Texas Instruments Derive 5 Reference manual

Introduction to

TI EXPLORATIONS

™SOFTWARE

by Bernhard Kutzler

& Vlasta Kokol-Voljc

™

Bernhard KUTZLER

Vlasta KOKOL-VOLJC

Introduction to

DERIVETM 5

A book for learning how to use DERIVE 5

The following Derive™ 5 documentation is being

provided as a courtesy of the authors, Bernhard

Kutzler / Vlasta Kokol and the publisher Texas

Instruments. We invite you to use the following

abbreviated document during your personal

evaluation of Derive™ 5. Use of the document for

any other purpose is strictly prohibited.

Kutzler, Bernhard & Kokol-Voljc, Vlasta

Introduction to DERIVE 5

2000

2000 Kutzler & Kokol-Voljc OEG, Austria

1. Edition, 1. Printing: March 2000

Typesetting: Bernhard Kutzler, Leonding, Austria

Cover art: Texas Instruments Incorporated, Dallas, Texas, USA

The author and publisher make no warranty of any kind, expressed or implied, with regard to the

documentation contained in this book. The author and publisher shall not be liable in any event for

incidental or consequental damages in connection with, or arising out of, the furnishing, performance

or use of this text.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any

form, or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior

permission, in writing, from the publisher.

DERIVE is a trademark of Texas Instruments Incorporated

WINDOWS, WINDOWS95, WINDOWSNT, and WIN32Sare registered trademarks of Microsoft Corp.

iii

Table of Contents

Introduction ............................................................................................................................................. 1

Chapter 1: First Steps ............................................................................................................................. 3

Chapter 2: Documenting Polynomial Zero Finding ........................................................................... 23

Chapter 3: The Whole and Its Parts – Subexpressions .....................................................................43

Chapter 4: Equations and Inequalities ................................................................................................ 63

Chapter 5: Approximate Versus Exact Computations ...................................................................... 83

Chapter 6: Sequences and Families of Curves ................................................................................... 95

Chapter 7: Investigations in Space .................................................................................................... 117

Chapter 8: What Is ‘Simple’? .............................................................................................................. 135

Chapter 9: Vectors, Matrices, and Sets ............................................................................................. 153

Chapter 10: Parametric Plots ............................................................................................................. 171

Chapter 11: Towards a Module for Analytical Geometry ............................................................... 185

Chapter 12: Some Calculus ................................................................................................................ 203

Chapter 13: More on Plotting ............................................................................................................. 221

Chapter 14: What Else Can DERIVE Do? ............................................................................................ 243

Learn More about DERIVE ................................................................................................................... 261

Appendix A: DERIVE Startup Options ................................................................................................ 263

Appendix B: Factory Default DERIVE ................................................................................................ 265

Index ..................................................................................................................................................... 269

Preface

The desire to make DERIVE 5 easily and quickly accessible led to this book.

Many thanks to Albert Rich and Theresa Shelby, the principal authors of DERIVE 5, for their

continuous support during the writing of this book.

Many thanks to Patricia Littlefield and David Stoutemyer who polished the language of this

book.

Bernhard Kutzler & Vlasta Kokol-Voljc, February 2000

Introduction

DERIVE is a mathematical computer program. It processes algebraic variables, expressions,

equations, functions, vectors, and matrices like a scientific calculator processes floating point

numbers. DERIVE can perform numeric and symbolic computations, algebra, trigonometry,

calculus, and plot graphs in 2 and 3 dimensions. The main strength of DERIVE are symbolic

algebra and powerful graphics. It is an excellent tool for doing and applying mathematics, for

documenting mathematical work, and for teaching and learning mathematics.

For a teacher and student, DERIVE is the ideal tool for supporting the teaching and the learning of

mathematics. By providing numeric, algebraic, and graphic capabilities together with seamless

integration of these, DERIVE enables new approaches in teaching, learning, and understanding

mathematics. You will find that many topics can be treated more efficiently and effectively than

by using traditional methods. Many problems that require extensive and laborious training at

school can be solved with a single keystroke using DERIVE: It eliminates the drudgery of

performing long mathematical calculations. While DERIVE takes the burden of doing the

mechanical/algorithmic parts of solving a problem, students can concentrate on the

mathematical meaning of concepts. Instead of teaching and learning boring technical skills,

teachers and students can concentrate on the exciting and useful techniques of problem solving.

It has proven to be highly supportive for the cognitive development of advanced mathematical

concepts.

For an engineer, DERIVE is the ideal tool for fast and effective access to numerous mathematical

operations and functions and for visualizing problems and their solutions in various ways. If you

use DERIVE for your everyday mathematical work, you will find it a tireless, powerful, and

knowledgeable mathematical assistant that is easy to use.

This book is for learning how to use DERIVE 5 by private study. Install DERIVE 5 on your

computer. Starting with the first chapter, you will learn step by step how to use the program.

Follow all instructions and examples. The text leads you through several mathematical topics

that are used for learning how to solve mathematical problems with DERIVE. Many of the

examples also provide ideas for using DERIVE during teaching; some of them are explained in

more detail in “Educator’s footnotes.” Paragraphs starting with the symbol give instructions

about what you should do on your computer. Hundreds of screen dumps ensure that you will not

get lost on this journey.

2 Introduction

By solving typical mathematical high school level problems, you will learn to handle DERIVE 5 as

much as necessary for everyday use and for teaching or learning mathematics. You will learn

how to use the major commands, keys, and functions. At the end of each chapter you will find a

summary of the features learned in that chapter. The Quick Reference Guide at the end of the

book is a summary of commands, keys, functions, and utility files, which is organized by tasks.

The index at the end is useful if you need to locate a particular portion of the text.

All you need to run DERIVE 5 is a PC compatible computer with WINDOWS 95, WINDOWS 98, or

WINDOWS NT.

It is assumed that you know how to use computers and the WINDOWS operating system. The

screen shots in this book were produced from DERIVE running on WINDOWS NT. If you are running

DERIVE on WINDOWS 95 or 98,some of the screens may appear slightly different.

This book introduces all features and functions that are required for routine use of DERIVE 5.

There is more functionality than can be described here. This book is not a reference manual for

DERIVE. A complete reference to all features is included with the software as online help. Some

of the chapters give examples of how to use the online help.

We plan to write additional texts on DERIVE 5. Please regularly look at the web site

http://series.bk-teachware.com for new texts and local dealer information.

Have fun reading and discovering.

Chapter 1: First Steps

DERIVE makes it easy to perform mathematical operations: Enter an expression, apply a

command, and a new expression is obtained. All expressions can be used for new

computations—just like on a piece of paper. This chapter teaches the basic techniques of using

DERIVE 5. Note: For simplicity, we will abbreviate DERIVE 5 as DERIVE throughout this text.

This text assumes that you use a factory default DERIVE. Only then will your screen images fully

match those in this book. If you just installed DERIVE, it is a factory default version. If you use a

version of DERIVE that was used by someone else, we recommend that you turn it into a factory

default version now. Appendix B gives instructions on how to do this.

Start DERIVE by double clicking on the DERIVE icon. If there is no DERIVE icon on your computer’s

desktop, you probably will find DERIVE on the Start menu or via Start>Programs.



The following screen appears after a few seconds:

The DERIVE screen comprises (from top to bottom):

•the Titlebar

•the Menu Bar

•the Command Toolbar

•a (currently empty) Algebra Window, also called the View

•the Status Bar

•the Expression Entry Toolbar, also called the entry line

•the Greek Symbol Toolbar and the Math Symbol Toolbar

4 Chapter 1: First Steps

Work with DERIVE by entering expressions and applying commands, thus creating a worksheet.

After starting DERIVE, the system is ready to accept user input via the Expression Entry Toolbar,

as is indicated by the blinking cursor in the toolbar’s entry field. Input mode can be implemented

with the Command Toolbar’s tenth button from the left, labeled .

Learn more about the button by moving the mouse pointer onto it.

The message Author Expression below the cursor is the button’s title. The Status Bar message

Enter new expression in active work sheet is the button’s function description.

Prepare for entering an expression: Move the mouse pointer onto , then click (i.e. press

and release) the left mouse button.

Enter the fraction: 2/3

End the input with the ‘Enter’-key (¢).

DERIVE displays this expression as a fraction with a horizontal line, a numerator, and a

denominator, i.e. in “2-dimensional” output format, as opposed to the “1-dimensional” or “linear”

input format used for entering the number. The expression’s unique label number, #1, is shown

to the left of the expression. DERIVE is again ready to accept the next input, i.e. input control (the

focus) is still in the entry line. Also observe that a copy of the input is still in the entry field and is

entirely highlighted. This has the same meaning as in text editors and word processors. You can

remove the highlighting with the right arrow key, then edit the string of symbols, or you can

replace the marked string by typing new symbols.

Kutzler & Kokol-Voljc: Introduction to DERIVE 55

Replace the last input by 11

+with an intentional typographical error:

Enter 1/2+1&3 (¢).

When a syntax error is detected, the cursor is moved to the location of the error and the cause of

the error is displayed in the Status Bar’s first pane. In the above example DERIVE discovered an

unexpected special character. In some cases (for example,when entering an opening

parenthesis instead of the division symbol) there are several errors possible, and DERIVE can only

guess.

Update the input to 1/2+1/3: Use the (Del) key (or the right arrow key (Æ) followed by

the backspace key (æ_)) to delete the incorrect character, then type the division operator.

Conclude with (¢).

The expression and its label, #2, are displayed. The new expression is highlighted in reverse

video. Expression #1 is no longer highlighted.

If you mistyped the input and want to delete the highlighted expression for a retry, use (Esc) to

move the focus into the algebra window, use the ‘Delete’ key (Del) to delete the highlighted

expression, then use the Author Expression button to move the focus back into the entry line.

An alternative technique for replacing an expression will be explained in Chapter 2.

Simplify expression #2 using the Command Toolbar’s Simplify button .

The result becomes the next expression with the label #3. By default, simplified expressions are

displayed centered. This makes it easy to distinguish between entry and result. As with many

other behaviors of DERIVE, this can be customized if desired.

Even after using the Simplify button, the focus still is in the entry line. Enter the next

expression, 24 . To enter the square root symbol, use the respective button on the Math

Symbol Toolbar:

Enter 24 as: 24 (¢)

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