AP Calculus AB, 2014-2015 Edition

450 Pages·2013·8.03 MB·English

by  Ma

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5 STEPS TO A 5
AP Calculus AB
2014--2015
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5
TM
5 STEPS TO A
AP Calculus AB
2014--2015
William Ma
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CONTENTS
AbouttheAuthor xi
Preface xii
Acknowledgments xiii
Introduction:TheFive-StepProgram xiv
Set Up Your Study Plan
STEP 1
1 WhatYouNeedtoKnowAbouttheAPCalculusABExam 3
1.1 WhatIsCoveredontheAPCalculusExam? 4
1.2 WhatIstheFormatoftheAPCalculusABExam? 4
1.3 WhatAretheAdvancedPlacementExamGrades? 5
HowIstheAPCalculusABExamGradeCalculated? 5
1.4 WhichGraphingCalculatorsAreAllowedfortheExam? 6
CalculatorsandOtherDevicesNotAllowedfortheAPCalculusABExam 7
OtherRestrictionsonCalculators 7
2 HowtoPlanYourTime 8
2.1 ThreeApproachestoPreparingfortheAPCalculusABExam 8
OverviewoftheThreePlans 8
2.2 CalendarforEachPlan 10
SummaryoftheThreeStudyPlans 13
Determine Your Test Readiness
STEP 2
3 TakeaDiagnosticExam 17
3.1 GettingStarted! 20
3.2 DiagnosticTest 20
3.3 AnswerstoDiagnosticTest 26
3.4 SolutionstoDiagnosticTest 27
3.5 CalculateYourScore 35
Short-AnswerQuestions 35
APCalculusABDiagnosticExam 35
Develop Strategies for Success
STEP 3
4 HowtoApproachEachQuestionType 39
4.1 TheMultiple-ChoiceQuestions 40
4.2 TheFree-ResponseQuestions 40
4.3 UsingaGraphingCalculator 41
4.4 TakingtheExam 42
WhatDoINeedtoBringtotheExam? 42
TipsforTakingtheExam 43
v
vi Contents
Review the Knowledge You Need to Score High
STEP 4
5 ReviewofPrecalculus 47
5.1 Lines 48
SlopeofaLine 48
EquationsofaLine 48
ParallelandPerpendicularLines 49
5.2 AbsoluteValuesandInequalities 52
AbsoluteValues 52
InequalitiesandtheRealNumberLine 53
SolvingAbsoluteValueInequalities 54
SolvingPolynomialInequalities 55
SolvingRationalInequalities 57
5.3 Functions 59
DefinitionofaFunction 59
OperationsonFunctions 60
InverseFunctions 62
TrigonometricandInverseTrigonometricFunctions 65
ExponentialandLogarithmicFunctions 68
5.4 GraphsofFunctions 72
IncreasingandDecreasingFunctions 72
InterceptsandZeros 74
OddandEvenFunctions 75
Shifting,Reflecting,andStretchingGraphs 77
5.5 RapidReview 80
5.6 PracticeProblems 81
5.7 CumulativeReviewProblems 82
5.8 SolutionstoPracticeProblems 82
5.9 SolutionstoCumulativeReviewProblems 85
6 LimitsandContinuity 86
6.1 TheLimitofaFunction 87
DefinitionandPropertiesofLimits 87
EvaluatingLimits 87
One-SidedLimits 89
SqueezeTheorem 92
6.2 LimitsInvolvingInfinities 94
InfiniteLimits(asx→a) 94
LimitsatInfinity(asx→±∞) 96
HorizontalandVerticalAsymptotes 98
6.3 ContinuityofaFunction 101
ContinuityofaFunctionataNumber 101
ContinuityofaFunctionoveranInterval 101
TheoremsonContinuity 101
Contents vii
6.4 RapidReview 104
6.5 PracticeProblems 105
6.6 CumulativeReviewProblems 106
6.7 SolutionstoPracticeProblems 107
6.8 SolutionstoCumulativeReviewProblems 109
7 Differentiation 111
7.1 DerivativesofAlgebraicFunctions 112
DefinitionoftheDerivativeofaFunction 112
PowerRule 115
TheSum,Difference,Product,andQuotientRules 116
TheChainRule 117
7.2 DerivativesofTrigonometric,InverseTrigonometric,
Exponential,andLogarithmicFunctions 118
DerivativesofTrigonometricFunctions 118
DerivativesofInverseTrigonometricFunctions 120
DerivativesofExponentialandLogarithmicFunctions 121
7.3 ImplicitDifferentiation 123
ProcedureforImplicitDifferentiation 123
7.4 ApproximatingaDerivative 126
7.5 DerivativesofInverseFunctions 128
7.6 HigherOrderDerivatives 130
7.7 RapidReview 131
7.8 PracticeProblems 132
7.9 CumulativeReviewProblems 132
7.10 SolutionstoPracticeProblems 133
7.11 SolutionstoCumulativeReviewProblems 136
8 GraphsofFunctionsandDerivatives 138
8.1 Rolle’sTheorem,MeanValueTheorem,andExtremeValueTheorem 138
Rolle’sTheorem 139
MeanValueTheorem 139
ExtremeValueTheorem 142
8.2 DeterminingtheBehaviorofFunctions 143
TestforIncreasingandDecreasingFunctions 143
FirstDerivativeTestandSecondDerivativeTestforRelativeExtrema 146
TestforConcavityandPointsofInflection 149
8.3 SketchingtheGraphsofFunctions 155
GraphingwithoutCalculators 155
GraphingwithCalculators 156
8.4 GraphsofDerivatives 158
8.5 RapidReview 163
8.6 PracticeProblems 165
8.7 CumulativeReviewProblems 168
viii Contents
8.8 SolutionstoPracticeProblems 168
8.9 SolutionstoCumulativeReviewProblems 175
9 ApplicationsofDerivatives 177
9.1 RelatedRate 177
GeneralProcedureforSolvingRelatedRateProblems 177
CommonRelatedRateProblems 178
InvertedCone(WaterTank)Problem 179
ShadowProblem 180
AngleofElevationProblem 181
9.2 AppliedMaximumandMinimumProblems 183
GeneralProcedureforSolvingAppliedMaximum
andMinimumProblems 183
DistanceProblem 183
AreaandVolumeProblems 184
BusinessProblems 187
9.3 RapidReview 188
9.4 PracticeProblems 189
9.5 CumulativeReviewProblems 191
9.6 SolutionstoPracticeProblems 192
9.7 SolutionstoCumulativeReviewProblems 199
10 MoreApplicationsofDerivatives 202
10.1 TangentandNormalLines 202
TangentLines 202
NormalLines 208
10.2 LinearApproximations 211
TangentLineApproximation(orLinearApproximation) 211
EstimatingthenthRootofaNumber 213
EstimatingtheValueofaTrigonometricFunctionofanAngle 213
10.3 MotionAlongaLine 214
InstantaneousVelocityandAcceleration 214
VerticalMotion 216
HorizontalMotion 216
10.4 RapidReview 218
10.5 PracticeProblems 219
10.6 CumulativeReviewProblems 220
10.7 SolutionstoPracticeProblems 221
10.8 SolutionstoCumulativeReviewProblems 225
11 Integration 227
11.1 EvaluatingBasicIntegrals 228
AntiderivativesandIntegrationFormulas 228
EvaluatingIntegrals 230