02 542257 FM.qxd 5/9/03 10:10 AM Page i CliffsQuickReview TM Calculus Anton/Bivens/Davis Version By Bernard V. Zandy, MA and Jonathan J. White, MS 02 542257 FM.qxd 5/9/03 10:10 AM Page ii About the Authors Publisher’s Acknowledgments Bernard V. Zandy, MA, Professor of Mathematics Editorial at Fullerton College in California has been teach- Project Editor: Brian Kramer ing secondary and college level mathematics for Acquisitions Editor: Sherry Gomoll 34 years. A co-author of the Cliffs PSAT and SAT Technical Editor: Dale Johnson Preparation Guides, Mr. Zandy has been a lecturer Composition and consultant for Bobrow Test Preparation Ser- Indexer: TECHBOOKS Production Services vices, conducting workshops at California State Proofreader: Joel K. Draper University and Colleges since 1977. Wiley Publishing Composition Services Jonathan J. White has a BA in mathematics from Coe College and an MS in mathematics from the University of Iowa. 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Some content that appears in print may not be available in electronic books. is a trademark of Wiley Publishing, Inc. 542257 FM.qxd 5/9/03 2:30 PM Page iii Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Why You Need This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Using Calculus 7e by Anton/Bivens/Davis (ABD) with CQR . . . . . . .2 Chapter 1: Review Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Interval Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 Chapter 2: Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Intuitive Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Evaluating Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 One-sided Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 Limits Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . .23 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 Chapter 3: The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Trigonometric Function Differentiation . . . . . . . . . . . . . . . . . . . . . . .34 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 Differentiation of Inverse Trigonometric Functions . . . . . . . . . . . . . .40 Differentiation of Exponential and Logarithmic Functions . . . . . . . .41 Chapter 4: Applications of the Derivative . . . . . . . . . . . . . . . . . . . . . . . .43 Tangent and Normal Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 Extreme Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 Increasing/Decreasing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .48 First Derivative Test for Relative Extrema . . . . . . . . . . . . . . . . . . . . . .49 Second Derivative Test for Relative Extrema . . . . . . . . . . . . . . . . . . .50 Concavity and Points of Inflection . . . . . . . . . . . . . . . . . . . . . . . . . . .51 Maximum/Minimum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52 Distance, Velocity, and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . .55 Related Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58 02 542257 FM.qxd 5/9/03 10:10 AM Page iv iv CliffsQuickReviewCalculus Chapter 5: Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 Antiderivatives/Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . .63 Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 Basic formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 Substitution and change of variables . . . . . . . . . . . . . . . . . . . . . . .66 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 Trigonometric integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 Distance, Velocity, and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . .73 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 Definition of definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .75 Properties of definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . .78 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . .80 Definite integral evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 Chapter 6: Applications of the Definite Integral . . . . . . . . . . . . . . . . . .88 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88 Volumes of Solids with Known Cross Sections . . . . . . . . . . . . . . . . . .93 Volumes of Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 Disk method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 Washer method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97 Cylindrical shell method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 CQR Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 CQR Resource Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113 Appendix: Using Graphing Calculators in Calculus . . . . . . . . . . . . . . .116 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121 02 542257 FM.qxd 5/9/03 10:10 AM Page v v Correlation Guide CORRELATION GUIDE: CLIFFSQUICKREVIEW CALCULUS WITH CALCULUS 7E BY ANTON, BIVENS, DAVIS CliffsQuickReview Calculus Calculus 7E Chapter Topic Title 1 Interval Notation Appendix A 1 Absolute Value Appendix B 1 Functions Sections 1.1, 1.2 1 Linear Equations Appendix C 1 Trigonometric Functions Appendix E 2 Intuitive Definition Section 2.1 2 Evaluating Limits Section 2.2 2 One-Sided Limits Sections 2.1, 2.2 2 Infinite Limits Sections 2.1, 2.2 2 Limits at Infinity Sections 2.1, 2.3 2 Limits Involving Trigonometric Functions Section 2.6 2 Continuity Section 2.5 3 Definition Section 3.2 3 Differentiation Rules Section 3.3 3 Trigonometric Function Differentiation Section 3.4 3 Chain Rule Section 3.5 3 Implicit Differentiation Section 3.6 3 Higher Order Derivatives Section 3.3 3 Differentiation of Inverse Trigonometric Functions Section 4.4 3 Differentiation of Exponential and Logarithmic Functions Section 4.3 (continued) 02 542257 FM.qxd 5/9/03 10:10 AM Page vi vi CliffsQuickReviewCalculus CliffsQuickReview Calculus Calculus 7E Chapter Topic Title 4 Tangent and Normal Lines Section 3.2 4 Critical Points Section 5.2 4 Extreme Value Theorem Section 5.5 4 Mean Value Theorem Section 5.8 4 Increasing/Decreasing Functions Section 5.1 4 First Derivative Test for Local Extrema Section 5.2 4 Second Derivative Test for Local Extrema Section 5.2 4 Concavity and Points of Inflection Section 5.1 4 Maximum/Minimum Problems Section 5.6 4 Distance, Velocity, Acceleration Section 5.4 4 Related Rates of Change Section 3.7 4 Differentials Section 3.8 5 Antiderivatives/Indefinite Integrals Section 6.2 5 Integration Techniques Section 6.2 5 Substitution and Change of Variables Section 6.3 5 Integration by Parts Section 8.2 5 Trigonometric Integrals Section 8.3 5 Trigonometric Substitutions Section 8.4 5 Distance, Velocity, and Acceleration Section 6.7 5 Definition of Definite Integrals Section 6.5 5 Properties of Definite Integrals Sections 6.5, 6.6 5 The Fundamental Theorem of Calculus Section 6.6 5 Definite Integral Evaluation Section 6.8 02 542257 FM.qxd 5/9/03 10:10 AM Page vii vii Correlation Guide Chapter Topic Title 6 Area Section 7.1 6 Volumes of Solids with Known Cross Sections Section 7.2 6 Disk Method Section 7.2 6 Washer Method Section 7.2 6 Cylindrical Shell Method Section 7.3 6 Arc Length Section 7.4 CORRELATION GUIDE: CLIFFSQUICKREVIEW CALCULUS WITH CALCULUS: EARLY TRANSCENDENTALS 7E BY ANTON, BIVENS, DAVIS CliffsQuickReview Calculus Calculus 7E Chapter Topic Title 1 Interval Notation Appendix A 1 Absolute Value Appendix B 1 Functions Sections 1.1, 1.2 1 Linear Equations Appendix C 1 Trigonometric Functions Appendix E 2 Intuitive Definition Section 2.1 2 Evaluating Limits Section 2.2 2 One-Sided Limits Sections 2.1, 2.2 2 Infinite Limits Sections 2.1, 2.2 2 Limits at Infinity Sections 2.1, 2.3 (continued) 02 542257 FM.qxd 5/9/03 10:10 AM Page viii viii CliffsQuickReviewCalculus CliffsQuickReview Calculus Calculus 7E Chapter Topic Title 2 Limits Involving Trigonometric Functions Section 2.6 2 Continuity Section 2.5 3 Definition Section 3.2 3 Differentiation Rules Section 3.3 3 Trigonometric Function Differentiation Section 3.4 3 Chain Rule Section 3.5 3 Implicit Differentiation Section 3.6 3 Higher Order Derivatives Section 3.3 3 Differentiation of Inverse Trigonometric Functions Section 4.4 3 Differentiation of Exponential and Logarithmic Functions Section 4.3 4 Tangent and Normal Lines Section 3.2 4 Critical Points Section 5.2 4 Extreme Value Theorem Section 5.5 4 Mean Value Theorem Section 5.8 4 Increasing/Decreasing Functions Section 5.1 4 First Derivative Test for Local Extrema Section 5.2 4 Second Derivative Test for Local Extrema Section 5.2 4 Concavity and Points of Inflection Section 5.1 4 Maximum/Minimum Problems Section 5.6 4 Distance, Velocity, Acceleration Section 5.4 4 Related Rates of Change Section 3.7 4 Differentials Section 3.8 02 542257 FM.qxd 5/9/03 10:10 AM Page ix ix Correlation Guide Chapter Topic Title 5 Antiderivatives/Indefinite Integrals Section 6.2 5 Integration Techniques Section 6.2 5 Substitution and Change of Variables Section 6.3 5 Integration by Parts Section 8.2 5 Trigonometric Integrals Section 8.3 5 Trigonometric Substitutions Section 8.4 5 Distance, Velocity, and Acceleration Section 6.7 5 Definition of Definite Integrals Section 6.5 5 Properties of Definite Integrals Sections 6.5, 6.6 5 The Fundamental Theorem of Calculus Section 6.6 5 Definite Integral Evaluation Section 6.8 6 Area Section 7.1 6 Volumes of Solids with Known Cross Sections Section 7.2 6 Disk Method Section 7.2 6 Washer Method Section 7.2 6 Cylindrical Shell Method Section 7.3 6 Arc Length Section 7.4 02 542257 FM.qxd 5/9/03 10:10 AM Page x x CliffsQuickReviewCalculus