GETTING STARTED WITH

ii

For use with Calculus WIZ 2.0 or later.

For information on additional copies of this documentation:contact Wolfram Research.

Comments on this manual are welcomed at: comments@wolfram.com.

Manual Credits:Manual author: Keith Stroyan • Designers: Jeremy Davis, Andy Hunt,André Kuzniarek, Richard Miske, Jamie Scott, Kara Wilson •Editorial staff: George Beck, Rebecca Bigelow, Mary JaneHarshbarger, Louise Holubek, Rebecca Kirk, Susan Kittivanichkulkrai,Marcia Krause, Connie Neil, Jay Peterson, Lorraine Selander,Caroline Small

Includes Dialogs and Utilities packages by John M. Novak.

For software credits, see About Calculus WIZ in the Calculus WIZfront end.

© 2003 Wolfram Research, Inc.

All rights reserved. No part of this document may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic,mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright holder.

Wolfram Research, Inc. is the holder of the copyright to the Calculus WIZ software system described in this document, including without limitation suchaspects of the system as its code, structure, sequence, organization, “look and feel”, programming language, and compilation of command names. Useof the system unless pursuant to the terms of a license granted by Wolfram Research or as otherwise authorized by law is an infringement of thecopyright. Wolfram Research makes no representations, express or implied, with respect to this documentation or the software it describes, includingwithout limitations, any implied warranties of merchantability, interoperability, or fitness for a particular purpose, all of which are expressly disclaimed.Users should be aware that included in the terms and conditions under which Wolfram Research is willing to license Calculus WIZ is a provision thatWolfram Research and its distribution licensees, distributors, and dealers shall in no event be liable for any indirect, incidental or consequentialdamages, and that liability for direct damages shall be limited to the amount of the purchase price paid for Calculus WIZ.

In addition to the foregoing, users should recognize that all complex software systems and their documentation contain errors and omissions. WolframResearch shall not be responsible under any circumstances for providing information on or corrections to errors and omissions discovered at any timein this document or the software it describes, whether or not they are aware of the errors or omissions. Wolfram Research does not recommend the useof the software described in this document for applications in which errors or omissions could threaten life, injury or significant loss.

Calculus WIZ is a trademark of Wolfram Research, Inc. Mathematica, “WIZ”, and the Beanie Design are registered trademarks of Wolfram Research,Inc. All other product names are trademarks of their producers.

Printed in the United States of America.

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Item: T4044 158803 0903.RCM

iii

General information:www.wolfram.com/wiz

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Technical support and bug reports:support@wolfram.comsupport.wolfram.comSupport is available only to registered users.

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Reseller support only.

Preface

Calculus is one of the greatest inventions of the human intellect. It has acted as the language of physical science and technology for 300 years. Now it has an expanding role in ecology, economics, finance, and other areas; but calculus is also notorious as a stumbling block for entrance into these fields. Calculus WIZ changes calculus for the beginner. I wrote Calculus WIZ to help beginners through the basics in the hope that it would encourage them to continue using calculus in the deep and interesting applications of the subject. Calculus WIZ uses Mathematica’s unrivaled computational power and unprecedented ease of use to help beginners solve the basic problems of calculus.

Best wishes, students; I hope you whiz through the basics of calculus and go on to make calculus a powerful tool in your professional and intellectual lives.

Keith Stroyan

July 2000

iv

Table of Contents

Getting Started

� What Is Calculus WIZ? ......................................................................... 1

� Running Calculus WIZ ......................................................................... 2

The Basics of Calculus WIZ

� Introducing Notebooks ........................................................................ 4

� Entering Text and Equations ................................................................ 9

Calculus with Calculus WIZ

� Introduction ........................................................................................ 18

� Derivatives ......................................................................................... 18

� Integrals ............................................................................................ 19

� Limits ................................................................................................ 20

� Infinite Series .................................................................................... 21

� Graphs .............................................................................................. 21

Finding Textbook Topics

� Calculus WIZ Help Browser ................................................................. 23

� Table of Contents .............................................................................. 23

� Table of Solvers ................................................................................ 24

� Calculus WIZ Index .......................................................................... 24

� The Master Index ............................................................................. 24

Homework with Calculus WIZ

� Introduction .................................................................................... 25

� A Tangent Line Exercise ................................................................. 25

� An Area Exercise ........................................................................... 27

v

About Professor Stroyan

Keith Stroyan is professor of mathematics at the University of Iowa. He received his Ph.D. in mathematics from the California Institute of Technology in 1971. He has had a distinguished career in mathematics research, including the publication of the widely acclaimed monographs Foundations of Infinitesimal Stochastic Analysis and Introduction to the Theory of Infinitesimals. Throughout his career Professor Stroyan has maintained a strong interest in computing and education, published books on computing laboratories for calculus and linear algebra in the 1970s, and run summer workshops in analytical geometry and computer graphics for high school teachers for over a decade. An outspoken advocate of calculus reform, Professor Stroyan has authored six undergraduate-level mathematics textbooks, including Calculus Using Mathematica, which was funded by the National Science Foundation as one of five major calculus reform projects in the 1980s. He has taught over 5,000 calculus students who have gone on to lead successful careers in medicine, law, engineering, mathematics, and other fields.

When not fighting for better calculus education, Professor Stroyan and his wife run Shohola Labradors, a retriever kennel at their home in the Iowa countryside.

To learn more about Professor Stroyan, visit the Calculus WIZ web site at www.wolfram.com/wiz.

vi

Getting Started

What Is Calculus WIZ?

Calculus WIZ is a tool that can help you solve most of the problems of a traditional first-year calculus course. You solve calculus problems simply by clicking a computer button and filling in information. Usually these steps give you the solutions to the textbook problems plus information beyond the textbook. This is possible because Mathematica, on which Calculus WIZ is based, has changed the kind of mathematics that computers can do. Thus Calculus WIZ changes calculus for the beginner.

Calculus WIZ is organized into chapters and sections that follow the outline of your regular calculus text. Chapters and sections are easily accessible from the Help Browser and built-in hyperlinks located throughout Calculus WIZ. Although Calculus WIZ is a complete reference to calculus, much like your calculus textbook, it is mainly designed to assist you with your homework from your regular calculus class.

The best way to use Calculus WIZ is to quickly search for the topic you are interested in, read through the explanation in the Help Browser, and then use Calculus WIZ solvers to do problems on your topic. (You can also see all the details of a solution by using the Calculus WIZ template buttons.)

The “Homework with the Calculus WIZ” notebook has examples of medium-difficulty textbook exercises. They are solved in three different ways: with automatic Calculus WIZ solvers, by textbook methods, and with short Calculus WIZ template programs. The three solutions let you compare the approaches. Calculus WIZ solvers and templates are also described in the printed manual.

1

Running Calculus WIZ

� Installation

Processor type 80386�compatible or later

Operating system Windows 95, 98, or Me; Windows NT 4.0, Windows 2000, or later

Hard disk space 150 MB recommended

System �memory ��RAM� Windows 95�98�Me:24 MB recommended16 MB minimum

Windows NT�2000:32 MB recommended32 MB minimum

System requirements for Windows.

To install Calculus WIZ on Windows:

1. Insert the Calculus WIZ CD. The CD window appears on your screen.

If the CD window does not appear, the CD autorun feature might be disabled on your computer. In that case, install Calculus WIZ by opening the CD-ROM drive icon, opening the PC folder, and double-clicking the MathInstaller icon.

2. Click the button labeled “Install Calculus WIZ to a hard disk”.

3. The “Welcome to MathInstaller” dialog box appears. Click Next to continue.

4. The “Choose Installation Location” dialog box appears. You may change the location, or accept the default location and click Next to continue.

5. The “Select Start Menu Folder” dialog box appears. Choose the name for the program folder, or accept the default folder name and click Next to continue.

6. If you are ready to install Calculus WIZ, click Next to begin the installation.

7. Register Calculus WIZ with Wolfram Research. See the registration card for details.

2

Operating system Mac OS 7.53 or later

Hard disk space 150 MB recommended

System �memory ��RAM� Front end :10 MB recommended5 MB minimum

Kernel :10 MB recommended5 MB minimum

System Requirements for Macintosh.

To install Calculus WIZ on the Macintosh:

1. Insert the Calculus WIZ CD. A new window appears on your screen.

2. Double-click the Calculus WIZ installer icon.

3. Click Continue in the splash screen that appears.

4. The “Calculus WIZ Installer” dialog box appears. Click Install to begin the installation process.

5. When the installation is complete, the installer reports “Installation was successful.” Click Quit.

6. Register Calculus WIZ with Wolfram Research. See the registration card for details.

� Running Calculus WIZ

To start and run Calculus WIZ, choose Calculus WIZ from your program group.

The first time you start Calculus WIZ after installation, you will see a window labeled “Building Help Browser Index.” This process will only take a moment or two and will not have to be repeated in subsequent sessions.

The basics of Calculus WIZ are presented in the next section. To start using Calculus WIZ to solve your calculus problems, go to “Calculus with Calculus WIZ” on page 18.

3

The Basics of Calculus WIZ

Introducing Notebooks

� Opening, Saving, and Closing a Notebook

To create a new notebook, click File � New.

To open a previously saved notebook, click File � Open and select the file from the dialog box.

To save a notebook, click the File � Save command.

The first time you save a notebook, a dialog box appears asking you to specify a name for the notebook and the directory in which it should be stored. It is a good practice to save your notebook frequently.

To print a notebook, click the File � Print command.

� Using Cells

All data entered into a notebook is organized into basic structural units called cells. Each cell has a specific style associated with it that defines all the attributes of the cell such as the size, color, and font of text within it. Each cell style has a name assigned to it, which usually indicates the role the cell will play in the notebook. Some common cell styles are “Input”, “Output”, “Section”, “Subsection”, and “Text”. The extent of any cell is indicated by a bracket on the right side of the notebook.

If you create a cell of a specified style, all text entered into it is automatically formatted according to the style defined for that cell. This allows you to make a heading or subheading for your document simply by entering text into a cell of the appropriate style. The consistent use of different cell styles provides a convenient way to organize your document in a modular fashion.

A notebook with different types of cells.

4

To create an input cell to evaluate:

When you open a notebook and start typing, a cell is automatically created. By default, all new cells are created as Input cells unless you specify otherwise. These cells can be evaluated by pressing ���. The results of the evaluation are then placed into Output cells just below the input.

To change the style of an existing cell:

1. Select the cell by clicking the cell bracket.

2. Click the Format � Style submenu. This brings up a list of all cell styles defined for your notebook.

3. Click any of the styles listed to convert the cell to that style.

To create a new cell of a given style:

1. Move your cursor below a cell until it becomes horizontal and then click once. A cell insertion bar appears on the screen in the form of a horizontal line.

2. Click the Format � Style menu. This brings up a list of all cell styles defined for your notebook.

3. Click any of the styles listed to select it. Any text you then enter in that cell will be of that style.

Cells can be arranged in hierarchical groups, with several cells of different styles being enclosed inside a cell of another style. The grouping of cells is indicated by nested brackets on the right.

A notebook displaying the hierarchy of cells.

5

A group of cells can be either open or closed. When a cell group is open, all cells within it can be seen explicitly. To close the group, double-click the outermost bracket. When a cell group is closed, only the first (or header) cell in the group can be seen.

A notebook with two closed cell groups.

When a group of cells is closed, the bracket enclosing it has an arrow at the bottom. Some closed cell groups also have a triangle on the left-hand side. You can double-click the arrow, the triangle, or the outer bracket itself to open the group. Large notebooks are often distributed with many closed groups of cells so that you see only an outline of the contents. You can then open the cell groups you are interested in viewing by double-clicking the appropriate brackets.

� Changing Text Styles

Once you have created a notebook, you can modify the style of text within it according to your preferences. Calculus WIZ allows you to specify properties like the size, font, and color of text, as well as the alignment and justification.

You can apply any of these changes at two different levels:

� a part of the text within a single cell

� an entire cell or group of cells

To change the style of text within a cell, simply select the text and apply the styles you want using the Format menu.

A cell with several different font styles applied inside it.

Most of the commands for altering text style are listed as a series of submenus under the Format menu. To apply any of these style options, select the chosen text and then click the option you want from the menu.

6

� You can alter the font of the selected text by selecting from the list provided under the Format � Font submenu. Calculus WIZ has a large number of built-in fonts, including the common fonts such as Times, Courier, or Helvetica.

Some font options. The list displayed will depend upon the configuration of your system.

� You can convert text to italics or bold letters using the Format � Face submenu.

Some face options.

� You can specify the text size, in points, using the Format � Size submenu.

Some examples of text in different font sizes.

� The color of the text and the background in which it appears can be changed by using the menu command Format � Text Color and Format � Background Color, respectively.

Some examples of text in different colors.

Other submenus listed under the Format menu allow you to:

� control the alignment of lines

� change the justification of text

� set the word wrapping of text

� insert lines of different thicknesses

7

All these submenus have self-descriptive names. You may want to experiment with these submenus to familiarize yourself with their functions.

� Finding and Replacing Text

Once you have created a notebook, you might want to locate all occurrences of a certain word or phrase in order to replace it with an alternative. To facilitate such searches, Calculus WIZ contains several useful functions which are listed under the Find menu.

To perform a search:

1. Click Find � Find. This brings up a Find window.

2. Type the word you want to find in the “Search for” text field.

You can make the search case insensitive by checking the “Ignore Case” box in the Find window.

3. You can then do one of the following:

� Click Next to jump to the next occurrence of the specified word in the notebook. The word will be highlighted to make it easily visible.

� Click Previous to jump to the previous occurrence of the specified word.

� Click Find All to highlight every occurrence of the phrase in the notebook.

When you use the last command, matches may be found in multiple cells throughout the notebook. If cell groups are closed, the selections will not be visible, but you can still copy or replace the selected matches. Choose Find � Open Selected if you want to see the matched text.

Once you have found one or more occurrences of a word using Find, you can replace it with another word using the Replace feature in the Find window.

To replace one word with another:

1. Type the new words in the “Replace with” text field.

2. Click Replace. The highlighted word(s) will be replaced by the words in the text field.

Using the Find window you can also replace all occurrences of a specified word in one step by clicking Replace All.

8

� Magnifying Notebooks

The last option that can be set from the Format menu involves the appearance of the notebook’s pages on the screen. You can change the magnification of the page by choosing a value from the Format � Magnification submenu.

Entering Text and Equations

� Performing Evaluations

With a notebook open, type the input 9.7^200. A new input cell is created. To evaluate this, make sure the cursor is in the cell and press ���. (You can also press the EVALUATE button in most of the examples to

evaluate input.)

Here is the result.

An example of evaluation.

Here is an example from algebra. The first bracket is a special color until the closing bracket is typed. To evaluate this, again press ���.

An example of a Calculus WIZ function.

9

Calculus WIZ obeys the following conventions.

2 � 3 2 � 3

2 � 3 2^3

These are standard arithmetic operations.

Sin[x]

Built-in functions are capitalized. Arguments to functions are wrapped with square brackets.

a*b a � b a(b+1)

2 x means 2*x.

Each of these represents multiplication.

{a, b, B}

Uppercase and lowercase letters are recognized as different. Lists are wrapped with curly brackets.

N[Pi, 50];

Built-in symbols are capitalized. Commas are used to separate arguments. A semicolon prevents output, but the command is still evaluated.

x = 5

xvalue = 3

Variables are usually lower case. Entire words can be used as variables.

� An Overview of Entering Text and Equations

Once a notebook is open, you can enter text in several different ways.

� Keyboard: Ordinary text and numbers can be typed into a notebook using the keyboard and number pad.

10

� Palettes: Special characters, like mathematical symbols and Greek letters, can be entered using palettes. A palette consists of a series of buttons, each labeled by a specific symbol or expression. Clicking any of these buttons causes the corresponding symbol to be entered into the notebook at the position of the cursor. Several different palettes are available from the File � Palettes submenu.

� Names for special characters: Most of the special characters appearing in the palettes can also be typed into the notebook directly from the keyboard. Each special character is assigned a name composed entirely of keyboard characters. Typing \, followed by the name of the character enclosed in square brackets, inserts the character into your notebook. For example, the Greek letter Α can be entered by

typing �[Alpha] and the integral symbol by typing �[Integral].

� Keyboard aliases: Finally, many of the special characters can be entered using keyboard aliases consisting of special keys like � and �, combined with ordinary letters and numbers. For example, the Greek

letter Α has the alias � a � and the integral symbol �� has the alias � int �. The � and � keys also provide a way to enter mathematical formulas involving subscripts, superscripts, square roots, powers, and fractions.

Note: The � key is represented by the symbol � in a notebook.

The following sections contain several examples to illustrate how you can use these methods, singly or in combination, to enter input of many different types.

� Greek Letters

Click File � Palettes � Complete Characters. A notebook appears with three different sections: “Letters”, “Letter-like Forms”, and “Operators”. Each section is further divided into subsections, with a palette corresponding to each. To open a palette, click the triangle symbol next to its name. Clicking Letters � Greek brings up the following palette.

Clicking any of the buttons causes the corresponding Greek letter to be entered into your notebook. Letters can be combined with ordinary text and numbers in any way.

In addition, there are two other ways to enter Greek letters using the keyboard.

� Type the full name of the character.

� Type the keyboard alias for the character.

11

to enter full name keyboard �alias �1 keyboard �alias �2

Α ��Alpha� � a � � alpha �

 ��Beta� � b � � beta �

��Pi� � p � � pi �

Some examples of how to enter Greek letters.

Note: the � key appears as � on the screen.

For a complete list of full names and keyboard aliases for all special characters, look in the Help Browser, available under Help � Help Browser.

� Two-dimensional Forms

You can enter two-dimensional input, such as xy , in two different ways—either using a palette or directly from the keyboard.

To enter two-dimensional forms using a palette:

1. Place the cursor at the point in your notebook where you want to enter the text.

2. Click File � Palettes � Basic Input to bring up the following palette.

3. Click on the button in the top left-hand corner to insert the symbol � in the notebook.

4. Type x to fill in the first box, called a placeholder.

12

5. Press � to move to the next placeholder.

6. Type y to fill in the second placeholder and get the superscripted symbol xy .

To enter two-dimensional forms directly from the keyboard:

Another way to enter two-dimensional expressions is by using special input forms based on control characters.

For example, ��^�, which corresponds to pressing the � and ^ keys simultaneously, moves the cursor into a superscript position. (The character � represents the space bar.)

key sequence displayed form

x ��_� y xy

x �� � � y x���y

x �� � � y � z x�������y�z

x �� � � y � ��� � � z x���y

� z

�� � � x � y��������������x � y

��� � x ��� � � y����x � y

Examples of two-dimensional expressions entered using control keys.

When you see a two-dimensional expression on the screen, you can edit it in all the ways you would edit text. You can, for example, place your cursor somewhere and start typing. Or you can select a part of the expression, remove it using the key, and insert new input.

13

� Subscripts, Bars, and Other Modifiers

In addition to subscripts and superscripts, you can also add modifiers such as overscripts, underscripts, bars, and

other special characters. To do this, start by clicking File � Palettes � Basic Input. This brings up the following palette.

You can select any of the buttons in this palette to input the corresponding modifier.

To input a letter with a subscript, such as xi :

1. Place your cursor at the point in the notebook where the letter is to be inserted.

2. Click the button with the subscript form in the bottom row of the palette. This places the symbol � in your notebook.

3. Type x in the position of the highlighted placeholder.

4. Press � to move to the position of the next placeholder. The subscript placeholder is now highlighted.

5. Type i to enter the subscript. The subscripted form xi is now entered in your notebook.

6. Press ���� to move the cursor out of the subscript position.

A similar procedure can be used for all the other buttons in this palette to enter any of the corresponding modifiers.

14

Another way to add modifiers is by using the special meaning assigned to control characters.

��_� or ��-� go to the position for a subscript

�� � � or �� � � go to the position underneath

��^ � or ��6� go to the position for a superscript

�� & � or ��7� go to the position on top

���� return from a special position

Keyboard shortcuts for subscripts, superscripts, underscripts, and overscripts.

� Mathematical Expressions

Using palettes, keyboard aliases, or specially defined long forms to represent symbols and operators, you can construct formulas with mathematical notation of any complexity.

To enter mathematical notation such as sums, products, derivatives, and integrals, you can again use a palette. Alternatively, you can use keyboard shortcuts like those introduced in the previous sections.

to enter long form keyboard alias

� � �Sum� � sum �

� � �Product� � prod �

� �Integral� � int �

� � �DifferentialD� � dd �

� � �PartialD� � pd �

Some mathematical symbols that can be entered using a palette or the keyboard.

Limits for sums, products, and definite integrals can be entered as underscripts and overscripts.

to enter type

a

b

f ��x� x �int� �� � � a ���� � �� & � �b ���� � f �x� �dd�x

�i�1

N

f ��i� �sum� �� � � i � 1 ���� �� & � N ���� f ��i�

Examples of mathematical expressions that can be entered using the keyboard.

15

Note: All the symbols shown in this table can also be entered directly from the “Operators” section of the CompleteCharacters palette listed under the File � Palettes submenu.

� Animating Graphics

You can run any sequence of graphics as a movie. The movie appears in the most visible cell in the sequence.

To animate a set of graphics:

1. Create each frame of the animation–for example, by using a Do loop.

16

An example of a group of graphics for an animation.

2. Select the cell bracket that contains all the graphics cells.

3. Double-click any graphic. The animation appears in the most visible cell of the selected graphics.

In most of the Calculus WIZ text, there will be an ANIMATE button that you can click, which will start the

animation.

� Hyperlinks

A hyperlink is a special kind of button that jumps to a different cell in a notebook, a different notebook, or a URL.

To follow a hyperlink:

Click on the underlined text. In most cases, the hyperlink will take you to the Help Browser (see “Finding Textbook Topics”). But if the hyperlink is to a URL, a browser window will be opened for you.

17

Calculus with Calculus WIZ

Introduction

In Calculus WIZ, basic calculus looks almost the same as in your textbook. In these sections on calculus, you will see formulas written as Calculus WIZ input commands and their values computed.

Derivatives

� Leibniz’s Notation

Leibniz’s notation for the derivative is dy�����dx

when y is a function of x . In Calculus WIZ there is an operator �������

that works as a command to find the derivative. The following is a sample result. To see the Calculus WIZ instructions online, click the triangle to the left of the Calculus WIZ example. If you would like to evaluate the code yourself, click the EVALUATE button.

A derivative example input.

� Function Notation for Derivatives

Function notation for derivatives is f � �x� for the function f �x� . Calculus WIZ can use this notation as a computation command as in the next example. To see the precise instructions to define a function and compute f � �x� , click the small triangle at the left of the Calculus WIZ example.

18

A derivative example input using functional notation.

Integrals

� Indefinite Integrals

Leibniz’s notation for the indefinite integral is y �x . Calculus WIZ uses this notation as a command to compute the integral as in the following example. (To see the Calculus WIZ instructions online, click the triangle to the left of the Calculus WIZ example.)

An integral example input.

19

� Definite Integrals

Leibniz’s notation for the definite integral is a

b y �x . Calculus WIZ uses this notation as a command to

compute the integral as in the following example.

A definite integral example input.

Limits

Calculus WIZ can also compute limits such as the following.

A limit example input.

20

Infinite Series

Calculus WIZ can find many infinite series exactly, like the following.

An infinite series example input.

Graphs

Calculus WIZ easily plots functions of one or two variables.

A two-dimensional plot.

21

A three-dimensional plot.

22

Finding Textbook Topics

Calculus WIZ Help Browser

Calculus WIZ is organized in the Help Browser by title, chapter, section, and subsection. For example, to get the Exercises (subsection) on graphing with the First Derivative (section) in the chapter on the Mean Value Theorem, click the Help Browser as shown in this picture.

The Help Browser with exercises displayed.

Table of Contents

Calculus WIZ’s Table of Contents is easily accessed from the Help Browser.

The Table of Contents consists of links that you can click to go to the various topics. You can browse through this list of chapters and sections by clicking the small triangles to open chapters.

23

Table of Solvers

The list of solvers of Calculus WIZ is easily accessed from the Help Browser by clicking the “Calculus WIZ” button and then the “Solvers” item.

Calculus WIZ Index

An index of Calculus WIZ is available by clicking the “Calculus WIZ” button, then clicking “Calculus WIZ Text”, and then the “Index” item. When you click the entries in the index, Calculus WIZ will find the topic in the Help Browser.

The Master Index

You can also look up any topic in Calculus WIZ by using the Master Index. For example, to look up information on approximations, click the Master Index button, type the word “approximations” into the lookup field at the top of the Help Browser, and then click the “Go To” button.

The Master Index displaying topics on approximations.

Notice that there may be several topics to choose from. Click the topic that you are interested in and then use the Back button at the top of the Help Browser to return to the previous screen.

24

Homework with Calculus WIZ

Introduction

Two main problems of calculus are to find a tangent line and to find the area under a curve. This section illustrates how your text solves these problems and how you can solve such problems with Calculus WIZ. To see this information online, choose the sections “A Tangent Line Exercise” and “An Area Exercise” from the Help Browser under Getting Started � Homework with Calculus WIZ.

A Tangent Line Exercise

If f �x� � x2

3� x

2, find the equation of the tangent line to y � f �x� at the point �a, b� � �2, 1

3� .

� Automatic Calculus WIZ Solver Solution of the Tangent Line Exercise

To solve the tangent line problem with Calculus WIZ, simply click the Tangent Line Solver button. Once the

new solver window appears, fill in the blank lines (or click the Example button) in the solver. Next, click the

Do It button to create a homework notebook and complete the solution.

To get information from a solver:

Every Calculus WIZ solver has a row of buttons that will give you different kinds of help.

A solver for tangent lines.

25

The Info? button finds reference information about the problem.

The Example button pastes an example into the solver for you to try.

The Calc Palette button brings up the calculus operators to help type formulas or commands.

The Input button finds help for you to type formulas.

The Clear button clears the input fields of the solver so that you can input a new problem.

Launch a solver and try out the buttons by clicking them.

� Textbook Solution of the Tangent Line Exercise

The following solution is similar to the way your text solves the tangent line problem that we solved above with the Calculus WIZ Tangent Line Solver.

Textbook exercise:

If f �x� � x2

3� x

2, find the equation of the tangent line to y � f �x� at the point �a, b� � �2, 1

3� .

Textbook solution:

1. The slope of the tangent to y � f �x� at a point is given by the derivative at that point, f � �a� .

2. To find the symbolic derivative, rules of calculus are applied to the formula.

y �x2

3

�x 2

� f �x��y �x

� 1 3

� � �x

��x2 � � 1 2

� �x �x

�y �x

� 2 3

�x2�1 � 1 2

� 1 x1�1

�y �x

� f '��x� � 2 3

�x � 1 2

3. The value x � a � 2 is substituted into the formula for f � �x� .

f � �a� � f � �2� �2 � 2

3�

1 2

�8 6

�3 6

�5 6

4. The value of x � a � 2 is substituted into the original function.

f �a� � f �2� �22

3

�2 2

�4 3

�3 3

�1 3

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5. These values are substituted into the tangent line formula.

y � f �a� � �x � a� f � �a�

y �5 �x � 2�

6�

1 3

This textbook exercise did not ask for the graphs, so we are done.

� Calculus WIZ Template Solution of the Tangent Line Exercise

To solve this textbook exercise with a partially prepared program, click the Sample Tangent Line Template

button. Clicking the template button will paste the template into a homework notebook. (If necessary, you will be asked to create a new homework notebook.)

To evaluate the expression in the template, press ��� with the cursor in the template cell.

An Area Exercise

Find the area bounded by the curves y � g� x� � x2 and y � f �x� � 2 x � x2 .

� Automatic Calculus WIZ Solution of the Area Exercise

First click the Area Between Solver button and then click the Example button in the solver that appears. Next

click Do It to see how the problem is solved.

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To get information from a solver:

Every Calculus WIZ solver has a row of buttons that will give you different kinds of help.

A solver for the area between two functions.

The Info? button finds reference information about the problem.

The Example button pastes an example into the solver for you to try.

The Calc Palette button brings up the calculus operators to help type formulas or commands.

The Input button finds help for you to type formulas.

The Clear button clears the input fields of the solver so that you can input a new problem.

Launch a solver and try out the buttons by clicking them in the solver.

� Textbook Solution of the Area Exercise

Textbook exercise:

Find the area bounded by the curves y � g �x� � x2 and y � f �x� � 2 x � x2 .

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Textbook solution:

The formula in your textbook for the area between the curves y � f �x� and y � g�x� is

�a

b �� f �x� � g�x�� � x

where x � a is the left intersection point of the graphs, x � b is the right intersection point, and f �x� is above g�x� over this interval. First, we need to find where the curves intersect. These points are the solutions to the equation.

g�x� � f �x�x2 � 2 x � x2

2 x2 � 2 x � 02 x �x � 1� � 0

So x � 0 and x � 1.

Next, we set up the integral and compute with rules of calculus.

�a

b

�� f �x� � g�x�� � x � �0

1

��2�x � x2 � x2 � �x � �0

1

��2�x � 2�x2 ��� x � 2��0

1

x x � 2��0

1

�x2 � x �

2 1 � 1

�x1�1 01 �2

2 � 1

�x2�1 01 � x2 01 �2 3

�x3 01 � �12 � 02 � �2 3

��13 � 03 � � 1 �2 3

�1 3

� Calculus WIZ Template Solution of the Area Exercise

Click the Area Between Template button and complete the program. Clicking the template button will paste the

template into a homework notebook. (If necessary, you will be asked to create a new homework notebook.)

To evaluate the expression in the template, press ��� with the cursor in the template cell.

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Who’s the Wiz?

With a little help, you’re the calculus wiz!

What’s Next?

Calculus WIZ is a self-explanatory tool for you to use in solving your tough homework problems. Use Calculus WIZ to do the following.

� Quickly search for the topic you want to study or the problems you want to solve.

� Review the methods of solving homework in Homework with the WIZ.

� Make your own formulas.

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