Calculus II Workbook For Dummies E-Book

Mark Zegarelli

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2023

Seitenzahl: 221

Beschreibung

Work your way through Calc 2 with crystal clear explanations and tons of practice Calculus II Workbook For Dummies is a hands-on guide to help you practice your way to a greater understanding of Calculus II. You'll get tons of chances to work on intermediate calculus topics such as substitution, integration techniques and when to use them, approximate integration, and improper integrals. This book is packed with practical examples, plenty of practice problems, and access to online quizzes so you'll be ready when it's test time. Plus, every practice problem in the book and online has a complete, step-by-step answer explanation. Great as a supplement to your textbook or a refresher before taking a standardized test like the MCAT, this Dummies workbook has what you need to succeed in this notoriously difficult subject. * Review important concepts from Calculus I and pre-calculus * Work through practical examples for integration, differentiation, and beyond * Test your knowledge with practice problems and online quizzes--and follow along with step-by-step solutions * Get the best grade you can on your Calculus II exam Calculus II Workbook For Dummies is an essential resource for students, alone or in tandem with Calculus II For Dummies.

Leseprobe

Calculus II Workbook For Dummies®

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Published simultaneously in Canada

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ISBN 978-1-394-18802-4; ISBN 978-1-394-18800-0 (ebk); ISBN 978-1-394-18801-7 (ebk)

Calculus II Workbook For Dummies®

To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Calculus II Workbook For Dummies Cheat Sheet” in the Search box.

Table of Contents

Cover

Title Page

Copyright

Introduction

About This Book

Foolish Assumptions

Icons Used in This Book

Beyond the Book

Where to Go from Here

Part 1: Introduction to Integration

Chapter 1: An Aerial View of the Area Problem

Measuring Area on the

xy

-Graph

Defining Area Problems with the Definite Integral

Calculating Area Defined by Functions and Curves on the

xy

-Graph

Answers and Explanations

Chapter 2: Forgotten but Not Gone: Review of Algebra and Pre-Calculus

Fractions

Factorials

Negative and Fractional Exponents

Simplifying Rational Functions

Trigonometry

Parent Functions

Parent Function Transformations

Sigma Notation for Series

Answers and Explanations

Chapter 3: Recent Memories: Calculus Review

Evaluating Limits

Derivatives of Common Functions and the Constant Multiple Rule

The Power Rule

The Sum Rule

The Product Rule and Quotient Rule

The Chain Rule

Answers and Explanations

Part 2: From Definite to Indefinite Integrals

Chapter 4: Approximating Area with Riemann Sums

Calculating Riemann Sums with Left and Right Rectangles

Getting a Better Estimate with Midpoint Rectangles

Improving Your Estimate with the Trapezoid Rule

Using Simpson’s Rule to Further Improve Your Approximation

Answers and Explanations

Chapter 5: The Fundamental Theorem of Calculus and Indefinite Integrals

Evaluating Definite Integrals Using FTC2

Anti-differentiation and Indefinite Integrals

Signed and Unsigned Area

Answers and Explanations

Part 3: Evaluating Indefinite Integrals

Chapter 6: Instant Integration

Antiderivatives of Common Functions and the Constant Multiple Rule

The Power Rule

The Sum Rule

Answers and Explanations

Chapter 7: Sharpening Your Integration Moves

Integrating Rational and Radical Functions

Using Algebra to Prepare Functions for Integration

Integrating Using Inverse Trig Functions

Using Trig Identities to Prepare Functions for Integration

Integrating Compositions of Functions with Linear Inputs

Answers and Explanations

Chapter 8: Here’s Looking at u-Substitution

Understanding the How of

u

-Substitution

When to Use

u

-Sub: The Simpler Case

When to Use

u

-Sub: The More Complex Case

Using Variable Substitution with Definite Integrals

Answers and Explanations

Part 4: Advanced Integration Techniques

Chapter 9: Integration by Parts

Using the Formula for Integration by Parts

Knowing How to Assign

u

and

dv

Applying Integration by Parts More Than Once

Answers and Explanations

Chapter 10: Trig Substitution

Integrating the Six Basic Trig Functions

Integrating Powers of Sines and Cosines

Integrating Powers of Tangents and Secants

Using Trig Substitution

Answers and Explanations

Chapter 11: Integration with Partial Fractions

Understanding Partial Fractions

Integrating Case 1: Rational Expressions That Have Distinct Linear Factors

Integrating Case 2: Repeated Linear Factors

Integrating Case 3: Distinct Quadratic or Higher-Degree Factors

Integrating Case 4: Repeated Quadratic Factors

Answers and Explanations

Part 5: Applications of Integrals

Chapter 12: Solving Area Problems

Breaking Definite Integrals in Two

Looking at Improper Integrals

Finding the Unsigned Area between Curves

Using the Mean Value Formula for Integration

Calculating Arclength

Answers and Explanations

Chapter 13: Pump up the Volume — Solving 3-D Problems

Using Disk and Washer Methods to Find the Volume of Revolution around a Horizontal Axis

Using Inverse Functions to Find Volume of Revolution around a Vertical Axis

Surface Area of Revolution

Shell Method

Answers and Explanations

Chapter 14: Differential Equations

Understanding Differential Equations

Solving Separable Differential Equations

Answers and Explanations

Part 6: Infinite Series

Chapter 15: Sequences and Series

Understanding Sequences

Understanding Series

Finding the Sequence of Partial Sums for a Series

Understanding and Evaluating Geometric Series

Understanding

p

-series

Answers and Explanations

Chapter 16: Convergent and Divergent Series

N

th-Term Test for Divergence

The Direct Comparison Test

The Limit Comparison Test

Integral Test

Ratio Test

Root Test

Answers and Explanations

Chapter 17: Taylor and Maclaurin Series

Expressing Functions as Maclaurin Series

Expressing Functions as Taylor Series

Answers and Explanations

Part 7: The Part of Tens

Chapter 18: Ten Mathematicians Who Anticipated Calculus before Newton and Leibniz

Zeno of Elea (495–430 BCE)

Eudoxus of Cnidus (408–355 BCE)

Archimedes (287–212 BCE)

Lui Hui (3rd Century CE)

Zu Chongzhi (425–500)

Hasan Ibn al-Haytham (965–1040)

Madhava of Sangamagrama (1340–1425)

Johannes Kepler (1571–1630)

Bonaventura Cavalieri (1598–1647)

Isaac Barrow (1630–1677)

Chapter 19: 10 Skills from Pre-Calculus and Calculus I That Will Help You to Do Well in Calculus II

Expressing Functions as Exponents Whenever Possible

Knowing the Basic Trig Identities Cold

Recalling the Most Common Functions and Their Basic Transformations

Using Sigma Notation for Series

Evaluating Limits

Differentiating the Most Common Functions

Knowing the Power Rule for Differentiation

Applying the Product Rule

Using the Chain Rule

Calculating Common Derivatives Quickly in Your Head

Index

About the Author

Advertisement Page

Connect with Dummies

End User License Agreement

List of Tables

Chapter 10

Table 10-1 The Three Cases for Trig Substitution

List of Illustrations

Chapter 2

FIGURE 2-1: Parent functions and graphs of four polynomial functions.

FIGURE 2-2: Parent functions and graphs for the square root and cube root funct...

FIGURE 2-3: Parent functions and graphs for the exponential and logarithmic fun...

FIGURE 2-4: Parent functions and graphs for the sine and cosine functions.

FIGURE 2-5: Parent functions and graphs for the absolute value and rational fun...

Guide

Cover

Title Page

Copyright

Table of Contents

Begin Reading

Index

About the Author

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Introduction

Calculus II is tough stuff by just about any standard.

In this book, I’ve done my best to explain the key topics that tend to hang students up as clearly and concisely as possible. I’ve also provided practice problems that I think will help you to make sense of this information, so you’ll be ready to take on similar problems that your teacher or professor might throw at you.

About This Book

Chapter by chapter, this workbook follows the typical order of topics in a standard Calculus II course. But please feel free to jump back and forth throughout the book in whatever order makes sense for you.

Chapter 1 provides an overview of the course, and then Chapters 2 and 3 give you a review of the Pre-Calculus and Calculus I topics you’ll need to know to move forward. In Parts 2 through 4, you learn a variety of integration methods, all of which are commonly taught in Calculus II. Part 5 focuses on basic applications of integration. Part 6 covers sequences and series. And in Part 7, I give you a couple of Top Ten lists related to calculus.

Additionally, for convenience, this workbook follows the same chapter-by-chapter format as Calculus II For Dummies, 3rd edition. Of course, you don’t have to buy that book to make good use of this one. But if you do want to use both books, you’ll find that corresponding numbered chapters cover the same topics, so you’ll be able to flip back and forth between them easily.

Foolish Assumptions

I assume that you either want or need to learn Calculus II. So, either you’re interested in the topic and want to study it on your own, or like many people, you’re taking a high-school or college course in the subject.

Icons Used in This Book

In this book, I use a variety of icons to give you a heads-up about what information is important and what you can safely skip over when you’re in a rush.

The example icon accompanies a sample question followed by a step-by-step solution. In most cases, examples should help you get a handle on difficult material in a way that makes sense.

This icon alerts you to key information that you may need to pay special attention to, especially if you’re currently studying for a test.

Tips provide you with a quick and easy way to work on a problem. Try them out as you work your way through the book, so you can use them in assignments and on tests.

The Technical Stuff icon marks information of a highly technical nature that you can normally skip if you are in a hurry.

This icon warns you of typical errors that students tend to fall into. Keep an eye on these little traps so that they don’t get you, too!

Beyond the Book

If you want to explore Calculus II in greater depth, or get an additional perspective on it, look no further than my book Calculus II For Dummies, 3rd edition. This workbook has been written in conjunction with that book to provide an even more complete picture of the topics taught here.

This book also includes accompanying online material in the form of a Cheat Sheet. To access it, go to www.dummies.com and type Calculus II Workbook For Dummies in the Search box.

Where to Go from Here

Success in a Calculus II class is built on a foundation of a whole lot of other math you’ve been studying since you learned how to count. To help you shore up this foundation, here are a few additional resources:

If you need more help with some of the foundational math than this book provides, an easy place to start is my book,

Basic Math and Pre-Algebra For Dummies.

If you need a refresher on any algebra concepts not covered here, check out

Algebra I For Dummies

and

Algebra II For Dummies

, both by Mary Jane Sterling.

If you find that your trig skills are in need of a makeover,

Trigonometry For Dummies

, also by Mary Jane Sterling, gives you wider and deeper coverage of the topic than you’ll find here.

Even if you’ve passed Calculus I, you may want a refresher. Check out

Calculus For Dummies

and its accompanying workbook by Mark Ryan for a closer look at basic calculus topics.

Part 1

Introduction to Integration

IN THIS PART …

See Calculus II as an ordered approach to finding the area of unusual shapes on the

xy

-graph

Use the definite integral to clearly define an area problem

Slice an irregularly shaped area into rectangles to approximate area

Review the math you need from Pre-Algebra, Algebra, Pre-Calculus, and Calculus I

Chapter 2

Forgotten but Not Gone: Review of Algebra and Pre-Calculus

IN THIS CHAPTER

Calculating with fractions and factorials

Working with exponents and simplifying rational expressions

Remembering radian measure

Proving trig identities

Understanding important parent functions and their transformations

Converting an infinite series from sigma notation to expanded notation

Most students have been studying math for at least 10 years before they enter their first calculus classroom. This fact leaves many students overwhelmed by all the math they should know, and perhaps did know at one time, but can’t quite recall.

Fortunately, you don’t need another 10 years of review to be ready for Calculus II. In this chapter, I get you back up to speed on the key topics from your Pre-Algebra, Algebra, and Pre-Calculus classes that will help you the most this semester.

To begin, you go all the way back to middle school for a quick review of fractions. I also give you some practice calculating factorials.

After that, I remind you how to work with exponents, and especially how to use negative and fractional exponents to express rational and radical functions. Then I cover a few important ideas from trigonometry that you’re sure to need, such as radian measure and trig identities.

Next, I give you an overview of how to sketch the most important parent functions on the xy-graph: polynomials, exponentials, radicals, logarithmic functions, and the sine and cosine functions. You use these to work with a variety of function transformations, such as vertical and horizontal transformations, as well as stretch, compress, and reflect transformations.

Fractions

When finding derivatives in Calculus I and integrals in Calculus II, you’ll often need to add 1 to (or subtract 1 from) a fraction. Here’s a trick for doing both of those operations quickly in your head without getting a common denominator:

Q. What is ?

A. . To do this calculation in your head, add the numerator and denominator, and then keep the denominator of 5:

Q. What is ?

A. . To calculate this value in your head, subtract the numerator minus the denominator, and then keep the denominator of 6.

1 Add 1 to the following fractions and express each answer as a proper or improper fraction (no mixed numbers).

a.

b.

c.

d.

2 Subtract 1 from the following fractions and express each answer as a proper or improper fraction (no mixed numbers).

a.

b.

c.

d.

Factorials

In Calculus II, when working with infinite series, you also may need to make use of factorials. Recall that the symbol for factorial is an exclamation point (!). The factorial of any positive integer is that number multiplied by every positive integer less than it. Thus:

Also, by definition, .

When you know how to expand factorials in this way, simplifying rational expressions that include them is relatively straightforward. Always look for opportunities to cancel factors in both the numerator and denominator.

Q. Simplify .

A. 10. Begin by expanding the factorials:

   Now, cancel factors in both the numerator and denominator, and simplify the result:

Q. Simplify .

A. . Expand the factorials as follows:

   Cancel factors in both the numerator and denominator, and simplify the result:

3 Simplify each of the following factorial expressions.

a.

b.

c.

d.

4 Simplify each expression in terms of n.

a.  b.

c.  d.

Negative and Fractional Exponents

When an expression has a negative exponent, you can rewrite it with a positive exponent and place it in the denominator of a fraction. For example:

When an expression has a fractional exponent, you can rewrite it as a radical. For example:

More complicated fractional exponents can be written in two separate and equally valid ways as a combination of a radical and an exponent. For example: