Acing AP Calculus AB and BC

1 E$ US ISBN 978-0-9754753-8-6 Acing AP CalculusMajor definitions, theorems, and important formulas are enclosed by a box in the beginning of each section, followed by examples and of exercises is included at the end of each chapter. These multiple choice and free response questions are grouped by section in order to help students master discrete concepts for the AP Calculus are 2 AB practice tests and 2 BC practice tests, each with 45 multiple choice questions and 6 free response questions. Many prep books use some of the same questions in their AB and BC tests, but our AB and BC practice tests never share Full-Length Practice Tests Hundreds of Examples and ExercisesDefinitions, Theorems, and Key Formulas* AP is a registered trademark of the College Entrance Examination Board.

2 Which is not affiliated with this bookCalculusAcing AP Understand the Basic Concepts of Calculus and Learn Problem-Solving TechniquesBCABandAB BCStudent Classroom EditionStudentClassroomEdition2017 Edition Acing AP Calculus AB and BC by Thomas Hyun GREENHALL PUBLISHING THOUSAND OAKS, CA Copyright 2016 by Greenhall Publishing All rights reserved. No part of this book may be reproduced in any manner without the written permission of the publisher. This edition published by Greenhall Publishing Greenhall Publishing Thousand Oaks, CA 91358 http:// Written by Thomas Hyun Cover design by Hespenheide Design Printed in the United States of America ISBN-13: 978-0-9754753-8-6 AP and Advanced Placement are registered trademarks of the College Entrance Examination Board, which does not affiliated this book.

3 Contents ABOUT THE EXAMS v PART A: MATH REVIEW 1 Chapter 1 Limits and Continuity 3 Rates of Change 3 The Limit of a Function and One Sided Limits 5 Calculating Limits Using the Limit Laws 11 Properties of Continuity and Intermediate Value Theorem 16 Limits and Asymptotes 20 Chapter 2 Differentiation 25 Definition of Derivatives and the Power

4 Rule 25 The Product and Quotient Rules and Higher Derivatives 32 The Chain Rule and the Composite Functions 37 Derivatives of Trigonometric Functions 41 Derivatives of Exponential and Logarithmic Functions 45 The Tangent Lines and the Normal Lines 50 Implicit Differentiation 54 Derivatives of an Inverse Function 59 Derivatives of Inverse Trigonometric Functions 62 Approximating a Derivative 65 Chapter 3 Applications of Differentiation 67 Related Rates 67 Position, Velocity, and Acceleration 74 The Roll s Theorem and The Mean Value Theorem 77 The First Derivative Test and the Extreme Values of Functions 82 The Second Derivative Test 91 Curves of f, f.

5 F and Curve Sketching 98 Optimization Problems 107 Tangent Line Approximation and Differentials 110 ii Contents Chapter 4 Integration 117 Antiderivatives and Indefinite Integrals 117 Riemann Sum and Area Approximation 121 Definite Integral, Area Under a Curve, and Application 126 Properties of Definite Integral 132 Trapezoidal Rule 135 The Fundamental Theorem of Calculus Part 1 139 The Fundamental Theorem of Calculus Part 2 143 Integration by Substitution 147 Integration of Exponential and Logarithmic Function 152 Chapter 5 Applications of Integration 155 Area of a Region between Two Curves 155

6 Volumes by Disk and Washers 158 Volumes of Solids with Known Cross Sections 164 The Total Change Theorem (Application of FTC) 169 Motion of a Particle, Distance, and Displacement 174 Average Value of a Function 181 Length of a Curve (Distance Traveled Along a Curve) BC 187 Chapter 6 Techniques of Integration 191 Basic Integration Rules 191 Trigonometric Integrals 196 Trigonometric Substitutions 201 L Hospital s Rule 205 Integration by Partial Fractions BC 209 Integration by Parts BC 213 Improper Integrals BC 219 Contents iii Chapter 7

7 Further Applications of Integration 223 Slope Field 223 Separable Differential Equations 227 Exponential Growth and Decay 232 Logistic Equations BC 235 Euler s Method BC 240 Chapter 8 Parametric Equations, Vectors, and Polar Coordinates BC 245 Slopes and Tangents to the Parametric Curves 245 Arc Length (Distance Traveled Along a Curve) in Parametric Form 251 Vector Valued Functions 254 Polar Coordinates and Slopes of Curves 260 Areas in Polar Coordinates 264 Chapter 9 Infinite Sequences and Series BC 269 Sequences and Series 269 The Integral Test and p-Series 273 The Comparison Test 277 Alternating Series and Error Bound 281 The Ratio Test

8 286 Convergence of Power Series 290 Representations of Functions as Power Series 294 Taylor Polynomial and Lagrange Error Bound 300 Taylor Series and Maclaurin Series 305 iv Contents PART B: PRACTICE TESTS 313 Calculus AB Practice Test 1 SECTION I Part A 315 SECTION I Part B 328 SECTION II Part A 333 SECTION II Part B 335 Calculus AB Practice Test 2 SECTION I Part A 339 SECTION I Part B 352 SECTION II Part A 358 SECTION II Part B

9 360 Calculus BC Practice Test 1 SECTION I Part A 363 SECTION I Part B 376 SECTION II Part A 382 SECTION II Part B 384 Calculus BC Practice Test 2 SECTION I Part A 387 SECTION I Part B 399 SECTION II Part A 404 SECTION II Part B 406 Answer Key 411 About the Exams v About the Calculus AB and Calculus BC Exams The AP exams in Calculus test your understanding of basic concepts in Calculus , as well as its methodology and applications.

10 The material covered by the Calculus AB exam is roughly equivalent to a one-semester introductory college course in Calculus . The Calculus BC exam is an extension of the AB material, adding on more advanced concepts such as improper integrals, series, logistic curves, and parametric and polar functions. It is important to note that both exams require a similar depth of understanding to the extent that they cover the same topics. S