Calculus Practice Problems For Dummies

1,001 Calculus Prac tice Problems 1,001 Calculus Prac tice Problems by PatrickJMT 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 (201) 748-6008, or online at http://www.wiley.com/go/permissions. 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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 vviiiiii 1,001 Calculus Practice Problems For Dummies 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