Texas instruments DERIVE 5 Introduction

Introduction to

™

by Bernhard Kutzler & Vlasta Kokol-Voljc

T I E X P L O R A T I O N S™ S O F T W A R E

Bernhard KUTZLER

Vlasta KOKOL-VOLJC

Introduction to

DERIVETM 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.

A book for learning how to use DERIVE 5

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 WIN32S are 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

v

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 5

5

Replace the last input by 1 + 1 with an intentional typographical error:

2 3

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 (¢)

6	Chapter 1: First Steps

Simplify using .

This is different from what an “ordinary” calculator would produce. A mathematician once asked: “How do you recognize a mathematician?” and suggested the following answer: “A mathematician considers expression #5 a beautiful result.” Most students strive to replace such an expression by the corresponding floating point approximation. DERIVE can do this as well: Highlight expression #4 so that you can apply a different command to it.

Highlight expression #4 by moving the mouse pointer anywhere in the row occupied by the expression, then clicking the left mouse button.

Selecting an expression with the mouse button is one technique of highlighting it. An alternative technique is first to move the focus into the algebra window (if necessary) using the (Esc) key, then using the cursor keys (½) or (¼) to move the highlighting one expression up or down.

Approximate using the Command Toolbar’s Approximate button .

While an expression is highlighted, the Status Bar’s second pane shows the automatically generated expression annotation. The third pane shows the computing time in case the expression was obtained as a result of a computation. For expression #6 this is:

The automatically generated annotation explains how the expression was obtained.

Approx(#4) means that the expression was obtained by applying the Approximate button (or command) to expression #4. The computation time displayed in the third pane, 0.000s, indicates that the calculation took less than 0.001 seconds (the time may be different on your computer).

Highlight expression #4, . . .

. . . then expression #5.,

The annotation of expression #4, User, means that it was entered by the user; the annotation of expression #5, Simp(#4), indicates that the expression was obtained by applying the Simplify button (or command) to expression #4. The first pane is always available for messages associated with a menu item, button, or command status.

DERIVE worksheets also can include text and other objects. The easiest way of entering text is via the Command Toolbar’s Insert Text button . New expressions are added at the end of the

Kutzler & Kokol-Voljc: Introduction to DERIVE 5

7

worksheet. Other objects (including text objects) are added after the highlighted object. To insert a text object above the square root of 24, first highlight the object that is now above it.

Highlight expression #3.

Display a function description of the Insert Text button by moving the mouse pointer onto it.

Insert a text object by clicking on the Insert Text button .

Highlighting of a text object is indicated by a frame around it. The blinking cursor inside indicates text editing mode.

Enter the text: We compute the square root of 24:

A text object allows simple text editing similar to what you can do in standard text editors. Later you will learn how to change the font size, alignment, color, etc.

As a next example compute 123456 . Due to the previous activity, the focus now is in the algebra window. Before you can enter another expression, move the focus into the entry line.

Enter 1234^56 by using the Author Expression button , then typing the respective string of digits followed by (¢). The exponentiation operator ^ can be found on both the keyboard and the Math Symbol Toolbar. (It is the sixth symbol from the left in the first row.)

Simplify using .

This is a very big number. For those who want to know the number of digits, there are two methods to find out: First, you can count them. Second, you can approximate the number.

8	Chapter 1: First Steps

Approximate using .

The answer is displayed in scientific notation. Since the count of whole digits is one more than the power of 10, the number has 173+1 = 174 digits.

In the next exercise, you will learn a different technique of entering expressions by using the buttons preceding the entry field.

Type into the entry line x/3+x/4 this time without concluding with (¢).

Note the five buttons left of the entry field. The usual technique of moving the mouse pointer onto a button reveals the first one, , as the Author Expression button. Selecting this button has the same effect as concluding the input with the (¢) key. Try it:

Enter the above expression with , then simplify as usual using the Command Toolbar’s

Simplify button .

Unlike ordinary calculators, DERIVE can perform nonnumeric (symbolic, algebraic) computations such as simplifying expression #10 into expression #11.

For the next example use the Expression Entry Toolbar’s second button, .

To simplify x + 2x immediately, type x+2x then select the entry line’s Simplify button .

This button simplified the entered expression immediately without the usual display of the unsimplified expression. Note the result’s annotation: Simp(User)

For the next example use the Expression Entry Toolbar’s third button, .

Enter and simplify xy + sin x by typing xy+sinx then using the entry line’s Author and Simplify button .

Kutzler & Kokol-Voljc: Introduction to DERIVE 5

9

This button produced two expressions, #13 and #14 and has the same effect as entering the unsimplified expression with (¢) or , then simplifying it with . It is, therefore, a convenient shortcut for the frequently used “enter and simplify.” This example also shows how convenient fast input is in DERIVE. You can enter expressions just as you would write them on paper. For ‘x times y ‘ simply enter xy. No multiplication operator is needed between x and y. For ‘Sine of x ‘ simply enter sinx. No parentheses are needed around x.

The Expression Entry Toolbar has buttons for entering, simplifying, entering & simplifying, approximating, and entering & approximating expressions.

The simplified expression #14 differs from the unsimplified expression #13 only in the order in which its terms are displayed. While unsimplified expressions are displayed as they were entered (except for the 2-dimensional pretty print format), simplified expressions are displayed in a standardized format using a certain term ordering.

Back to how simple it is to enter expressions. A consequence of the convenient fast input, such as xy+sinx for x y + sin(x) , is that variable names can consist of only one character (for example x and y). This suffices most of the time, but if you need to use multicharacter variable names, DERIVE allows this, too (for example time or x12). Using multicharacter variable names will be explained in Chapter 14.

Clearly, you cannot omit all parentheses. For example, you will need to parenthesize the

2

denominator to enter a rational expression such as x +1 . If the parentheses are omitted in this example, the resulting expression has a different meaning.

Enter: 2/x+1

Oops—the expression on the screen looks different from the intended expression! DERIVE applies operations in the conventional order, for example multiplication and division before addition and subtraction. As you can see from the above example, the 2-dimensional screen display of an input provides you with valuable feedback about the soundness of your input.1

1 Educator’s footnote: A very simple educational exercise with DERIVE, therefore, consists of asking the students to input expressions given to them on the chalkboard or a piece of paper. Because DERIVE features 2-dimensional output of expressions, the students get an immediate feedback. If the expression on the screen looks different from the one on the chalkboard or paper, then the input was wrong, and they must try again. When the teacher lets students input expressions of increasing complexity, they learn how to “linearize” expressions by trying and experimenting (trial and error), and learn to understand the structure of expressions. In this way, they improve their competence in recognizing structures, which is one of the basic mathematical skills important in many areas.

10	Chapter 1: First Steps

When correcting the most recent input, you can take advantage of the fact that a copy of the

most recent input and the focus are still in the entry line.

To edit the expression use the right arrow key (Æ) to remove the highlighting. Change the input to 2/(x+1) by adding the parentheses, then enter the expression with (¢).

Now it looks correct. Since you don’t need expression #15 any more, delete it.

Prepare for deletion: Highlight expression #15 either with the mouse or with the keyboard’s arrow keys after moving the focus into the algebra window using (Esc).

Delete expression #15: Use the Delete Object button or press the (Del) key.

The expression that was expression #15 disappeared. The expression that was expression #16 has become expression #15. By default, automatic renumbering adjusts expression numbers so that they begin with #1 and have no gaps.

Errors such as omitting a whole pair of parentheses may change the meaning of an expression, as was the case in the previous example. If only one parenthesis is omitted, the input becomes a meaningless character string, and DERIVE will issue a warning in the form of an appropriate syntax error message:

Enter 4x-1/x-5) after moving the focus into the entry line with .

DERIVE attempts to position the cursor in front of the expected error. Since a superfluous closing parenthesis can be spotted while a missing opening parenthesis obviously cannot, the first alternative is used for the error message. Depending on how the expression should look, you have to either delete the closing parenthesis or insert an opening parenthesis somewhere before it. For the above example there are six possible repairs:

input

4x-1/x-5

4x-1/x-(5)

4x-1/(x-5)

4x-(1/x-5)

4(x-1/x-5)

(4x-1/x-5)

output

1

1

1

1

1

1

4x −

− 5

4x −

− 5

4x −

4x −

− 5

4 x −

− 5

4x −

− 5

x

x

x − 5

x

x

x

To choose the third variant insert an opening parenthesis between the division operator and the

variable x.

Kutzler & Kokol-Voljc: Introduction to DERIVE 5

11

Edit the input string to 4x-1/(x-5) then press (¢).2

When working with DERIVE, focus can be either in the entry line or in the algebra window (View). When focus is in the entry line, (Esc) will move focus into the View. When focus is in the View, the Author Expression button or its hot key equivalent, (F2), moves it into the entry line. Another method to move focus is using the mouse. Focus is where one last moved the mouse pointer to and then pressed the left mouse button.

Ensure that focus is in the entry line by moving the mouse pointer into the entry line’s entry field, then clicking with the left mouse button.

The disadvantage of this method is that it removes highlighting if there was any, so now you cannot simply replace the old input with a new one by starting to type the new input string. You could use the backspace key several times to delete the old string, but a more elegant way is to use the tab key.

Highlight the contents of the entry line with the tab key (ÿ).

Enter and simplify √x^2. It is up to you to either use the ‘Enter’ key followed by the Simplify button or to use the entry line’s Enter and Simplify button. The square root symbol √ can be obtained from the Math Symbol Toolbar () or entered as (Ctrl)-(Q).

Type √x^2 then press (Ctrl)+(¢). This is the same as , i.e. this is a simple way to perform an “enter and simplify” operation without using the mouse.

As an alternative, introduce a pair of parentheses around x^2.

Enter and simplify: √(x^2)

2 Educator’s footnote: This is another example for an elementary educational use of DERIVE. Ask students how many different expressions they can generate by inserting 1, 2 (or more) pairs of parentheses into a valid string of characters. This is another excellent exercise to help students gain an understanding of the structure of expressions.

12 Chapter 1: First Steps

The last two examples are remarkable for two reasons. First, they demonstrate the importance

2	(meaning (	2
of using parentheses to differentiate between x		x ) ) and x2 (meaning

(x2 ) ). Second, expression #20 shows how carefully DERIVE simplifies expressions.

The third power of α −1 is entered as follows:

Enter (α-1)^3. (Insert Alpha with the Greek Symbol Toolbar button .)

Try to expand expression #21, first by simplifying with .

This did not change anything. Now you have an opportunity to apply one of those commands for

which there is no equivalent Command Toolbar button.

Prepare for opening the Simplify menu by moving the mouse pointer above the Menu Bar’s

Simplify command.

Open the Simplify menu by clicking the left mouse button.

This menu offers several commands. The Expand command is appropriate for expanding an

expression.

Select this command by moving the mouse pointer above the word Expand . . .

Kutzler & Kokol-Voljc: Introduction to DERIVE 5

13

. . . then invoke the command by clicking on it with the left mouse button.

DERIVE opens the Expand Expression dialog box. You will obtain similar dialog boxes with all commands that require specification of parameters. The above dialog box requires the specification of the expansion variable and the amount of expansion. Often it is enough to accept the default specifications and immediately exit the dialog box with the ‘Enter’ key or by clicking the default button, which here is (_Expand_). (The default button is the one prominently displayed.) Use the (_Cancel_) button or the (Esc) key to cancel the command. Use (_OK_) if you want an unsimplified application of the EXPAND function.

Perform the expansion with the suggested parameters by using (_Expand_) (either press

(¢) because this is the default button or click on (_Expand_).)

A keyboard alternative for selecting the Expansion command from the Simplify menu is the following standard WINDOWS technique: (Alt)+(S) opens the Simplify menu (use (S) because of the underscore under the letter S in Simplify), then press (E) (again the letter with the underscore, but now without the (Alt), which is used only to open menus.) This technique works for all menu commands.

For all buttons from the Command Toolbar there exist corresponding menu commands. Use commands for the next example. Enter, simplify, then approximate sin (π 4) .

To enter the above expression, select the Author>Expression command, then type sin(¹/4) (¢) . (Obtain π from either the Greek or the Math Symbol Toolbar. A button

for this frequently used character is in both of these toolbars.)

Simplify expression #24 with the Simplify>Basic command.