AP Calculus BC, 2014-2015 Edition

499 Pages·2013·3.43 MB·English

by  Ma

SAME POWERFUL STUDY GUIDES
NEW PLATFORMS TO HELP YOU SUCCEED!
ISBN 978-0-07-180239-0
MHID 0-07-180239-8
51895
www.mhpracticeplus.com
9 780071 802390
5 STEPS TO A 5
TM
AP Calculus BC
2014--2015
This page intentionally left blank
5
TM
5 STEPS TO A
AP Calculus BC
2014--2015
William Ma
Parts of Review Chapters by Carolyn Wheater
NewYork Chicago SanFrancisco Athens London Madrid
MexicoCity Milan NewDelhi Singapore Sydney Toronto
Copyright © 2013 by McGraw-Hill Education. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of
this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written
permission of the publisher.
ISBN: 978-0-07-180240-6
MHID: 0-07-180240-1
The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-180239-0,
MHID: 0-07-180239-8.
All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names
in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear
in this book, they have been printed with initial caps.
McGraw-Hill Education eBooks are available at special quantity discounts to use as premiums and sales promotions or for use in corporate training
programs. To contact a representative, please visit the Contact Us page at www.mhprofessional.com.
TERMS OF USE
This is a copyrighted work and McGraw-Hill Education and its licensors reserve all rights in and to the work. Use of this work is subject to these terms.
Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse
engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it
without McGraw-Hill Education’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly
prohibited. Your right to use the work may be terminated if you fail to comply with these terms.
THE WORK IS PROVIDED “AS IS.” McGRAW-HILL EDUCATION AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS
TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY
INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY
WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS
FOR A PARTICULAR PURPOSE. McGraw-Hill Education and its licensors do not warrant or guarantee that the functions contained in the work will
meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill Education nor its licensors shall be liable to you or
anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill Education has no
responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill Education and/or its licensors be
liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of
them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or
cause arises in contract, tort or otherwise.
CONTENTS
AbouttheAuthors xii
Preface xiii
Acknowledgments xiv
Introduction:TheFive-StepProgram xv
Set Up Your Study Plan
STEP 1
1 WhatYouNeedtoKnowAbouttheAPCalculusBCExam 3
1.1 WhatIsCoveredontheAPCalculusBCExam? 4
1.2 WhatIstheFormatoftheAPCalculusBCExam? 4
1.3 WhatAretheAdvancedPlacementExamGrades? 5
HowIstheAPCalculusBCExamGradeCalculated? 5
1.4 WhichGraphingCalculatorsAreAllowedfortheExam? 6
CalculatorsandOtherDevicesNotAllowedfortheAPCalculusBCExam 7
OtherRestrictionsonCalculators 7
2 HowtoPlanYourTime 8
2.1 ThreeApproachestoPreparingfortheAPCalculusBCExam 8
OverviewoftheThreePlans 8
2.2 CalendarforEachPlan 10
SummaryoftheThreeStudyPlans 13
Determine Your Test Readiness
STEP 2
3 TakeaDiagnosticExam 17
3.1 GettingStarted! 21
3.2 DiagnosticTest 21
3.3 AnswerstoDiagnosticTest 27
3.4 SolutionstoDiagnosticTest 28
3.5 CalculateYourScore 38
Short-AnswerQuestions 38
APCalculusBCDiagnosticExam 38
Develop Strategies for Success
STEP 3
4 HowtoApproachEachQuestionType 41
4.1 TheMultiple-ChoiceQuestions 42
4.2 TheFree-ResponseQuestions 42
4.3 UsingaGraphingCalculator 43
4.4 TakingtheExam 44
WhatDoINeedtoBringtotheExam? 44
TipsforTakingtheExam 45
v
vi Contents
Review the Knowledge You Need to Score High
STEP 4
5 LimitsandContinuity 49
5.1 TheLimitofaFunction 50
DefinitionandPropertiesofLimits 50
EvaluatingLimits 50
One-SidedLimits 52
SqueezeTheorem 55
5.2 LimitsInvolvingInfinities 57
InfiniteLimits(as x → a) 57
LimitsatInfinity(as x → ±∞) 59
HorizontalandVerticalAsymptotes 61
5.3 ContinuityofaFunction 64
ContinuityofaFunctionataNumber 64
ContinuityofaFunctionoveranInterval 64
TheoremsonContinuity 64
5.4 RapidReview 67
5.5 PracticeProblems 69
5.6 CumulativeReviewProblems 70
5.7 SolutionstoPracticeProblems 70
5.8 SolutionstoCumulativeReviewProblems 73
6 Differentiation 75
6.1 DerivativesofAlgebraicFunctions 76
DefinitionoftheDerivativeofaFunction 76
PowerRule 79
TheSum,Difference,Product,andQuotientRules 80
TheChainRule 81
6.2 DerivativesofTrigonometric,InverseTrigonometric,
Exponential,andLogarithmicFunctions 82
DerivativesofTrigonometricFunctions 82
DerivativesofInverseTrigonometricFunctions 84
DerivativesofExponentialandLogarithmicFunctions 85
6.3 ImplicitDifferentiation 87
ProcedureforImplicitDifferentiation 87
6.4 ApproximatingaDerivative 90
6.5 DerivativesofInverseFunctions 92
6.6 HigherOrderDerivatives 94
6.7 IndeterminateForms 95
L’Hoˆpital’sRuleforIndeterminateForms 95
6.8 RapidReview 95
6.9 PracticeProblems 97
6.10 CumulativeReviewProblems 98
Contents vii
6.11 SolutionstoPracticeProblems 98
6.12 SolutionstoCumulativeReviewProblems 101
7 GraphsofFunctionsandDerivatives 103
7.1 Rolle’sTheorem,MeanValueTheorem,andExtremeValueTheorem 103
Rolle’sTheorem 104
MeanValueTheorem 104
ExtremeValueTheorem 107
7.2 DeterminingtheBehaviorofFunctions 108
TestforIncreasingandDecreasingFunctions 108
FirstDerivativeTestandSecondDerivativeTestforRelativeExtrema 111
TestforConcavityandPointsofInflection 114
7.3 SketchingtheGraphsofFunctions 120
GraphingwithoutCalculators 120
GraphingwithCalculators 121
7.4 GraphsofDerivatives 123
7.5 Parametric,Polar,andVectorRepresentations 128
ParametricCurves 128
PolarEquations 129
TypesofPolarGraphs 129
SymmetryofPolarGraphs 130
Vectors 131
VectorArithmetic 132
7.6 RapidReview 133
7.7 PracticeProblems 137
7.8 CumulativeReviewProblems 139
7.9 SolutionstoPracticeProblems 140
7.10 SolutionstoCumulativeReviewProblems 147
8 ApplicationsofDerivatives 149
8.1 RelatedRate 149
GeneralProcedureforSolvingRelatedRateProblems 149
CommonRelatedRateProblems 150
InvertedCone(WaterTank)Problem 151
ShadowProblem 152
AngleofElevationProblem 153
8.2 AppliedMaximumandMinimumProblems 155
GeneralProcedureforSolvingAppliedMaximum
andMinimumProblems 155
DistanceProblem 155
AreaandVolumeProblem 156
BusinessProblems 159
8.3 RapidReview 160
viii Contents
8.4 PracticeProblems 161
8.5 CumulativeReviewProblems 163
8.6 SolutionstoPracticeProblems 164
8.7 SolutionstoCumulativeReviewProblems 171
9 MoreApplicationsofDerivatives 174
9.1 TangentandNormalLines 174
TangentLines 174
NormalLines 180
9.2 LinearApproximations 183
TangentLineApproximation(orLinearApproximation) 183
EstimatingthenthRootofaNumber 185
EstimatingtheValueofaTrigonometricFunctionofanAngle 185
9.3 MotionAlongaLine 186
InstantaneousVelocityandAcceleration 186
VerticalMotion 188
HorizontalMotion 188
9.4 Parametric,Polar,andVectorDerivatives 190
DerivativesofParametricEquations 190
Position,Speed,andAcceleration 191
DerivativesofPolarEquations 191
VelocityandAccelerationofVectorFunctions 192
9.5 RapidReview 195
9.6 PracticeProblems 196
9.7 CumulativeReviewProblems 198
9.8 SolutionstoPracticeProblems 199
9.9 SolutionstoCumulativeReviewProblems 204
10 Integration 207
10.1 EvaluatingBasicIntegrals 208
AntiderivativesandIntegrationFormulas 208
EvaluatingIntegrals 210
10.2 IntegrationbyU-Substitution 213
TheU-SubstitutionMethod 213
U-SubstitutionandAlgebraicFunctions 213
U-SubstitutionandTrigonometricFunctions 215
U-SubstitutionandInverseTrigonometricFunctions 216
U-SubstitutionandLogarithmicandExponentialFunctions 218
10.3 TechniquesofIntegration 221
IntegrationbyParts 221
IntegrationbyPartialFractions 222
10.4 RapidReview 223
10.5 PracticeProblems 224