definition 2-2, R-1

entering 1-19

gamma 13-13

intersection point 3-5

math menu R-13, R-17

slope 3-5

exclusive OR 13-20

exiting views 1-19

EXP2HYP 14-63

EXP2POW 14-29

EXPAND 14-12

syntax 13-2

tracing 2-8

Function aplet 2-20, 3-1

function variables

area 21-31

EXPANDMOD 14-52

expansion 14-25, 14-27

EXPLN 14-29

axes 21-31

connect 21-31

exponent

fastres 21-32

fit 10-13

grid 21-32

minus 1 13-10

in menu map R-7

indep 21-33

of value 13-17

raising to 13-5

isect 21-33

exponentials 14-30, 14-63

labels 21-34

expression

Recenter 21-34

defining 2-1, R-1

entering in HOME 1-19

evaluating in aplets 2-3

literal 13-18

root 21-34

ycross 21-37

plot 3-3

GAMMA 14-64

GCD 14-47, 14-56

GCDMOD 14-53

extended greatest common divisor

14-55

I-4

hp40g+.book Page 5 Friday, December 9, 2005 1:03 AM

glossary R-1

adjusting 10-16

range 10-18

graph

setting min/max values for bars

21-32

analyzing statistical data in 10-19

auto scale 2-14

width 10-18

box-and-whisker 10-16

capture current display 21-21

cobweb 6-1

history 1-2, 14-8, 21-25

Home 1-1

comparing 2-5

calculating in 1-19

display 1-2

connected points 10-17

defining the independent variable

21-36

evaluating expressions 2-4

reusing lines 1-23

variables 17-1, 17-7, R-6

home 14-7

drawing axes 2-7

expressions 3-3

grid points 2-7

horizontal zoom 21-38

hyperbolic

histogram 10-15

in Solve aplet 7-7

one-variable statistics 10-18

overlaying 2-15

maths functions 13-10

hyperbolic trigonometry

ACOSH 13-9

scatter 10-15, 10-17

split-screen view 2-14

splitting into plot and close-up

2-13

ALOG 13-10

ASINH 13-9

ATANH 13-9

COSH 13-10

splitting into plot and table 2-13

stairsteps 6-1

EXP 13-10

EXPM1 13-10

statistical data 10-15

t values 2-6

LNP1 13-10

SINH 13-10

tickmarks 2-6

TANH 13-10

tracing 2-8

hypothesis

two-variable statistics 10-18

Graphic commands

→GROB 21-21

alternative 11-2

inference tests 11-8

null 11-2

DISPLAY→ 21-21

GROBNOT 21-21

GROBOR 21-21

tests 11-2

GROBXOR 21-22

MAKEGROB 21-22

PLOT→ 21-22

i 13-8, 14-45

IABCUV 14-64

IBERNOULLI 14-65

IBP 14-18

REPLACE 21-22

SUB 21-22

ZEROGROB 21-22

ICHINREM 14-65

IDIV2 14-48

graphics

copying 20-6

IEGCD 14-48

copying into Sketch view 20-6

storing and recalling 20-6, 21-21

greatest common divisor 14-56

ILAP 14-65

IM 13-7

implied multiplication 1-20

importing

graphics 20-6

notes 20-8

increasing display contrast 1-2

indefinite integral

HALFTAN 14-40

HERMITE 14-56

histogram 10-15

I-5

hp40g+.book Page 6 Friday, December 9, 2005 1:03 AM

using symbolic variables 13-23

independent values

ISPRIME? 14-50

adding to table 2-19

independent variable

defined for Tracing mode 21-33

inference

confidence intervals 11-15

hypothesis tests 11-8

One-Proportion Z-Interval 11-17

One-Sample Z-Interval 11-15

One-Sample Z-Test 11-8

Two-Proportion Z-Interval 11-17

Two-Proportion Z-Test 11-11

Two-Sample T-Interval 11-19

Two-Sample Z-Interval 11-16

infinite result R-20

keyboard

editing keys 1-5

entry keys 1-5

inactive keys 1-8

list keys 19-2

math functions 1-7

menu keys 1-4

Notepad keys 20-8

shifted keystrokes 1-6

labeling

axes 2-7

parts of a sketch 20-5

LAP 14-67

initial guess 7-5

input forms

resetting default values 1-9

setting Modes 1-11

Laplace transform 14-65

Laplace transform, inverse 14-66

LCM 14-50, 14-57

LDEC 14-35

insufficient memory R-20

insufficient statistics data R-20

integer rank

least common multiple 14-50, 14-57

matrix 18-12

LEGENDRE 14-57

integer scaling 2-14, 2-15

letters, typing 1-6

integral

library, managing aplets in 22-6

lim 14-21

definite 13-6

indefinite 13-23

limits 14-21

integration 13-6, 14-18, 14-24

LIN 14-30

interpreting

linear fit 10-13

intermediate guesses 7-7

intersection 3-11

Linear Solver aplet 8-1

linear systems 14-35

linearize 14-30, 14-43

LINSOLVE 14-35

INTVX 14-19

invalid

dimension R-20

statistics data R-20

list

syntax R-21

arithmetic with 19-7

calculate sequence of elements

19-8

calculating product of 19-8

composed from differences 19-7

concatenating 19-7

counting elements in 19-9

creating 19-1, 19-3, 19-4, 19-5

deleting 19-6

deleting list items 19-3

displaying 19-4

displaying list elements 19-4

editing 19-3

inverse hyperbolic cosine 13-9

inverse hyperbolic functions 13-10

inverse hyperbolic sine 13-9

inverse hyperbolic tangent 13-9

inverse Laplace transform 14-66

inverting matrices 18-8

INVMOD 14-53

IQUOT 14-49

IREMAINDER 14-49

isect variable 21-33

ISOLATE 14-34

I-6

hp40g+.book Page 7 Friday, December 9, 2005 1:03 AM

finding statistical values in list ele-

ments 19-9

generate a series 19-8

list function syntax 19-6

list variables 19-1

returning position of element in

19-8

logical operators 13-19

menu 1-7

polynomial 13-11

probability 13-12

real-number 13-14

symbolic 13-17

trigonometry 13-20

reversing order in 19-8

sending and receiving 19-6

sorting elements 19-9

storing elements 19-1, 19-4, 19-5

storing one element 19-6

LNCOLLECT 14-31

logarithm 13-4

MATH menu 13-1

math operations 1-19

enclosing arguments 1-21

in scientific notation 1-20

negative numbers in 1-20

matrices

adding rows 21-24

addition and subtraction 18-6

arguments 18-10

logarithmic

fit 10-13

functions 13-4

arithmetic operations in 18-6

assembly from vectors 18-1

changing row position 21-25

column norm 18-10

comma 19-7

logarithms 14-31

logical operators

AND 13-19

equals (logical test) 13-19

greater than 13-19

greater than or equal to 13-19

IFTE 13-19

commands 18-10

condition number 18-11

create identity 18-13

creating 18-3

less than 13-19

creating in Home 18-5

deleting 18-4

less than or equal to 13-19

NOT 13-19

deleting columns 21-24

deleting rows 21-24

determinant 18-11

not equal to 13-19

OR 13-19

XOR 13-20

display eigenvalues 18-11

displaying 18-5

logistic fit 10-13

loop commands

displaying matrix elements 18-5

dividing by a square matrix 18-8

dot product 18-11

BREAK 21-23

DO...UNTIL...END 21-23

FOR I= 21-23

editing 18-4

WHILE...REPEAT...END 21-23

extracting a portion 21-25

finding the trace of a square ma-

trix 18-13

loop functions

ITERATE 13-10

RECURSE 13-11

inverting 18-8

summation 13-11

low battery 1-1

matrix calculations 18-1

multiplying and dividing by scalar

18-7

lowercase letters 1-6

multiplying by vector 18-7

multiplying row by value and add-

ing result to second row 21-25

multiplying row number by value

21-25

negating elements 18-8

opening Matrix Editor 21-28

raised to a power 18-7

mantissa 13-15

math functions

complex number 13-7

hyperbolic 13-10

in menu map R-13, R-17

keyboard 13-3

I-7

hp40g+.book Page 8 Friday, December 9, 2005 1:03 AM

redimension 21-24

searching 1-9

replacing portion of matrix or vec-

tor 21-25

minimum real number 13-8

mixed fraction format 1-11

modes

sending or receiving 18-4

singular value decomposition

18-13

singular values 18-13

size 18-12

spectral norm 18-13

spectral radius 18-13

start Matrix Editor 21-24

storing elements 18-3, 18-5

storing matrix elements 18-6

swap column 21-25

swap row 21-25

transposing 18-13

variables 18-1

angle measure 1-10

CAS 14-5

decimal mark 1-11

number format 1-10

MODSTO 14-53

modular arithmetic 14-51

multiple solutions

plotting to find 7-7

multiplication 13-4, 14-28

implied 1-20

MULTMOD 14-54

matrix functions 18-10

COLNORM 18-10

COND 18-11

name conflict R-21

naming

CROSS 18-11

programs 21-4

DET 18-11

natural exponential 13-4, 13-10

natural log plus 1 13-10

natural logarithm 13-4

negation 13-5

DOT 18-11

EIGENVAL 18-11

EIGENVV 18-11

IDENMAT 18-11

INVERSE 18-11

LQ 18-11

negative numbers 1-20

NEXTPRIME 14-51

no equations checked R-21

non-rational 14-6

Normal Z-distribution, confidence in-

tervals 11-15

LSQ 18-11

LU 18-12

MAKEMAT 18-12

QR 18-12

RANK 18-12

note

ROWNORM 18-12

RREF 18-12

copying 20-8

editing 20-2

SCHUR 18-12

importing 20-8

SIZE 18-12

printing 21-26

SPECNORM 18-13

SPECRAD 18-13

SVD 18-13

viewing 20-1

writing 20-1

Notepad 20-1

SVL 18-13

catalog keys 20-7

creating notes 20-6

writing in 20-6

TRACE 18-13

TRN 18-13

maximum real number 1-22, 13-8

nth root 13-6

memory R-20

null hypothesis 11-2

number format

clearing all R-3

organizing 17-9

out of R-21

engineering 1-11

fixed 1-10

saving 1-25, 22-1

viewing 17-1

fraction 1-11

in Solve aplet 7-5

menu lists

I-8

hp40g+.book Page 9 Friday, December 9, 2005 1:03 AM

mixed fraction 1-11

scientific 1-10

permutations 13-13

pictures

Standard 1-10

attaching in Sketch view 20-3

numeric precision 17-9

Numeric view

plot

analyzing statistical data in 10-19

auto scale 2-14

adding values 2-19

automatic 2-16

box-and-whisker 10-16

cobweb 6-1

build your own table 2-19

display defining function for col-

umn 2-17

comparing 2-5

connected points 10-17, 10-19

decimal scaling 2-14

defining the independent variable

21-36

drawing axes 2-7

expressions 3-3

grid points 2-7

recalculating 2-19

setup 2-16, 2-19

off

automatic 1-1

histogram 10-15

in Solve aplet 7-7

integer scaling 2-14

one-variable statistics 10-18

overlay plot 2-13

overlaying 2-15, 4-3

scaling 2-13

power 1-1

on/cancel 1-1

One-Proportion Z-Interval 11-17

One-Sample T-Interval 11-18

One-Sample T-Test 11-12

One-Sample Z-Interval 11-15

One-Sample Z-Test 11-8

online help 14-8

scatter 10-15, 10-17

sequence 2-6

order of precedence 1-21

overlaying plots 2-15, 4-3

setting up 2-5, 3-2

split-screen view 2-14

splitting 2-14

splitting into plot and close-up

2-13

π 13-8

PA2B2 14-67

splitting into plot and table 2-13

stairsteps 6-1

paired columns 10-11

parametric variables

axes 21-31

statistical data 10-15

statistics parameters 10-18

t values 2-6

connect 21-31

tickmarks 2-6

grid 21-32

to capture current display 21-21

tracing 2-8

in menu map R-8

indep 21-33

trigonometric scaling 2-14

two-variable statistics 10-18

plotting resolution

and tracing 2-8

labels 21-34

recenter 21-34

ycross 21-37

parentheses

plot-view variables

area 21-31

to close arguments 1-21

to specify order of operation 1-21

PARTFRAC 14-13, 14-57

partial derivative 14-16

partial fraction expansion 14-13

partial integration 14-18

pause 21-29

connect 21-31

fastres 21-32

function 21-31

grid 21-32

hmin/hmax 21-32

hwidth 21-33

isect 21-33

I-9

hp40g+.book Page 10 Friday, December 9, 2005 1:03 AM

labels 21-34

! 13-13

recenter 21-34

root 21-34

COMB 13-12

RANDOM 13-13

UTPC 13-13

UTPF 13-13

UTPN 13-13

UTPT 13-14

s1mark-s5mark 21-34

statplot 21-35

tracing 21-33

umin/umax 21-35

ustep 21-35

polar variables

program

commands 21-4

axes 21-31

copying 21-8

connect 21-31

creating 21-4

grid 21-32

debugging 21-7

in menu map R-9

indep 21-33

deleting 21-9

delimiters 21-1

labels 21-34

editing 21-5

recenter 21-34

naming 21-4

ycross 21-37

pausing 21-29

polynomial

printing 21-26

sending and receiving 21-8

structured 21-1

coefficients 13-11

evaluation 13-11

form 13-12

roots 13-12

Taylor 13-7

prompt commands

beep 21-26

create choose box 21-26

create input form 21-28

display item 21-27

display message box 21-29

halt program execution 21-29

insert line breaks 21-29

prevent screen display being up-

dated 21-28

polynomial functions

POLYCOEF 13-11

POLYEVAL 13-11

POLYFORM 13-12

POLYROOT 13-12

ports 22-5

position argument 21-21

power (x raised to y) 13-5

powers 14-6

POWEXPAND 14-31

POWMOD 14-54

precedence 1-22

predicted values

statistical 10-20

PREVAL 14-23

set date and time 21-27

store keycode 21-28

PROPFRAC 14-58

PSI 14-67

Psi 14-68

PTAYL 14-58

quadratic

PREVPRIME 14-51

prime factors 14-47

prime numbers 14-50, 14-51

primitive 14-23, 14-24

extremum 3-6

fit 10-13

function 3-4

QUOT 14-58

QUOTE 14-13

quotes

contents of display 21-25

name and contents of variable

21-26

in program names 21-4

object in history 21-25

variables 21-26

probability functions

random numbers 13-13

I-10

hp40g+.book Page 11 Friday, December 9, 2005 1:03 AM

RE 13-8

rigorous 14-6

RISCH 14-24

root

real number

maximum 13-8

minimum 13-8

real part 13-8

real-number functions 13-14

% 13-16

interactive 3-10

nth 13-6

variable 21-34

root-finding

%CHANGE 13-16

%TOTAL 13-16

CEILING 13-14

DEGtoRAD 13-14

FNROOT 13-14

HMSto 13-15

displaying 7-7

interactive 3-9

operations 3-10

variables 3-10

INT 13-15

S1mark-S5mark variables 21-34

scaling

MANT 13-15

MAX 13-15

automatic 2-14

decimal 2-10, 2-14

integer 2-10, 2-14, 2-15

options 2-13

MIN 13-15

MOD 13-15

RADtoDEG 13-16

ROUND 13-16

SIGN 13-16

resetting 2-13

trigonometric 2-14

scatter plot 10-15, 10-17

connected 10-17, 10-19

SCHUR decomposition 18-12

scientific number format 1-10, 1-20

scrolling

TRUNCATE 13-17

XPON 13-17

reatest common divisor 14-47

recalculation for table 2-19

receive error R-21

receiving

in Trace mode 2-8

aplet 22-5

searching

lists 19-6

menu lists 1-9

matrices 18-4

speed searches 1-9

secant 13-20

programs 21-8

redrawing

Sending 22-5

table of numbers 2-18

reduced row echelon 18-12

regression

analysis 10-17

fit models 10-13

formula 10-12

user-defined fit 10-13

relative error

sending

aplets 22-4

lists 19-6

programs 21-8

sequence

definition 2-2

sequence variables

Axes 21-31

statistical 10-18

REMAINDER 14-59

REORDER 14-68

resetting

Grid 21-32

in menu map R-10

Indep 21-33

Labels 21-34

aplet 22-3

Recenter 21-34

Ycross 21-37

calculator R-3

memory R-3

serial port connectivity 22-5

SERIES 14-24

result

copying to edit line 1-22

reusing 1-22

setting

I-11

hp40g+.book Page 12 Friday, December 9, 2005 1:03 AM

date 21-27

time 21-27

aplets in chronological order 22-6

elements in a list 19-9

SEVAL 14-68

spectral norm 18-13

spectral radius 18-13

square root 13-5

SIGMA 14-68

SIGMAVX 14-69

SIGN 14-46

sign reversal 7-6

SIMPLIFY 14-32

simplify 14-68, 14-70

SINCOS 14-31, 14-40

sine 13-4

stack history

printing 21-25

stairsteps graph 6-1

standard number format 1-10

statistics

analysis 10-1

analyzing plots 10-19

angle mode 10-12

calculate one-variable 21-30

calculate two-variable 21-30

data set variables 21-40

data structure 21-40

define one-variable sample 21-30

define two-variable data set’s de-

pendent column 21-30

define two-variable data set’s in-

dependent column 21-30

defining a fit 10-12

defining a regression model

10-12

inverse hyperbolic 13-9

singular value decomposition

matrix 18-13

singular values

matrix 18-13

sketches

creating 20-5

creating a blank graphic 21-22

creating a set of 20-5

erasing a line 21-20

labeling 20-5

opening view 20-3

sets 20-5

storing in graphics variable 20-5

deleting data 10-11

editing data 10-11

frequency 21-30

slope 3-10

soft key labels 1-2

SOLVE 14-37

inserting data 10-11

plot type 10-18

solve

error messages 7-7

initial guesses 7-5

interpreting intermediate guesses

7-7

plotting data 10-15

predicted values 10-20

regression curve (fit) models

10-12

interpreting results 7-6

plotting to find guesses 7-7

setting number format 7-5

solve variables

saving data 10-10

sorting data 10-11

specifying angle setting 10-12

toggling between one-variable

and two-variable 10-12

tracing plots 10-19

troubleshooting with plots 10-19

zooming in plots 10-19

statistics variables

axes 21-31

connect 21-31

fastres 21-32

grid 21-32

in menu map R-11

indep 21-33

Axes 21-31

labels 21-34

Connect 21-31

recenter 21-34

Grid 21-32

ycross 21-37

Hmin/Hmax 21-32

Hwidth 21-33

SOLVEVX 14-38

sorting 22-6

in menu map R-12

Indep 21-33

aplets in alphabetic order 22-6

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hp40g+.book Page 13 Friday, December 9, 2005 1:03 AM

Labels 21-34

Recenter 21-34

TABVAR 14-27

TAN2CS2 14-40

TAN2SC 14-41

TAN2SC2 14-41

tangent 13-4

inverse hyperbolic 13-9

Taylor polynomial 13-7

TAYLOR0 14-27

TCHEBYCHEFF 14-59

TCOLLECT 14-41

tests 14-61

S1mark-S5mark 21-34

Ycross 21-37

step size of independent variable

21-36

step-by-step 14-6

STORE 14-14

storing

list elements 19-1, 19-4, 19-5,

19-6

matrix elements 18-3, 18-5, 18-6

results of calculation 17-2

value 17-2

TEXPAND 14-15, 14-42

tickmarks for plotting 2-6

time 13-15

strings

setting 21-27

literal in symbolic operations

time, converting 13-15

times sign 1-20

TLIN 14-43

13-18

STURMAB 14-69

SUBST 14-15

tmax 21-36

tmin 21-36

substitution 14-14

SUBTMOD 14-55

subtract 13-4

summation function 13-11

symbolic

too few arguments R-21

TOOL menu 15-1

tracing

functions 2-8

calculations in Function aplet

13-21

more than one curve 2-8

not matching plot 2-8

plots 2-8

defining expressions 2-1

differentiation 13-21

displaying definitions 3-8

evaluating variables in view 2-3

setup view for statistics 10-12

symbolic calculations 14-1

symbolic functions

| (where) 13-18

equals 13-17

transcendental expressions 14-42

transmitting

lists 19-6

matrices 18-4

programs 21-8

transposing a matrix 18-13

Triangle Solver aplet 9-1

TRIG 14-43

ISOLATE 13-17

LINEAR? 13-18

TRIGCOS 14-44

trigonometric

QUAD 13-18

QUOTE 13-18

fit 10-13

Symbolic view

functions 13-20

scaling 2-10, 2-14, 2-15

trigonometry functions

ACOS2S 14-38

ACOT 13-20

defining expressions 3-2

syntax 13-2

syntax errors 21-7

ACSC 13-20

ASEC 13-20

table

ASIN2C 14-39

ASIN2S 14-39

navigate around 3-8

numeric values 3-7

numeric view setup 2-16

ASIN2T 14-39

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hp40g+.book Page 14 Friday, December 9, 2005 1:03 AM

COT 13-20

CSC 13-20

CAS 14-4

categories 17-7

HALFTAN 14-40

SEC 13-20

clearing 17-3

definition 17-1, 17-7, R-2

in equations 7-10

in Symbolic view 2-3

independent 14-6, 21-36

local 17-1

SINCOS 14-40

TAN2CS2 14-40

TAN2SC 14-41

TAN2SC2 14-41

TRIGCOS 14-44

TRIGSIN 14-44

TRIGTAN 14-44

TRIGSIN 14-44

TRIGTAN 14-44

TRUNC 14-28

previous result (Ans) 1-23

printing 21-26

root 21-34

root-finding 3-10

step size of independent 21-36

types 17-1, 17-7

use in calculations 17-3

truncating values to decimal places

variation table 14-27

VARS menu 17-4, 17-5

vectors

13-17

TSIMP 14-70

tstep 21-36

column 18-1

Two-Proportion Z-Interval 11-17

Two-Proportion Z-Test 11-11

Two-Sample T-Interval 11-19

Two-Sample T-test 11-14

Two-Sample Z-Interval 11-16

typing letters 1-6

cross product 18-11

definition of R-2

VER 14-70

verbose 14-6

version 14-70

views 1-18

configuration 1-18

definition of R-3

UNASSIGN 14-15

UNASSUME 14-61

undefined

warning symbol 1-8

where command ( | ) 13-18

name R-21

result R-21

un-zoom 2-11

upper-tail chi-squared probability

13-13

Xcross variable 21-36

XNUM 14-32

XQ 14-32

upper-tail normal probability 13-13

upper-tail Snedecor’s F 13-13

upper-tail student’s t-probability

13-14

Ycross variable 21-37

USB connectivity 22-5

user defined

regression fit 10-13

Z-Interval 11-15

zoom 2-17

axes 2-12

value

box 2-9

center 2-9

examples of 2-11

factors 2-13

recall 17-3

storing 17-2

variables

aplet 17-1

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hp40g+.book Page 15 Friday, December 9, 2005 1:03 AM

in 2-9

square 2-10

options 2-9, 3-8

options within a table 2-18

out 2-9

un-zoom 2-11

within Numeric view 2-18

X-zoom 2-9

redrawing table of numbers op-

tions 2-18

Y-zoom 2-10

I-15

hp40g+.book Page 16 Friday, December 9, 2005 1:03 AM

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