Calculus For Dummies

626 Pages·2014·10.94 MB·English

by  Jones, Patrick

1,001 Calculus
Practice Problems

by PatrickJMT

1,001 Calculus
Practice Problems

1,001 Calculus Practice Problems For Dummies®

,

Published by: John Wiley & Sons, Inc., 111 River St., Hoboken, NJ 07030-5774, www.wiley.com

Copyright © 2014 by John Wiley & Sons, Inc., Hoboken, New Jersey

Media and software compilation copyright © 2014 by John Wiley & Sons, Inc. All rights reserved.

Published simultaneously in Canada

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, scanning or otherwise, except
as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior
written permission of the Publisher. Requests to the Publisher for permission should be addressed
to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201)
748-6011, fax [phone], or online at http://www.wiley.com/go/permissions.

Trademarks: Wiley, For Dummies, the Dummies Man logo, Dummies.com, Making Everything Easier, and
related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc., and may not be
used without written permission. All other trademarks are the property of their respective owners. John
Wiley & Sons, Inc., is not associated with any product or vendor mentioned in this book.

LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: WHILE THE PUBLISHER AND AUTHOR HAVE
USED THEIR BEST EFFORTS IN PREPARING THIS BOOK, THEY MAKE NO REPRESENTATIONS OR
WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF
THIS BOOK AND SPECIFICALLY DISCLAIM ANY IMPLIED WARRANTIES OF MERCHANTABILITY
OR FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED
BY SALES REPRESENTATIVES OR WRITTEN SALES MATERIALS. THE ADVICE AND STRATEGIES
CONTAINED HEREIN MAY NOT BE SUITABLE FOR YOUR SITUATION. YOU SHOULD CONSULT WITH
A PROFESSIONAL WHERE APPROPRIATE. NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE
LIABLE FOR DAMAGES ARISING HEREFROM.

For general information on our other products and services, please contact our Customer Care
Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993, or fax [phone]. For
technical support, please visit www.wiley.com/techsupport.

Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material
included with standard print versions of this book may not be included in e-books or in print-on-demand.
If this book refers to media such as a CD or DVD that is not included in the version you purchased, you
may download this material at http://booksupport.wiley.com. For more information about Wiley
products, visit www.wiley.com.

Library of Congress Control Number: 2013954232

ISBN 978-1-118-49671-8 (pbk); ISBN 978-1-118-49670-1 (ebk); ISBN 978-1-118-49673-2 (ebk)

Manufactured in the United States of America

10 9 8 7 6 5 4 3 2 1

Contents at a Glance

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

Part I: The Questions................................................................ 5

Chapter 1: Algebra Review .................................................................................................................. 7
Chapter 2: Trigonometry Review...................................................................................................... 17
Chapter 3: Limits and Rates of Change............................................................................................ 29
Chapter 4: Derivative Basics ............................................................................................................. 43
Chapter 5: The Product, Quotient, and Chain Rules...................................................................... 49
Chapter 6: Exponential and Logarithmic Functions and Tangent Lines...................................... 55
Chapter 7: Implicit Differentiation.................................................................................................... 59
Chapter 8: Applications of Derivatives ............................................................................................ 63
Chapter 9: Areas and Riemann Sums ............................................................................................... 75
Chapter 10: The Fundamental Theorem of Calculus and the Net Change Theorem.................. 79
Chapter 11: Applications of Integration........................................................................................... 87
Chapter 12: Inverse Trigonometric Functions, Hyperbolic Functions,
and L’Hôpital’s Rule......................................................................................................................... 99
Chapter 13: U-Substitution and Integration by Parts ................................................................... 107
Chapter 14: Trigonometric Integrals, Trigonometric Substitution,
and Partial Fractions...................................................................................................................... 113
Chapter 15: Improper Integrals and More Approximating Techniques..................................... 121
Part II: The Answers ............................................................ 125

Chapter 16: Answers and Explanations ......................................................................................... 127
Index .................................................................................. 595

Table of Contents

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

What You’ll Find ..............................................................................................1
Beyond the Book .............................................................................................1
What you’ll find online ..........................................................................2
How to register.......................................................................................2
Where to Go for Additional Help ...................................................................2

Part I: The Questions .................................................... 5

Chapter 1: Algebra Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

The Problems You’ll Work On .......................................................................7
What to Watch Out For...................................................................................7
Simplifying Fractions.......................................................................................8
Simplifying Radicals ........................................................................................8
Writing Exponents Using Radical Notation..................................................9
The Horizontal Line Test................................................................................9
Find Inverses Algebraically ............................................................................9
The Domain and Range of a Function and Its Inverse ..............................10
Linear Equations............................................................................................10
Quadratic Equations .....................................................................................10
Solving Polynomial Equations by Factoring...............................................11
Absolute Value Equations ............................................................................11
Solving Rational Equations...........................................................................11
Polynomial and Rational Inequalities .........................................................12
Absolute Value Inequalities..........................................................................12
Graphing Common Functions ......................................................................12
Domain and Range from a Graph.................................................................13
End Behavior of Polynomials.......................................................................14
Adding Polynomials ......................................................................................14
Subtracting Polynomials...............................................................................14
Multiplying Polynomials...............................................................................15
Long Division of Polynomials.......................................................................15

Chapter 2: Trigonometry Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

The Problems You’ll Work On .....................................................................17
What to Watch Out For.................................................................................17
Basic Trigonometry.......................................................................................18
Converting Degree Measure to Radian Measure .......................................18
Converting Radian Measure to Degree Measure .......................................19
Finding Angles in the Coordinate Plane......................................................19
Finding Common Trigonometric Values.....................................................21
Simplifying Trigonometric Expressions......................................................21
Solving Trigonometric Equations................................................................22

1,001 Calculus Practice Problems For Dummies
viii

Amplitude, Period, Phase Shift, and Midline..............................................23
Equations of Periodic Functions..................................................................23
Inverse Trigonometric Function Basics......................................................26
Solving Trigonometric Equations using Inverses......................................26

Chapter 3: Limits and Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . .29

The Problems You’ll Work On .....................................................................29
What to Watch Out For.................................................................................29
Finding Limits from Graphs..........................................................................30
Evaluating Limits ...........................................................................................31
Applying the Squeeze Theorem...................................................................32
Evaluating Trigonometric Limits.................................................................33
Infinite Limits .................................................................................................33
Limits from Graphs........................................................................................36
Limits at Infinity.............................................................................................37
Horizontal Asymptotes.................................................................................38
Classifying Discontinuities ...........................................................................38
Continuity and Discontinuities ....................................................................39
Making a Function Continuous ....................................................................40
The Intermediate Value Theorem ...............................................................41

Chapter 4: Derivative Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

The Problems You’ll Work On .....................................................................43
What to Watch Out For.................................................................................43
Determining Differentiability from a Graph ...............................................44
Finding the Derivative by Using the Definition..........................................45
Finding the Value of the Derivative Using a Graph ...................................46
Using the Power Rule to Find Derivatives..................................................47
Finding All Points on a Graph Where Tangent Lines
Have a Given Value ....................................................................................48

Chapter 5: The Product, Quotient, and Chain Rules . . . . . . . . . . . . . . . .49

The Problems You’ll Work On .....................................................................49
What to Watch Out For.................................................................................49
Using the Product Rule to Find Derivatives...............................................50
Using the Quotient Rule to Find Derivatives..............................................51
Using the Chain Rule to Find Derivatives...................................................52
More Challenging Chain Rule Problems .....................................................53

Chapter 6: Exponential and Logarithmic Functions
and Tangent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

The Problems You’ll Work On .....................................................................55
What to Watch Out For.................................................................................55
Derivatives Involving Logarithmic Functions ............................................56
Logarithmic Differentiation to Find the Derivative ...................................56
Finding Derivatives of Functions Involving
Exponential Functions...............................................................................57
Finding Equations of Tangent Lines............................................................57
Finding Equations of Normal Lines .............................................................58

viii

Table of Contents
ixix

Chapter 7: Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59

The Problems You’ll Work On .....................................................................59
What to Watch Out For.................................................................................59
Using Implicit Differentiation to Find a Derivative....................................60
Using Implicit Differentiation to Find a Second Derivative ......................60
Finding Equations of Tangent Lines Using Implicit
Differentiation.............................................................................................61

Chapter 8: Applications of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . .63

The Problems You’ll Work On .....................................................................63
What to Watch Out For.................................................................................63
Finding and Evaluating Differentials ...........................................................64
Finding Linearizations...................................................................................64
Using Linearizations to Estimate Values ....................................................64
Understanding Related Rates.......................................................................64
Finding Maxima and Minima from Graphs .................................................66
Using the Closed Interval Method...............................................................67
Finding Intervals of Increase and Decrease ...............................................68
Using the First Derivative Test to Find Local Maxima
and Minima .................................................................................................68
Determining Concavity .................................................................................68
Identifying Inflection Points .........................................................................69
Using the Second Derivative Test to Find Local Maxima
and Minima .................................................................................................69
Applying Rolle’s Theorem ............................................................................69
Using the Mean Value Theorem...................................................................70
Applying the Mean Value Theorem to Solve Problems............................70
Relating Velocity and Position.....................................................................70
Finding Velocity and Speed..........................................................................70
Solving Optimization Problems ...................................................................71
Doing Approximations Using Newton’s Method .......................................73
Approximating Roots Using Newton’s Method..........................................73

Chapter 9: Areas and Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . .75

The Problems You’ll Work On .....................................................................75
What to Watch Out For.................................................................................75
Calculating Riemann Sums Using Left Endpoints......................................76
Calculating Riemann Sums Using Right Endpoints ...................................76
Calculating Riemann Sums Using Midpoints..............................................77
Using Limits and Riemann Sums to Find Expressions
for Definite Integrals..................................................................................77
Finding a Definite Integral from the Limit
and Riemann Sum Form ............................................................................78
Using Limits and Riemann Sums to Evaluate
Definite Integrals........................................................................................78

1,001 Calculus Practice Problems For Dummies
x

Chapter 10: The Fundamental Theorem of Calculus
and the Net Change Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

The Problems You’ll Work On .....................................................................79
What to Watch Out For.................................................................................79
Using the Fundamental Theorem of Calculus
to Find Derivatives.....................................................................................80
Working with Basic Examples of Definite Integrals...................................80
Understanding Basic Indefinite Integrals ...................................................81
Understanding the Net Change Theorem...................................................84
Finding the Displacement of a Particle Given the Velocity......................85
Finding the Distance Traveled by a Particle
Given the Velocity......................................................................................85
Finding the Displacement of a Particle Given Acceleration.....................86
Finding the Distance Traveled by a Particle
Given Acceleration.....................................................................................86

Chapter 11: Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . .87

The Problems You’ll Work On .....................................................................87
What to Watch Out For.................................................................................87
Areas between Curves ..................................................................................88
Finding Volumes Using Disks and Washers ...............................................89
Finding Volume Using Cross-Sectional Slices ............................................91
Finding Volumes Using Cylindrical Shells ..................................................92
Work Problems ..............................................................................................94
Average Value of a Function.........................................................................97

Chapter 12: Inverse Trigonometric Functions,
Hyperbolic Functions, and L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . .99

The Problems You’ll Work On .....................................................................99
What to Watch Out For.................................................................................99
Finding Derivatives Involving Inverse

Trigonometric Functions ........................................................................100
Finding Antiderivatives by Using Inverse
Trigonometric Functions ........................................................................101
Evaluating Hyperbolic Functions Using Their Definitions .....................101
Finding Derivatives of Hyperbolic Functions...........................................102
Finding Antiderivatives of Hyperbolic Functions....................................102
Evaluating Indeterminate Forms Using L’Hôpital’s Rule ........................103

Chapter 13: U-Substitution and Integration by Parts . . . . . . . . . . . . . .107

The Problems You’ll Work On ...................................................................107
What to Watch Out For...............................................................................107
Using u-Substitutions ..................................................................................108
Using Integration by Parts..........................................................................109

Table of Contents
xi

Chapter 14: Trigonometric Integrals, Trigonometric
Substitution, and Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113

The Problems You’ll Work On ...................................................................113
What to Watch Out For...............................................................................114
Trigonometric Integrals..............................................................................114
Trigonometric Substitutions......................................................................116
Finding Partial Fraction Decompositions
(without Coefficients)..............................................................................117
Finding Partial Fraction Decompositions
(Including Coefficients)...........................................................................118
Integrals Involving Partial Fractions .........................................................118
Rationalizing Substitutions ........................................................................119

Chapter 15: Improper Integrals and More
Approximating Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121

The Problems You’ll Work On ...................................................................121
What to Watch Out For...............................................................................121
Convergent and Divergent Improper Integrals........................................122
The Comparison Test for Integrals ...........................................................123
The Trapezoid Rule.....................................................................................124
Simpson’s Rule.............................................................................................124

Part II: The Answers ................................................. 125

Chapter 16: Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . .127

Index....................................................................... 595

1,001 Calculus Practice Problems For Dummies
xii

Introduction

T

his book is intended for a variety of calculus students. Perhaps you want a supplement
to your current calculus class or you’re looking to brush up on a course you took long
ago. Or maybe you’re teaching yourself and need a comprehensive book of extra practice
problems.

The 1,001 questions in this book cover calculus concepts that a high school student would
encounter in a calculus course in preparation for the AP exam. It also covers most of the
concepts that a calculus student could expect to see in the first two semesters of a three-
semester calculus course. The types of questions are questions that I regularly assigned
when teaching both as homework questions or are questions that a student could’ve
expected to see on a quiz or test.

Jump around the book as you like. You can find a robust algebra and trigonometry review at
the beginning of the book to make sure that you’re prepared for calculus. The number-one
reason students have difficulty in calculus is not calculus itself but having a weak back-
ground in algebra and trigonometry. If you’re rusty on the fundamentals, spend time on
those first two chapters before jumping into the rest of the text!

As with many things worth doing in life, there’s no shortcut to becoming proficient in
mathematics. However, by practicing the problems in this book, you’ll be on your way
to becoming a much stronger calculus student.

What You’ll Find

The 1,001 calculus practice problems in the book are divided into 15 chapters, with each
chapter providing practice of the mechanical side of calculus or of applications of calcu-
lus. Some of the questions have a diagram or graph that you need in order to answer the
question.

The end of the book provides thorough and detailed solutions to all the problems. If you
get an answer wrong, try again before reading the solution! Knowing what not to do is often
a great starting point in discovering the correct approach, so don’t worry if you don’t
immediately solve each question; some problems can be quite challenging.

Beyond the Book

This book provides a lot of calculus practice. If you’d also like to track your progress online,
you’re in luck! Your book purchase comes with a free one-year subscription to all 1,001
practice questions online. You can access the content whenever you want. Create your own
question sets and view personalized reports that show what you need to study most.