HP 40gs User Manual

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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 WARRANTY 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.

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

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Contents

Preface

Manual conventions .............................................................. P-1

Notice ................................................................................. P-2

1 Getting started

On/off, cancel operations......................................................1-1

The display ..........................................................................1-2

The keyboard .......................................................................1-3

Menus .................................................................................1-8

Input forms ...........................................................................1-9

Mode settings.....................................................................1-10

Setting a mode...............................................................1-11

Aplets (E-lessons).................................................................1-12

Aplet library ..................................................................1-16

Aplet views....................................................................1-16

Aplet view configuration..................................................1-18

Mathematical calculations ....................................................1-19

Using fractions....................................................................1-25

Complex numbers ...............................................................1-29

Catalogs and editors ...........................................................1-30

2 Aplets and their views

Aplet views ..........................................................................2-1

About the Symbolic view ...................................................2-1

Defining an expression (Symbolic view) ..............................2-1

Evaluating expressions ......................................................2-3

About the Plot view...........................................................2-5

Setting up the plot (Plot view setup).....................................2-5

Exploring the graph..........................................................2-7

Other views for scaling and splitting the graph ..................2-13

About the numeric view...................................................2-16

Setting up the table (Numeric view setup) ..........................2-16

Exploring the table of numbers.........................................2-17

Building your own table of numbers..................................2-19

“Build Your Own” menu keys...........................................2-20

Example: plotting a circle ................................................2-20

3 Function aplet

About the Function aplet ........................................................3-1

Getting started with the Function aplet.................................3-1

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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 key first, are represented as follows:

CLEAR, MODES, ACOS, etc.

• 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

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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

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Getting started

On/off, cancel operations

To turn on Press to turn on the calculator.

To cancel When the calculator is on, the key cancels the

current operation.

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 Bat message, 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 and (or ) to increase (or decrease) the contrast.

To clear the display • Press CANCEL to clear the edit line.

• Press display history.

CLEAR to clear the edit line and the

Parts of the display

History

Edit line

Menu key or soft key labels. The labels for the menu keys’ current meanings. is the label for the first menu key in this picture. “Press ” 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.

T and S symbols indicate whether there is more

The history in the HOME display. Press the and to scroll in the HOME display.

Title

Menu key labels

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

α Alpha in effect for next keystroke.

((•)) Low battery power.

Data is being transferred.

The keyboard

Shift in effect for next keystroke. To cancel, press again.

To cancel, press again.

Busy.

HP 40gs Graphing Calculator

Menu Key Labels

Menu Keys

Aplet Control Keys

Alpha Key

Shift Key

Getting started 1-3

Cursor Keys

Enter Key

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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

(

CANCEL)

CLEAR

, , ,

Cancels the current operation if the 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 or is present as a menu key, acts the same as pressing or

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 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

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Key Meaning (Continued)

CHARS Displays a menu of all available

characters. To type one, use the arrow keys to highlight it, and press

select each and press , then press .

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 . (MODES is labeled in blue above the key). You do not need to hold down when you press HOME. This action is depicted in this manual as “press

To cancel a shift, press again.

. To select multiple characters,

MODES.”

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

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HELPWITH 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.

Example 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 SYNTAX will show the CAS help menu.

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 ) functions. Press to complete the operation: 256 displays 16.

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.

HINT

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 appears. There is no beep.

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.

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To search a menu • Press or to scroll through the list. If you press

or , 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 or 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 . To go down a page, press .

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 ( ). Some input forms include items to check ( ). See below for examples input forms.

Reset input form values

Getting started 1-9

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.

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Mode settings

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

HINT

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 and ).

SETUP keys (

Press

MODES to access the HOME MODES input

form.

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

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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.

Getting started 1-11

MODES to open the HOME MODES input

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 to return to HOME.

HINT

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

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symbolic views of the aplets in the following table. See “Aplet view configuration” on page 1-18 for further information.

Aplet

Use this aplet to explore:

name

Function Real-valued, rectangular functions y in

terms of x. Example: .

y 2x23x 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 24θ()cos=

Sequence Sequence functions U in terms of n, or

in terms of previous terms in the same or another sequence, such as and

. Example: , and

n 2–

U10= U21=

n1–

Solve Equations in one or more real-valued

variables. Example: .

x 1+ x

n 1–

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

HINT

The Quad Explorer aplet is used to investigate the behaviour of as the values of a, h and

yaxh+()

v+=

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.

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 keys, and the 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 and key moves between subexpressions, while pressing the and key changes their values.

Pressing allows the user to select whether all three sub-expressions will be explored at once or only one at a time.

A 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!

1-14 Getting started

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Trig Explorer aplet The Trig Explorer aplet is used to investigate the

behaviour of the graph of 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 and

keys transforms the graph, with these transformations reflected in the equation.

The button labelled is a toggle between and . When is chosen, the ‘point of control’ is at the origin (0,0) and the and keys control vertical and horizontal transformations. When is chosen the ‘point of control’ is on the first extremum of the graph (i.e. for the sine graph at .

ya bxc+()d+sin=

Origin

π 21,⁄()

The arrow keys change the amplitude and frequency of

Extremum

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 or key changes the parameter’s values.

The default angle setting for this aplet is radians. The angle setting can be changed to degrees by pressing

Getting started 1-15

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Aplet library

Aplets are stored in the Aplet library.

To open an aplet Press to display the Aplet library menu. Select the

aplet and press or . 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 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.

Plot view 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.

1-16 Getting started

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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

Getting started 1-17

The VIEWS menu contains the Overlay Plot view.

Select Overlay Plot Plots the current

expression(s) without erasing any pre-existing plot(s).

See “Other views for scaling and splitting the graph” on page 2-13 for further information.

hp40g+.book Page 18 Friday, December 9, 2005 1:03 AM

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

Plot Setup Press SETUP-PLOT.

Sets parameters to plot a graph.

Numeric Setup Press SETUP-NUM. Sets

parameters for building a table of numeric values.

Symbolic Setup This view is only available in

the Statistics aplet in mode, where it plays an important role in choosing data models. Press

MODES view.

SETUP-SYMB.

1-18 Getting started

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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 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”.)

CMDS.

• 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 1-19

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

Example Calculate :

23214 8–

---------------------------3–

8 3

45()ln

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

Scientific notation (powers of 10)

Example Calculate

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.

A number like or is 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 using 10 .)

410

×()610

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

510

× 3.21 107–×

13–

310

= 25, whereas –52 =

EEX. (This is easier than

×()

5–

23 3 EEX

Explicit and implicit multiplication

1-20 Getting started

Implied multiplication takes place when two operands appear with no operator in between. If you enter AB, for example, the result is A*B.

EEX

hp40g+.book Page 21 Friday, December 9, 2005 1:03 AM

However, for clarity, it is better to include the multiplication sign where you expect multiplication in an expression. It is clearest to enter AB as A*B.

HINT

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 A at the value B+4’, and function A does 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...

45 π sin (45 + π)

45 π sin (45) + π

85 9

Getting started 1-21

85 9×

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

Algebraic precedence order of evaluation

Largest and smallest numbers

Clearing numbers

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.

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. AN D a nd NOT.

8. OR and XOR.

9. Left argument of | (where).

10.Equals, =.

The smallest number the HP 40gs can represent is

–499

1×10 zero. The largest number is 9.99999999999 × 10 (1E499). A greater result is displayed as this number.

• 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.

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

499

CANCEL ( ) clears the edit line.

•

Using previous results

1-22 Getting started

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.

CLEAR clears all input and output in the

display, including the display history.

Input

Last input

Edit line

Output Last output

hp40g+.book Page 23 Friday, December 9, 2005 1:03 AM

When you highlight a previous input or result (by pressing

), the and menu labels appear.

To copy a previous line

To reuse the last result

Highlight the line (press ) and press . The number (or expression) is copied into the edit line.

Press ANS (last answer) to put the last result from the HOME display into an expression. 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 calculation is repeated iteratively.

Example See how

(50), and updates 50 25

You can use the last result as the first expression in the edit line without pressing

, (or other operators that require a preceding

argument) automatically enters 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

ANS is different from the numbers in HOME’s

display history. A value in full precision of the calculated result, whereas the displayed numbers match the display mode.

ANS is a variable that

ANS, the

ANS retrieves and reuses the last result

ANS (from 50 to 75 to 100).

ANS. Pressing , , , or

ANS before the operator.

ANS is stored internally with the

Getting started 1-23

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

HINT

Storing a value in a variable

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 evaluates (or re-evaluates) the last input, whereas pressing into the edit line.

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.

45 8 3

2. Store the result in the A variable.

ANS copies the last result (as ANS)

3. Perform another calculation using the A variable.

95 2 A

1-24 Getting started

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Accessing the display history

Clearing 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.

CLEAR

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.

Clears all lines of display history and the edit line.

Using fractions

To work with fractions in HOME, you set the number format to Fraction or Mixed Fraction, as follows:

Setting Fraction mode

Getting started 1-25

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

MODES

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

2. Select Number Format, press to display the options, and highlight Fraction or 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 set the precision. Press to return

to HOME. See “Setting fraction precision” below for more

information.

Setting fraction precision

1-26 Getting started

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:

hp40g+.book Page 27 Friday, December 9, 2005 1:03 AM

• 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, 11/2, you enter it in the format (1+

/2).

For example, to perform the following calculation: 3(23/4 + 57/8)

1. Set the Number format mode to Fraction or Mixed Fraction and specify a precision value of

In this example, we’ll select Fraction as our

format.)

MODES

Select

Fraction

Getting started 1-27

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

2. Enter the calculation. 323

457 8

Note: Ensure you are in the HOME view.

3. Evaluate the calculation.

Note that if you had selected Mixed

Fraction instead of Fraction as 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

hp40g+.book Page 29 Friday, December 9, 2005 1:03 AM

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 returns (0,1).

1–

To enter complex numbers

Storing 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, :

• (x, y) or

• x + iy.

To enter i:

• press or

• press , or keys to select Constant,

select i, and .

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 .

1–

to move to the right column of the menu, to

Z 0

Getting started 1-29

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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 ()

Sketch editor ( SKETCH)

List (

LIST)

Matrix (

MATRIX)

Notepad (

NOTEPAD)

Program (

PROGRM)

Equation Writer ()

Aplets.

Sketches and diagrams, See Chapter 20, “Notes and sketches”.

Lists. In HOME, lists are enclosed in {}. See Chapter 19, “Lists”.

One- and two-dimensional arrays. In HOME, arrays are enclosed in []. See Chapter 18, “Matrices”.

Notes (short text entries). See Chapter 20, “Notes and sketches”.

Programs that you create, or associated with user-defined aplets. See Chapter 21, “Programming”.

The editor used for creating expressions and equations in CAS. See Chapter 15, “Equation Writer”.

1-30 Getting started

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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 or to 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 . All expressions are plotted.

Aplets and their views 2-1

selected

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).

2-2 Aplets and their views

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

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.

B F1

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 2-3

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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 9 in place of X into 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.

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Key Meaning (Continued)

CHARS Displays special characters. To enter

CLEAR Deletes all expressions in the list or

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.

Displays the menu for entering math operations.

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.

clears the edit line.

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

or .

– If there is an option to choose, press ,

highlight your choice, and press or . As a shortcut to , just highlight the field to change and press to cycle through the options.

– If there is an option to select or deselect, press

to check or uncheck it.

2. Press to view more settings.

3. When done, press to view the new plot.

Aplets and their views 2-5

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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 Parametric aplet: Specifies the t-

values (T) for the graph.

θRNG Polar aplet: Specifies the angle (θ)

value range for the graph.

NRNG Sequence aplet: Specifies the

index (N) values for the graph.

TSTEP For Parametric plots: the increment

for the independent variable.

θSTEP For Polar plots: the increment

value for the independent variable.

SEQPLOT For Sequence aplet: Stairstep or

Cobweb types.

XTICK Horizontal spacing for tickmarks.

YTICK Vertical spacing for tickmarks.

Those items with space for a checkmark are settings you can turn on or off. Press to display the second page.

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.

2-6 Aplets and their views

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Field Meaning (Continued)

CONNECT Connect the plotted points. (The

LABELS Label the axes with XRNG and

AXES Draw the axes.

GRID Draw grid points using XTICK

Sequence aplet always connects them.)

YRNG values.

and YTICK spacing.

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.

or Interrupts plotting.

Continues plotting if interrupted.

Aplets and their views 2-7

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Key Meaning (Continued)

Turns menu-key labels on and off. When the labels are off, pressing

• Pressing once displays the

• Pressing a second time

• Pressing a third time

Displays the ZOOM menu list. Turns trace mode on/off. A white box

appears over the on . 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 to restore the menu.

turns them back on.

full row of labels. removes the row of labels to

display only the graph. displays the coordinate mode.

Trace a graph You can trace along a function using the or 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

2-8 Aplets and their views

If there is more than one relation displayed, press or

to move between relations.

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

To jump directly to a value

To jump straight to a value rather than using the Trace function, use the menu key. Press , then enter a value. Press 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 . Zooming redraws the plot on a larger or smaller scale. It is a shortcut for changing the Plot Setup.

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

is not displayed, press .) Not all options are available in all aplets.

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.

In 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.

Aplets and their views 2-9

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Option Meaning (Continued)

Y-Zoom In Divides vertical scale only, using

Y-factor.

Y-Zoom Out Multiplies vertical scale only, using

Y-factor.

Square 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 Rescales horizontal axis only,

making each pixel =1 unit. (Not available in Sequence or Statistics aplets.)

Trig Rescales horizontal axis so

1 pixel = π/24 radians, 7.58, or

/3grads; rescales vertical axis

8 so 1 pixel = 0.1 unit. (Not in Sequence or Statistics aplets.)

2-10 Aplets and their views

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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 . Plot of

Zoom In:

3 xsin

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.

Aplets and their views 2-11

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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 to turn on the menu-key labels.

2. Press and select Box...

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

4. Use the cursor keys ( , etc.) to drag to

the opposite corner.

5. Press to zoom in on the boxed area.

2-12 Aplets and their views

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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 Trig to 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

Press , select an option, and press .

options

Option Meaning

PlotDetail

Plot-Table Splits the screen into the plot and

Overlay Plot

Aplets and their views 2-13

Splits the screen into the plot and a close-up.

the data table. 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 Rescales horizontal axis only,

making each pixel=1 unit. (Not available in Sequence or Statistics aplets.)

Trig Rescales horizontal axis so

1 pixel=π/24 radian, 7.58, or

8 1 pixel = 0.1 unit. (Not in Sequence or Statistics aplets.)

/3 grads; rescales vertical axis so

Split the screen The Plot-Detail view can give you two simultaneous views

of the plot.

1. Press . Select Plot-Detail and press . The graph is plotted twice. You can now zoom in on the right side.

2. Press select the zoom method

and press or

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).

2-14 Aplets and their views

. This zooms the

hp40g+.book Page 15 Friday, December 9, 2005 1:03 AM

– moves the leftmost cursor to the

screen’s left edge and moves the rightmost cursor to the screen’s right edge.

– The menu key copies the right plot to the left

plot.

3. To un-split the screen, press . The left side takes over the whole screen.

The Plot-Table view gives you two simultaneous views of the plot.

1. Press . Select Plot-Table and

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 Plot instead of

. Note that tracing follows only the current

functions from the current aplet.

Decimal 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 Integer scaling compresses the axes so that each pixel is

and the origin is near the screen center.

11×

Trigonometric scaling

Aplets and their views 2-15

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

After entering and selecting (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 or .

– Shortcut: Press the key to copy values

from the Plot Setup into NUMSTART and NUMSTEP. Effectively, the menu key allows you to make the table match the pixel columns in the graph view.

2. When done, press to view the table of numbers.

Numeric view settings

2-16 Aplets and their views

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

NUMZOOM Allows you to zoom in or out on a

or Build Your Own. To build your own table, you must type each independent value into the table yourself.

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

Aplets and their views 2-17

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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

In 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

NUMZOOM factor in Numeric Setup.

Decimal Changes intervals for the

independent variable to 0.1 units. Starts at zero. (Shortcut to changing

NUMSTART and NUMSTEP.)

Integer Changes intervals for the

independent variable to 1 unit. Starts at zero. (Shortcut to changing

NUMSTEP.)

Trig Changes intervals for independent

variable to π/24 radian or 7.5 degrees or 8 zero.

Un-zoom Returns the display to the previous

zoom.

/3 grads. Starts at

The display on the right is a Zoom In of the display on the left. The ZOOM factor is 4.

HINT

2-18 Aplets and their views

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 NUMTYPE is “Automatic”, which fills the table with data for regular intervals of the independent (X, T, θ, or N) variable. With the NUMTYPE option 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 (

5. Enter the independent values in the left-hand column. Type in a number and press . You do not have

to enter them in order, because the function can rearrange them. To insert a number between two others, use .

CLEAR).

F1 and F2

You enter numbers into the X column

entries are generated automatically

Clear data Press CLEAR, to erase the data from a table.

Aplets and their views 2-19

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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 .

CLEAR

Example: plotting a circle

Plot the circle, x

y 9 x2–±=

To plot both the positive and negative y values, you need to define two equations as follows:

y 9 x2–= y 9 x2––=

1. In the Function aplet, specify the functions.

and

Toggles between two character sizes.

Displays the defining function expression for the highlighted column.

Deletes the highlighted row.

Clears all data from the table.

+ y

= 9. First rearrange it to read

2-20 Aplets and their views

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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.

Aplets and their views 2-21

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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 .

Once you have defined a function you can:

• create graphs to find roots, intercepts, slope, signed

• create tables to evaluate functions at particular

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.

y 2x 3+=

area, and extrema

values.

Getting started with the Function aplet

The following example involves two functions: a linear function and a quadratic equation

yx3+()

Open the

1. Open the Function aplet.

Function aplet

Function aplet 3-1

y 1 x–=

2–=

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.

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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 to

clear all lines.)

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

5. Plot the functions.

functions

3-2 Function aplet

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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.

Function aplet 3-3

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Analyse graph with FCN functions

To find a root of the quadratic function

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.

10.Move the cursor to the graph of the quadratic equation by pressing the or key. Then move the cursor so that it is near by pressing the

or key.

Select

Root

x 1–=

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 intersection of the two functions

3-4 Function aplet

11.Find the 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

To find the signed area of the two functions

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.

14.To find the area between the two functions in the range –2 ≤ x ≤ –1, first move the cursor to

F1 x() 1 x–=

Select Signed area

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

and select the signed area option.

Function aplet 3-5

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16.Press to accept using F2(x) = (x + 3) other boundary for the integral.

17.Choose the end value for x.

The cursor jumps to x = –1 on the linear function.

– 2 as the

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.

3-6 Function aplet

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HINT

Display the

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.

20.Display the numeric view.

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

23.Display the table of values.

table

Function aplet 3-7

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To navigate around a table

To go directly to a value

To access the zoom options

24.Move to X = –5.9.

6 times

25.Move directly to X = 10. 1 0

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

3-8 Function aplet

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

Function aplet 3-9

The FCN variables are contained on the VARS menu. To access FCN variables in HOME:

Select Plot FCN

or to choose a

variable

To access FCN variable in the Function aplet’s Symbolic view:

Select Plot FCN

or to choose a variable

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FCN functions The FCN functions are:

Function Description

Root Select Root to find the root of the

Extremum Select Extremum to find 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.

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 Slope to find the numeric

derivative at the current position of the cursor. The result is saved in a variable named SLOPE.

Signed area Select Signed area to 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.

3-10 Function aplet

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Function Description (Continued)

Intersection Select Intersection to 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 to plot the functions.

4. Press or to position the cursor at the starting point of the area you want to shade.

5. Press .

6. Press , then select Signed area and press

7. Press , choose the function that will act as the boundary of the shaded area, and press .

8. Press the or key to shade in the area.

9. Press 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 3-11

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Plotting a piecewise-defined function

Suppose you wanted to plot the following piecewisedefined function.

⎧

x 2 x 1–≤;+

⎪

fx()

1. Open the Function

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

⎨

⎪

4 xx1≥;–

⎩

1– x 1≤<;

aplet.

Select

Function

expression. (You can press to delete an existing line, or

CLEAR to clear all lines.)

2 CHARS ≤

CHARS > 1

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 and .

ygt()=

Getting started with the Parametric aplet

The following example uses the parametric equations

xt() 3 t yt() 3 tcos=

sin=

xft()=

Note: This example will produce a circle. For this example to work, the angle measure must be set to degrees.

Open the Parametric aplet

Define the expressions

Parametric aplet 4-1

1. Open the Parametric aplet.

Select

Parametric

2. Define the expressions.

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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, TRNG and TSTEP. TRNG specifies the range of t values. TSTEP specifies the step value between t values.

5. Set the TRNG and TSTEP so that t steps from 0° to 360° in 5° steps.

360

Plot the expression

4-2 Parametric aplet

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 Overlay Plot

A triangle is displayed rather than a circle (without changing the equation) because the changed value of TSTEP ensures 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 27 for further information.

Display the

9. Display the table of values.

numbers

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 4-3

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Polar aplet

Getting started with the Polar aplet

Open the Polar aplet

Define the expression

Specify plot settings

1. Open the Polar aplet. Select Polar

Like the Function aplet, the Polar aplet opens in the Symbolic view.

2. Define the polar equation .

2 π

3. Specify the plot settings. In this example, we will use the default settings, except for the θRNG fields.

SETUP-PLOT

CLEAR

4 π

r 2πθ2⁄()θ()

coscos=

Plot the

4. Plot the expression.

expression

Polar aplet 5-1

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Explore the graph

Display the numbers

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.

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.

5-2 Polar aplet

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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

– A Cobweb graph plots U

axis and Un on the vertical axis.

on the vertical axis.

n–1

on the horizontal

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 6-1

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Open the Sequence aplet

Define the expression

1. Open the Sequence aplet. Select

Sequence

The Sequence aplet starts in the Symbolic view.

2. Define the Fibonacci sequence, in which each term

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

U11= U21=

In the Symbolic view of the Sequence aplet, highlight the U

1 1

Note: You can use the

equations.

, , for .

+= n 3>

n 1–

n2–

1(1) field and begin defining your sequence.

, , ,

, and menu keys to assist in the entry of

Specify plot settings

6-2 Sequence aplet

3. In Plot Setup, first set the SEQPLOT option to

Stairstep. Reset the default plot settings by

clearing the Plot Setup view.

SETUP-PLOT

CLEAR

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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 6-3

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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:

U22AD+=

Open the Solve aplet

Define the equation

Enter known variables

1. Open the Solve aplet. Select Solve

The Solve aplet starts in the symbolic view.

2. Define the equation.

A D

Note: You can use the menu key to assist in the entry of equations.

3. Display the Solve numeric view screen.

7-2 Solve aplet

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4. Enter the values for the known variables.

2 7 7 8 1 6 6 7

1 0 0

HINT

If the Decimal Mark setting in the Modes input form ( MODES) is set to Comma, use instead of .

Solve the unknown variable

Plot the equation

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.

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 is . One of these is , with , that is,

Y 771.7284=

The other graph will be , with

U 16.67= D 100= Y 200A 277.8889+=

desired solution is the value of A where these two lines intersect.

U22AD+=

= V 27.78=

. This graph will be a horizontal line.

U22AD+=

and , that is,

. This graph is also a line. The

Solve aplet 7-3

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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 when done.

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 when done.

Finds a solution for the highlighted variable, based on the values of the other variables.

7-4 Solve aplet

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Key Meaning (Continued)

CLEAR Resets all variable values to zero or

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 . Solve starts looking for a solution at the initial guess.

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.

Clears highlighted variable to zero or deletes current character in edit line, if edit line is active.

clears the edit line, if cursor is in edit line.

HINT

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 Fixed 2 would be appropriate in this case.

Solve aplet 7-5

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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 to clear the message.

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

HINT

The Root-Finder at work

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.

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.

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 ) and start over with a different initial guess.

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:

TVX +=

Solve aplet 7-7

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where X is distance, V0 is initial velocity, T is time, and A is acceleration. This is actually two equations, Y = X and

T + (AT 2) / 2.

Y = V

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

X V T

T 2

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.

30 2

4 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 two graphs: one for and one for

X = V x T + A x T

/2. Since we have set in

this example, one of the graphs will be .

/2, the plot will produce

YX=

X 30=

Y 30=

Therefore, make the YRNG –5 to 35. Keep the XRNG default of – 6.5 to 6.5.

SETUP-PLOT

5 35

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.

3 50

Solve aplet 7-9

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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

Xmin and 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.

HINT

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

CLEAR to reset all

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 the Solve aplet to solve the equation X

7-10 Solve aplet

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

+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=

be in the form . 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 threeequation sets).

The hp40gs will alert you if no solution can be found, or if there is an infinite number of solutions.

. In a three-equation set, each equation must

ax by cz++ k=

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

Choose the equation set

Linear Solver aplet 8-1

1. Open the Linear Sequence aplet.

Select Linear

Solver

The Linear Equation Solver opens.

2. If the last time you used the Linear Solver aplet you solved for two equations, the twoequation input form is displayed (as in the

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

example in the previous step). To solve a threeequation set, press . Now the input form

displays three equations.

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 or

4. The cursor moves to the next co-efficient. Enter that coefficient, press or , 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 and begin entering the name. The 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.

8-2 Linear Solver aplet

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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 8-3

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HP 40gs User Manual

Specifications and Main Features

Frequently Asked Questions

User Manual