Cliffsap Calculus Ab & Bc

Cliffsap Calculus Ab & Bc

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Full Description


A focused review to help you score high and earn college credit on the Calculus AB & BC Advanced Placement Program exam. This hard-hitting guide features: * Helpful test-taking strategies * Focus sections on specific topic areas, including precalculus, limits and continuity, derivatives, and integrals * Sample multiple choice and free-response questions * A discussion of calculators to use during the exam, including which are the best types Advanced Placement Program and AP are registered trademarks of the College Board, which was not involved in the production of, and does not endorse this product.

Table of Contents

Preface                                            ix
Study Guide Checklist ix
PART I: INTRODUCTION
Introduction 3 (16)
Questions Commonly Asked About the AP 3 (2)
Calculus AB and BC Exams
Strategies for the Exam 5 (1)
Multiple-Choice Questions 6 (2)
Free-Response Questions 8 (2)
Calculator Questions 10 (3)
Topics Covered on a Recent AP Calculus AB 13 (6)
or BC Exam
Precalculus 13 (1)
Calculus 13 (6)
PART II: SPECIFIC TOPICS
Precalculus Topics 19 (56)
Functions and Function Notation 19 (7)
Definition of a Function 19 (1)
Vocabulary 20 (1)
Notation 20 (1)
Zeros or Roots 21 (1)
Symmetry 21 (1)
Definition of Even and Old Functions 22 (1)
Graphing a Function 23 (1)
Shifts and Distortions of Graphs 23 (2)
Absolute-Value Distortions 25 (1)
Inverse Functions 26 (2)
Definition of Inverse Functions 27 (1)
Notation 27 (1)
Polynomial Functions 28 (2)
Degree Test 29 (1)
Leading Coefficient Test 29 (1)
Trigonometric Functions 30 (6)
Graphs, Domain and Range, Periods 30 (1)
Shifts and Distortions for Trig Functions 31 (1)
Trig Identities 32 (1)
Solving Trig Equations 33 (1)
Inverse Trig Functions 34 (2)
Exponential and Logarithmic Functions 36 (4)
Exponential Functions 36 (1)
Logarithmic Functions 37 (1)
Special Log Functions 38 (1)
Properties of Logarithms 38 (2)
Relationship Between Logs and Exponentials 40 (1)
Rational Functions 40 (3)
Definition of a Rational Function 40 (1)
Graphing a Rational Function 41 (2)
Conic Sections 43 (3)
Completing the Square 44 (2)
Algebra 46 (2)
Interval Notation 46 (1)
Lines 47 (1)
Interval Testing 47 (1)
Absolute Value 48 (1)
Parametric Equations (BC Only) 48 (1)
Domain of t 49 (1)
Eliminating the Parameter 49 (4)
Particle Problems 52 (1)
Polar Coordinates and Graphs (BC Only) 53 (8)
Notation 54 (1)
Conversion 54 (1)
Calculators 54 (1)
Polar Graphs 55 (1)
Symmetry 55 (1)
Intersections 55 (1)
Converting Equations from Rectangular to 56 (1)
Polar Form
Converting Equations from Polar to 57 (2)
Rectangular Form
Special Polar Graphs 59 (2)
Vectors and Vector-Valued Functions (BC 61 (5)
Only)
Component Form of a Vector 61 (1)
Magnitude and Direction of a Vector in 62 (1)
Standard Position
Vectors Operations 62 (1)
Alternate Component Form 63 (1)
Magnitude/Direction to Component Form 64 (1)
Vector-Valued Functions 65 (1)
Sample Multiple-Choice Questions: 66 (3)
Precalculus
Answers to Multiple-Choice Questions 69 (4)
Sample Free-Response Question: Precalculus 73 (1)
Answer to Free-Response Question 73 (2)
Limits and Continuity 75 (30)
Intutive Definition of a Limit 75 (4)
Algebraic Techniques for Finding Limits 79 (1)
One-Sided Limits 80 (3)
Infinite Limits 83 (2)
Theorem for Infinite Limits 84 (1)
Limits at Infinity 85 (5)
Theorem for Limits at Infinity 88 (2)
Special Trig Limits 90 (1)
Continuity 90 (4)
Definition of Continuity at a Point 91 (1)
Definition of Continuity on an Open 92 (1)
Interval
Definition of Continuity on a Closed 92 (2)
Interval
Sample Multiple-Choice Questions: Limits 94 (3)
and Continuity
Answers to Multiple-Choice Questions 97 (3)
Sample Free-Response Questions: Limits and 100(1)
Continuity
Answers to Free-Response Questions 101(4)
Derivatives 105(48)
Definition of the Derivative 105(4)
Differentiation Rules 109(6)
Constant Rule 109(1)
Constant Multiple Rule 109(1)
Sum and Difference Rule 110(1)
Power Rule 110(2)
Trigonometric Rules 112(1)
Product Rule 112(1)
Quotient Rule 113(2)
The Chain Rule 115(4)
Chain Rule 116(1)
Alternative Form of the Chain Rule 117(1)
Power Rule with Chain 118(1)
Trig Rules with Chain 118(1)
Higher-Order Derivatives 119(1)
Derivatives of Exponential Functions 119(2)
Exponential Rule 120(1)
Base-a Exponentials Rule 120(1)
Derivatives of Logarithmic Functions 121(2)
Natural Log Rule 122(1)
Base-a Logarithms Rule 122(1)
Derivatives of Inverse Trigonometric 123(3)
Functions
Inverse Trig Rules 124(1)
Derivatives of Inverses 124(2)
Implicit Differentiation 126(2)
Logarithmic Differentiation 128(2)
Parametric Derivatives (BC Only) 130(2)
Parametric From of the Derivative 130(1)
Parametric From of the Second Derivative 130(2)
Differentiation with Polar Curves (BC Only) 132(2)
Slope in Polar 132(2)
Limits and Continuity of Vector-Valued 134(1)
Functions
Limit and Continuity of a Vector-Valued 135(1)
Function
Differentiation of Vector-Valued Function 135(2)
(BC Only)
Derivative of a Vector-Valued Function 135(2)
Differentiability and Continuity 137(3)
Differentiability Implies Continuity 137(3)
Sample Multiple-Choice Questions: 140(4)
Derivatives
Answers to Multiple-Choice Questions 144(7)
Sample Free-Response Question: Derivatives 151(1)
Answer to Free-Response Question 151(2)
Applications of the Derivative 153(60)
Tangent and Normal Lines 153(3)
Position, Velocity, and Acceleration 156(5)
Theorem for PVA 157(4)
Related Rates 161(10)
Sample Problem 161(10)
Relative Extrema and the First Derivative 171(4)
Test
Theorem on Increasing and Decreasing 171(2)
First Derivative Test for Extrema 173(2)
Concavity and the Second Derivative Test 175(6)
Definition of Concavity 176(1)
Theorem: Test for Concavity 177(1)
Definition of Point of Inflection 178(2)
Second Derivative Test for Extrema 180(1)
Absolute Extrema and Optimization 181(7)
Extreme Value Theorem 182(2)
Sample Problem 184(4)
Rolle's Theorem and the Mean Value Theorem 188(3)
Rolle's Theorem 188(1)
Mean Value Theorem 189(2)
Differentials 191(2)
L'Hopital's Rule (BC Only) 193(5)
Theorem: L'Hopital's Rule 193(4)
Summary of Forms Requiring L'Hospital's 197(1)
Rule
Sample Multiple-Choice Questions: 198(4)
Applications of the Derivative
Answers to Multiple-Choice Questions 202(8)
Sample Free-Response Questions: 210(1)
Applications of the Derivative
Answers to Free-Response Questions 211(2)
Antiderivatives and Definite Integrals 213(46)
Antiderivatives 213(4)
Definition of an Antiderivative 213(1)
Theorem on Antiderivatives 213(1)
Notation 214(1)
Vocabulary 214(1)
Constant Multiple Rule 214(1)
Sum and Difference Rule 214(1)
Trig Rules 215(1)
Power Rule 215(2)
The Chain Rule for Antiderviatives 217(4)
Chain Rule for Antiderivatives 217(4)
Exponential Antiderivatives 221(2)
Exponential Antiderivative 221(2)
Antiderivative of Other Exponentials 223(1)
Antiderivatives and the Natural Log 223(3)
Antiderivatives Rule for the Natural Log 224(2)
Trig Integrals 226(2)
Antiderivatives of Inverse Trig Functions 228(3)
Inverse Trig Antiderivatives 229(2)
Integration by Parts (BC Only) 231(3)
Theorem: Integration by Parts 231(3)
Integration by Partial Fractions (BC Only) 234(2)
Improper Integrals (BC Only) 236(7)
Examples of Improper Integrals 237(1)
Improper Integrals with Infinite Bounds 238(2)
of Integration
Improper Integrals with Infinite 240(3)
Discontinuities
Sample Multiple-Choice Questions: 243(6)
Antiderivatives
Answers to Multiple-Choice Questions 249(10)
Applications of Antiderivatives and Definite 259(84)
Integrals
The Fundamental Theorem of Calculus 259(2)
Fundamental Theorem of Calculus 259(1)
Notation 260(1)
Definition of the Definite Integral 261(8)
Definition of the Definite Integral 261(3)
Theorem: Trapezoidal Rule 264(2)
Theorem: Area Between Curves 266(3)
Properties of the Definite Integral 269(5)
Four Properties of the Definite Integral 269(3)
Property for Even and Odd Functions 272(1)
Variable Bounds Property 273(1)
Average-Value Property 274(1)
Volume 274(18)
Vertical Axis of Revolution 275(2)
Horizontal Axis of Revolution 277(3)
Known Cross-Sectional Area 280(1)
Solids of Revolution 281(8)
Solids with Known Cross Sections 289(3)
Differential Equations 292(3)
Numerical Solutions to Differential 295(5)
Equations: Euler's Method
Euler's Method as a Recursive Formula 299(1)
Differential Equations and Slope Fields 300(5)
PVA with Antiderivatives 305(6)
Theorem for PVA 305(3)
Exponential Decay and Newton's Law of 308(1)
Cooling
Bounded Growth 309(2)
Logistic Growth (BC Only) 311(3)
Spread of a Rumor 312(1)
Spread of a Disease 312(1)
Population Growth 312(2)
Work (BC Only) 314(7)
Work Along a Strainght Line 315(3)
Liquid Work 318(3)
Length of an Arc (BC Only) 321(2)
Length of an Arc 321(2)
Arc Length for Parametric Curves (BC Only) 323(1)
Arc Length in Parametric 323(1)
Area and Arc Length for Polar Curves (BC 324(5)
Only)
Area in Polar 324(4)
Arc Length in Polar 328(1)
Integration of Vector-Valued Functions (BC 329(1)
Only)
Sample Multiple-Choice Questions: 330(3)
Applications of Antiderivatives and
Definite Integrals
Answers to Multiple-Choice Questions 333(8)
Sample Free-Response Question: Applications 341(1)
of Antiderivatives and Definite Integrals
Answer to Free-Response Question 342(1)
Sequences, Infinite Series, and Polynomial 343(56)
Approximations (BC Only)
Sequences 343(2)
Using Current Technology 344(1)
Series 345(17)
Tests of Convergence or Divergence of an 346(13)
Infinite Series
Other Useful Information Regarding 359(1)
Infinite Series
Summary of Tests for Convergence or 359(1)
Divergence of Infinite Series
Alternating Series Error Bound 360(2)
Radius and Interval of Convergence of Power 362(3)
Series
Sample Multiple-Choice Questions: Sequences 365(2)
and Infinite Series
Answers to Multiple-Choice Questions 367(7)
Sample Free-Response Question: Sequences 374(1)
and Series
Answer to Free-Response Question 375(1)
Polynomial Approximations of Functions 376(2)
(Taylor and Maclaurin Polynomials)
Differentiation and Integration of Power 378(7)
Series
Lagrange From of the Remainder of a Taylor 385(3)
Polynomial
Taylor's Theorem 385(2)
Power Series for Some Common Functions 387(1)
Sample Multiple-Choice Questions: 388(2)
Polynomial Approximations of Functions
(Taylor and Maclaurin Polynomials)
Answers to Multiple-Choice Questions 390(3)
Sample Free-Response Question: Polynomial 393(1)
Approximation of Functions (Taylor and
Maclaurin Polynomials)
Answer to Free-Response Question 393(4)
PART III: AP CALCULUS AB AND BC PRACTICE TESTS
Answer Sheet for Practice Test 1 - AB 397(1)
Taking and Grading the Practice Exams 398(1)
Practice Test 1 - AB 399(34)
Section I: Multiple-Choice Questions 399(8)
Section IA 399(5)
Section IB 404(3)
Section II: Free-Response Questions 407(2)
Section IIA 407(1)
Section IIB 408(1)
Answer Key for Practice Test 1 - AB 409(2)
Section I: Multiple-Choice Questions 409(1)
Section II: Free-Response Questions 410(1)
Practice Test I Scoring Worksheet 410(1)
Answers and Explanations for Practice Test 411(20)
1 - AB
Section I: Multiple-Choice Questions 411(10)
Section II: Free-Response Questions 421(10)
Answer Sheet for Practice Test 2 - BC 431(2)
Practice Test 2- BC 433(34)
Section I: Multiple-Choice Questions 433(8)
Section IA 433(5)
Section IB: BC 438(3)
Section II: Free-Response Questions 441(2)
Section IIA 441(1)
Section IIB 442(1)
Answer Key for Practice Test 2 - BC 443(2)
Section I: Multiple-Choice Questions 443(1)
Section II: Free-Response Questions 444(1)
Practice Test 2 Scoring Worksheet 444(1)
Answers and Explanations for Practice Test 445(22)
2 - BC
Section I: Multiple-Choice Questions 445(14)
Section II: Free-Response Questions 459(8)
Appendix: Calculus Dictionary 467