Texas instruments DERIVE 5 Introduction
™
Introduction to
by Bernhard Kutzler
& Vlasta Kokol-Voljc
TI EXPLORATIONS™SOFTWARE
Bernhard KUTZLER
Vlasta KOKOL-VOLJC
Introduction to
TM
D
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 D
ERIVE
5
ERIVE
5
Kutzler, Bernhard & Kokol-Voljc, Vlasta
Introduction to D
ERIVE
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.
ERIVE
is a trademark of Texas Instruments Incorporated
D
INDOWS
W
, W
INDOWS
95, W
INDOWS
NT, and WIN32S are registered trademarks of Microsoft Corp.
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
ERIVE
Chapter 14: What Else Can D
ERIVE
Learn More about D
Appendix A: D
Appendix B: Factory Default D
Index ..................................................................................................................................................... 269
ERIVE
................................................................................................................... 261
Startup Options ................................................................................................ 263
Do? ............................................................................................ 243
ERIVE
................................................................................................ 265
iii
Preface
v
The desire to make D
Many thanks to Albert Rich and Theresa Shelby, the principal authors of D
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
ERIVE
5 easily and quickly accessible led to this book.
ERIVE
5, for their
Introduction
ERIVE
D
is a mathematical computer program. It processes algebraic variables, expressions,
equations, functions, vectors, and matrices like a scientific calculator processes floating point
numbers. D
calculus, and plot graphs in 2 and 3 dimensions. The main strength of D
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, D
mathematics. By providing numeric, algebraic, and graphic capabilities together with seamless
integration of these, D
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 D
performing long mathematical calculations. While D
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, D
operations and functions and for visualizing problems and their solutions in various ways. If you
use D
knowledgeable mathematical assistant that is easy to use.
ERIVE
can perform numeric and symbolic computations, algebra, trigonometry,
ERIVE
are symbolic
ERIVE
is the ideal tool for supporting the teaching and the learning of
ERIVE
enables new approaches in teaching, learning, and understanding
ERIVE
: It eliminates the drudgery of
ERIVE
takes the burden of doing the
ERIVE
is the ideal tool for fast and effective access to numerous mathematical
ERIVE
for your everyday mathematical work, you will find it a tireless, powerful, and
This book is for learning how to use D
ERIVE
5 by private study. Install D
ERIVE
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 D
ERIVE
examples also provide ideas for using D
during teaching; some of them are explained in
ERIVE
. Many of the
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 D
ERIVE
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 D
INDOWS
W
NT.
ERIVE
5 is a PC compatible computer with W
It is assumed that you know how to use computers and the W
screen shots in this book were produced from D
D
ERIVE
on W
INDOWS
95 or 98, some of the screens may appear slightly different.
ERIVE
running on W
This book introduces all features and functions that are required for routine use of D
There is more functionality than can be described here. This book is
ERIVE
. A complete reference to all features is included with the software as online help. Some
D
INDOWS
95, W
INDOWS
operating system. The
INDOWS
not
a reference manual for
INDOWS
98, or
NT. If you are running
ERIVE
5.
of the chapters give examples of how to use the online help.
ERIVE
We plan to write additional texts on D
http://series.bk-teachware.com
for new texts and local dealer information.
5. Please regularly look at the web site
Have fun reading and discovering.
Chapter 1: First Steps
ERIVE
D
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
ERIVE
D
5. Note: For simplicity, we will abbreviate D
This text assumes that you use a factory default D
match those in this book. If you just installed D
version of D
ERIVE
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.
ERIVE
Start D
by double clicking on the D
desktop, you probably will find D
ERIVE
ERIVE
on the
The following screen appears after a few seconds:
ERIVE
ERIVE
. Only then will your screen images fully
ERIVE
, it is a factory default version. If you use a
icon. If there is no D
Start
menu or via
5 as D
ERIVE
throughout this text.
ERIVE
icon on your computer’s
Start>Programs
.
ERIVE
The D
•
•
•
•
•
•
•
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 D
After starting D
ERIVE
by entering expressions and applying commands, thus creating a worksheet.
ERIVE
, 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
The message
Author Expression
Enter new expression in active work sheet
Prepare for entering an expression: Move the mouse pointer onto
by moving the mouse pointer onto it.
below the cursor is the button’s title. The Status Bar message
is the button’s function description.
.
, then click (i.e. press
and release) the left mouse button.
Enter the fraction:
2/3
End the input with the ‘Enter’-key
ERIVE
D
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
ERIVE
to the left of the expression. D
focus
) is still in the entry line. Also observe that a copy of the input is still in the entry field and is
is again ready to accept the next input, i.e. input control (the
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 D
11
(¢)
+
23
.
Replace the last input by
Enter
1/2+1&3
55
ERIVE
with an intentional typographical error:
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 D
ERIVE
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 D
ERIVE
can only
guess.
Update the input to
the backspace key
Conclude with
(¢)
1/2+1/3
(æ_)
: Use the
) to delete the incorrect character, then type the division operator.
(Del)
key (or the right arrow key
.
(Æ)
followed by
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
move the focus into the algebra window, use the ‘Delete’ key
expression, then use the
Author Expression
button to move the focus back into the entry line.
(Del)
to delete the highlighted
(Esc)
to
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
ERIVE
other behaviors of D
Even after using the
24
expression,
. To enter the square root symbol, use the respective button on the Math
, this can be customized if desired.
Simplify
button, the focus still is in the entry line. Enter the next
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
“How do you recognize a mathematician?”
asked:
mathematician considers expression #5 a beautiful result.”
and suggested the following answer:
Most students strive to replace
such an expression by the corresponding floating point approximation. D
ERIVE
can do this as
“A
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
then using the cursor keys
(½)
or
(¼)
to move the highlighting one expression up or down.
(Esc)
key,
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 expres-
sion 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
command) to expression #4. The computation time displayed in the third pane,
Approximate
0.000s
button (or
,
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,
expression #5,
Simp(#4)
User
, means that it was entered by the user; the annotation of
, 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.
ERIVE
worksheets also can include text and other objects. The easiest way of entering text is via
D
the Command Toolbar’s
Insert Text
button . New expressions are added at the end of the
Kutzler & Kokol-Voljc: Introduction to D
57
ERIVE
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
56
. Due to the previous activity, the focus now is in the algebra
1234
window. Before you can enter another expression, move the focus into the entry line.
1234^56
Enter
by using the
of digits followed by
Author Expression
(¢)
. The exponentiation operator ^ can be found on both the
button , then typing the respective string
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
has the same effect as concluding the input with the
Enter the above expression with
Simplify
Unlike ordinary calculators, D
button .
ERIVE
can perform nonnumeric (symbolic, algebraic) computations
Author Expression
(¢)
button. Selecting this button
key. Try it:
, then simplify as usual using the Command Toolbar’s
such as simplifying expression #10 into expression #11.
For the next example use the Expression Entry Toolbar’s second button,
To simplify
+
2xx
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:
For the next example use the Expression Entry Toolbar’s third button,
Enter and simplify
Simplify
button .
+
sinxy x
by typing
xy+sinx
Simp(User)
.
then using the entry line’s
Author and
Kutzler & Kokol-Voljc: Introduction to D
59
ERIVE
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 D
ERIVE
. 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
xy+sinx
as
for
⋅+
sin( )xy 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, D
ERIVE
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
denominator to enter a rational expression such as
2
. If the parentheses are omitted in this
+
1x
example, the resulting expression has a different meaning.
Enter:
2/x+1
Oops—the expression on the screen looks different from the intended expression! D
ERIVE
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
Educator’s footnote:
the students to input expressions given to them on the chalkboard or a piece of paper. Because D
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.
A very simple educational exercise with D
ERIVE
, therefore, consists of asking
1
ERIVE
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
input to
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
2/(x+1)
by adding the parentheses, then enter the expression with
(Æ)
to remove the highlighting. Change the
(¢)
(Esc)
.
.
Delete expression #15: Use the
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 D
syntax error message:
ERIVE
D
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)
Enter
attempts to position the cursor in front of the expected error. Since a superfluous closing
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)
after moving the focus into the entry line with
Delete Object
ERIVE
will issue a warning in the form of an appropriate
button or press the
(Del)
.
key.
output
To choose the third variant insert an opening parenthesis between the division operator and the
variable x.
−−
45x
1
x
−−
45x
1
x
45x
1
−
−
x
45x
−−
1
x
1
−−
45x
x
45x
1
−−
x
Kutzler & Kokol-Voljc: Introduction to D
Edit the input string to
4x-1/(x-5)
511
ERIVE
2
then press
(¢)
.
When working with D
When focus is in the entry line,
Author Expression
the
ERIVE
, focus can be either in the entry line or in the algebra window (View).
(Esc)
will move focus into the View. When focus is in the View,
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
x^2
Enter and simplify
button or to use the entry line’s
obtained from the Math Symbol Toolbar (
Type
√
x^2
√
. It is up to you to either use the ‘Enter’ key followed by the
then press
Enter and Simplify
(Ctrl)+(¢)
) or entered as
. This is the same as
button. The square root symbol √ can be
(ÿ)
.
(Ctrl)-(Q)
.
, i.e. this is a simple way to
Simplify
perform an “enter and simplify” operation without using the mouse.
As an alternative, introduce a pair of parentheses around
Enter and simplify:
2
Educator’s footnote:
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.
(x^2)
√
This is another example for an elementary educational use of D
x^2
.
ERIVE
. Ask
12 Chapter 1: First Steps
The last two examples are remarkable for two reasons. First, they demonstrate the importance
of using parentheses to differentiate between
2
x
). Second, expression #20 shows how carefully D
()
α
The third power of
−
is entered as follows:
1
2
x
(meaning
ERIVE
2
x
) and
()
simplifies expressions.
2
x
(meaning
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.
(α-1)^3
Enter
Try to expand expression #21, first by simplifying with
Prepare for opening the
Simplify
Open the
. (Insert Alpha with the Greek Symbol Toolbar button
command.
Simplify
Simplify
menu by clicking the left mouse button.
menu by moving the mouse pointer above the Menu Bar’s
.)
.
This menu offers several commands. The
expression.
Select this command by moving the mouse pointer above the word
Expand
command is appropriate for expanding an
Expand
. . .
Kutzler & Kokol-Voljc: Introduction to D
. . . then invoke the command by clicking on it with the left mouse button.
513
ERIVE
D
ERIVE
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
xpand_)
the default button, which here is
displayed.) Use the
(_Cancel_)
(_E
button or the
you want an unsimplified application of the
Perform the expansion with the suggested parameters by using
(¢)
because this is the default button or click on
A keyboard alternative for selecting the
following standard W
the underscore under the letter S in
underscore, but now without the
INDOWS
technique:
Simplify
(Alt)
. (The default button is the one prominently
(Esc)
EXPAND
Expansion
(Alt)+(S)
), then press
key to cancel the command. Use
function.
(_E
xpand_)
(_E
xpand_)
command from the
opens the
(E)
.)
Simplify
Simplify
menu (use
(again the letter with the
(_OK_)
(either press
menu is the
(S)
because of
, which is used only to open menus.) This technique
if
works for all menu commands.
For all buttons from the Command Toolbar there exist corresponding menu commands. Use
π
commands for the next example. Enter, simplify, then approximate
To enter the above expression, select the
sin(¹/4)
(¢)
. (Obtain π from either the Greek or the Math Symbol Toolbar. A button
Author>Expression
sin 4
command, then type
.
()
for this frequently used character is in both of these toolbars.)
Simplify expression #24 with the
Simplify>Basic
command.
Loading...