Teacher’s Resource Binder to accompany ROGAWSKI’S CALCULUS for AP* Early Transcendentals Second Edition Jon Rogawski Ray Cannon by Lin McMullin * AP is a trademark registered and/or owned by the College Board, which was not involved in the publication of and does not endorse this product. © 2012 by W.H. Freeman and Company ISBN-13: 978-1-4292-8629-9 ISBN-10: 1-4292-8629-6 All rights reserved. Printed in the United States of America First Printing W.H. Freeman and Company 41 Madison Avenue New York, NY 10010 Houndmills, Basingstoke RG21 6XS, England www.whfreeman.com TABLE OF CONTENTS Preface v Chapter 1 Precalculus Review 1 1.1 Real Numbers, Functions, and Graphs 1 1.2 Linear and Quadratic functions 8 1.3 The Basic Classes of Functions 14 1.4 Trigonometric Functions 18 1.5 Inverse Functions 24 1.6 Exponential and Logarithmic Functions 27 1.7 Technology: Calculators and Computers 31 Chapter 2 Limits 37 2.1 Limits, Rates of Change, and Tangent Lines 38 2.2 Limits: A Numerical and Graphical Approach 43 2.3 Basic Limit Laws 48 2.4 Limits and Continuity 51 2.5 Evaluating Limits Algebraically 56 2.6 Trigonometric Limits 60 2.7 Limits at Infinity 63 2.8 Intermediate Value Theorem 69 2.9 The Formal Definition of a Limit 72 Chapter 3 Differentiation 83 3.1 Definition of the Derivative 85 3.2 The Derivative as a Function 90 3.3 The Product and Quotient Rules 93 3.4 Rates of Change 96 3.5 Higher Derivatives 102 3.6 Trigonometric Functions 106 3.7 The Chain Rule 109 3.8 Derivatives of Inverse Functions 112 3.9 Derivatives of Exponential and Logarithmic Functions 118 3.10 Implicit Differentiation 121 3.11 Related Rates 127 Chapter 4 Applications of the Derivative 145 4.1 Linear Approximation and Applications 146 4.2 Extreme Values 151 4.3 The Mean Value Theorem and Monotonicity 157 4.4 The Shape of a Graph 162 4.5 L’Hopital’s Rule 167 4.6 Graph Sketching and Asymptotes 173 4.7 Applied Optimization 179 4.8 Newton’s Method 186 4.9 Antiderivatives 190 Chapter 5 The Integral 209 5.1 Limits: Approximating and Computing Area 210 5.2 The Definite Integral 218 5.3 The Fundamental Theorem of Calculus, Part I 224 5.4 The Fundamental Theorem of Calculus, Part II 229 5.5 Net or Total Change as the Integral of a Rate 237 5.6 Substitution Method 243 5.7 Further Transcendental Functions 249 5.8 Exponential Growth and Decay 252 Chapter 6 Applications of the Integral 267 6.1 Area Between Two Curves 268 6.2 Setting Up Integrals: Volume, Density, Average Value 275 6.3 Volumes of Revolution 281 6.4 The Method of Cylindrical Shells 291 6.5 Work and Energy 299 Chapter 7 Techniques of Integration 313 7.1 Integration by Parts 314 7.2 Trigonometric Integrals 320 7.3 Trigonometric Substitution 326 7.4 Integrals of Hyperbolic and Inverse Hyperbolic Functions 331 7.5 The Method of Partial Fractions 335 7.6 Improper Integrals 340 7.7 Probability and Integration 345 7.8 Numerical Integration 346 Chapter 8 Further Applications 357 8.1 Arc Length and Surface Area 358 8.2 Fluid Pressure and Force 361 8.3 Center of Mass 367 8.4 Taylor Polynomials 375 Chapter 9 Introduction to Differential Equations 387 9.1 Solving Differential Equations 388 9.2 Models involving y’ = k (y-b) 393 9.3 Graphical and Numerical Methods 397 9.4 The Logistic Equation 403 9.5 First-Order Linear Equations 408 Chapter 10 Infinite Series 415 10.1 Sequences 416 10.2 Summing an Infinite Series 422 10.3 Convergence of Series with Positive Terms 428 10.4 Absolute and Conditional Convergence 435 10.5 The Ratio and Root Tests 440 10.6 Power Series 445 10.7 Taylor Series 451 Chapter 11 Parametric Equations, Polar Coordinates, and Vector Functions 467 11.1 Parametric Equations 468 11.2 Arc Length and Speed 474 11.3 Polar Coordinates 477 11.4 Area and Arc Length in Polar coordinates 483 11.5 Vectors in the Plane 487 11.6 Dot Product and the Angle Between Two Vectors 495 11.7 Calculus of Vector-Valued Functions 500 Flashcards 513 Preface Written to support Calculus for AP* Early Transcendentals, Second Edition, by John Rogawski and Ray Cannon, this Teacher’s Resource Binder offers both supplementary and complementary material. We understand a teacher's time is precious; as a result, we try to keep our notes succinct for ease of reference. Our overarching goal is to be concise, practical, and easy to use. Hopefully it will save some time in planning classes! Regardless of a teacher's level of experience, we hope that our readers share a similar enjoyment of teaching and exploring calculus. We developed a number of features to help the veteran instructor or the first time teacher. Many teachers find different techniques effective in conveying the key topics in calculus. In this TRB, we made an effort to address the different approaches that may be employed. We make suggestions and provide guidance to those who prefer lectures and also supply material for instructors who prefer other approaches. We hope this resource will help you to try new methods in approaching a difficult topic. On the following pages we note the features in this guide and provide brief descriptions about how each feature should best be utilized. We would like to acknowledge the editorial staff at W. H. Freeman and Company. © W. H. Freeman & Company Rogawski’s Calculus for AP* , Early Transcendentals v Features Each Section includes the following information: 1) Class Time - covering all of the material for the AP exams is difficult. • We make suggestions about how much time should be spent on each section for AB and BC calculus. • We also note how critical each section is to a student's understanding of calculus. 2) Key Points - It may be difficult to discern some of the main ideas in the text. We provide • a streamlined list of all the important topics from each chapter. • concise points for quick reference. • a bulleted list that identifies the main ideas in each section. 3) Lecture Material - In this feature, we take a more theoretical approach to the section's material to foster a conceptual understanding for the student. The material • is concise in its presentation. • is based on our own teaching experiences. • intertwines key examples and exercises from the text. • guides the teacher in a lecture or lecture and discussion setting. 4) Discussion Topics/Class Activities - With any class, some of the more interesting topics require some deviation from the main concepts, and there are some common issues that repeatedly give students problems. These topics • are engaging examples from our own experience and also examples and problems from the text. • will force a student to think and reflect on the material, allowing the student to formulate a distinct understanding of the material at hand. • provide an opportunity to get outside the typical lecture. 5) Suggested Problems - There a large number and variety of problems at the end of each section. To help the teacher identify good problems, especially those that are in the AP style, we provide the following: • a quick reference guide for homework problems. • suggestions about some core problems that cover a variety of topics and problem types. • problems that cover graphical, numerical, abstract, and algebraic genres. Often, we also note the difficulty, and if the problem relates to a specific topic in that particular section. 6) Worksheet and Worksheet Solutions - We provide material that can be distributed in class. The best way to learn calculus is by doing calculus. It is very helpful, especially to struggling students, when the first attempt at a type of problem takes place in class, with the teacher available to help. Thus, we provide problems that relate directly to the material. • Exercises that provide additional practice to students who are having trouble with specific topics. • Exercises that provide feedback for both students and the instructor. Each Chapter includes the following: AP Style Questions - After talking to many AP teachers, we understand that there can never be enough practice and preparation for the exam. As a result, we developed multiple choice and free response questions that correspond with sections of Rogawski's Calculus. These can be used as practice quizzes or for testing material. vi Preface © W. H. Freeman & Company Teacher’s Resource Manual Author Team Lin McMullin has taken the lead role in developing the TRB for both the first and second editions of Jon Rogawski’s Calculus. He is an author and consultant working in mathematics education. He taught high school mathematics, including AB and BC Calculus for 34 years. He has led many workshops and institutes for AP Calculus teachers in the United States and Europe. He served as an AP Calculus exam reader and table leader for 14 years. He is the author of Teaching AP Calculus, which is a resource book for teachers based on the material he teaches in his one-week summer institutes. Ray Cannon (Baylor University) wrote the chapter overviews that begin each chapter. Ray has long been interested in the articulation between high school and college mathematics and has served the AP Calculus program in a variety of ways: as a Reader of the exams, as a Table Leader, as Exam Leader (both AB and BC0, and, finally, through four years as Chief Reader. He has also served on the College Board’s Test Development Committee for AP Calculus. Ray is a frequent consultant for the College Board, presenting at workshops and leading week-long summer institutes. Additionally, Ray served on Mathematical Association of America (MAA) committees concerned with the issue of proper placement of students in precalculus and calculus courses. Ray has won numerous awards for his teaching and service, including university-wide teaching awards from the University of North Carolina and Baylor University. He was named a Piper Professor in the state of Texas in 1997 and has twice been given awards by the Southwestern region of the College Board for outstanding contributions to the Advanced Placement Program. AP Question Writers for the Teacher’s Resource Binder John Jensen is currently the Faculty Chair in Mathematics at Rio Salado College in Tempe, Arizona. Before arriving at Rio Salado, he taught high school mathematics for 30 years in the Paradise Valley School District in Phoenix, Arizona. For 25 years, he taught Advanced Placement Calculus. John has been an AP Calculus reader and table leader for 17 years and has conducted over 150 workshops and institutes in the United States, Canada, Europe, and Asia. During the course of his career, he has received the following honors: Presidential Award for Excellence in Teaching Mathematics in 1987; the first Siemens Advanced Placement Award in 1998; the Distinguished Service Award (1998) and the Exemplar Award (2001) by the College Board; and the Tandy Technology (Radio Shack) Award in 1997. John is also a former fellow of the Woodrow Wilson Mathematics Institute at Princeton University and holds a National Board Adolescence and Young Adulthood Certificate in Mathematics. Haika Karr teaches AP Calculus AB and BC at Liberty Hill High School in Liberty Hill, Texas. Her 12 years of teaching experience include 7 years of teaching Calculus. She is currently the Mathematics Department Chairperson, coaches UIL Mathematics and Number Sense, and sponsors her school's chapter of the National Honor's Society. Bret Norvilitis has taught high school and middle school math for the past 16 years, the past 10 at Orchard Park High School in Orchard Park, NY, where he is currently the department chair. He became a Calculus teacher five years ago. He has attended many weekend AP seminars and two week-long AP conferences in both AB and BC. He has taught the AB curriculum seven times and the BC curriculum twice. © W. H. Freeman & Company Rogawski’s Calculus for AP* , Early Transcendentals vii Correlation to The College Board’s AP* Topic Outline (AB and BC) AB BC College Board Topic Outline Rogawski ET Rogawski LT AB BC I. Function, Graphs, and Limits AB BC Analysis of graphs 1.7 1.5 AB BC Limits of Functions (including one-sided limits) Ch. 2 Ch. 2 AB BC  Intuitive understanding of the limiting process 2.1-2.2 2.1-2.2 AB BC  Calculating limits using algebra 2.3, 2.5-2.6 2.3, 2.5-2.6 AB BC  Estimating limits from graphs or tables of data 2.2, 2.6 2.2, 2.6 AB BC Asymptotic and unbounded behavior 4.6 4.5  Understanding asymptotes in terms of graphical AB BC behavior 4.6 4.5  Describing asymptotic behavior in terms of AB BC limits involving infinity 4.6 4.5  Comparing relative magnitudes of functions and AB BC their rates of change 4.5 7.7 AB BC Continuity as a property of functions 2.4 2.4 AB BC  Intuitive understanding of continuity 2.4 2.4 AB BC  Understanding continuity in terms of limits 2.4 2.4  Geometric understanding of graphs of AB BC continuous functions 2.8 2.8 BC Parametric, polar and vector functions 11.5 12.5 AB BC II. Derivatives Chs 3-4 Chs 3-4 AB BC Concept of the derivative 3.1 3.1  Derivative presented graphically, numerically, AB BC and analytically 3.1, 3.3 3.1, 3.3  Derivative interpreted as an instantaneous rate AB BC of change 3.4 3.4  Derivative defined as the limit of the difference AB BC quotient 3.1 3.1  Relationship between differentiability and AB BC continuity 3.2 3.2 AB BC Derivative at a point 3.1 3.1 AB BC  Slope of a curve at a point 3.1 3.1  Tangent line to a curve at a point and local AB BC linear approximation 4.1 4.1  Instantaneous rate of change as the limit of AB BC average rate of change 3.1 3.1  Approximate rate of change from graphs and AB BC tables of values 3.1 3.1 AB BC Derivative as a function 3.2 3.2  Corresponding characteristics of graphs of f and AB BC f' 3.2, 4.2, 4.4, 4.6 3.2, 4.2, 4.4-4.5  Relationship between the increasing and AB BC decreasing behavior of f and the sign of f' 4.2, 4.4, 4.6 4.2, 4.4-4.5 viii Preface © W. H. Freeman & Company  The Mean Value Theorem and its geometric AB BC interpretation 3.9, 4.2-4.4, 4.6 4.2-4.5, 7.1, 7.3 AB BC  Equations involving derivatives 4.4, 4.6 4.4-4.5 AB BC Second derivatives 4.4, 4.6 4.4-4.5  Corresponding characteristics of the graphs of f, AB BC f',and f" 4.4, 4.6 4.4-4.5  Relationship between the concavity of f and the AB BC sign of f" 4.4, 4.6 4.4-4.5  Points of inflection as places where concavity AB BC changes 4.4, 4.6 4.4-4.5 AB BC Applications of derivatives AB BC  Analysis of curves 4.3-4.4, 4.6 4.3-4.5  Analysis of planar curves given in parametric 4.6, 11.1, 11.3, 4.5, 12.1, 12.3, BC form, polar form, and vector form 11.5 12.5  Optimization, both absolute (global) and relative AB BC (local) extrema 4.7 4.6  Modeling rates of change, including related AB BC rates 3.11 3.9  Use of implicit differentiation to find the AB BC derivative of an inverse function 3.8, 3.10 3.8, 7.2  Interpretation of the derivative as a rate of AB BC change in varied applied contexts 3.4 3.4  Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for AB BC differential equations 9.3 10.2  Numerical solution of differential equations BC using Euler's method 9.2 7.6 BC  L'Hospital's Rule 4.5, 7.6, 10.3 7.7, 8.6, 11.3 AB BC Computation of derivatives 3.3, 3.5 3.3, 3.5 AB BC  Knowledge of derivatives of basic functions 3.2-3.3, 3.5-3.6 3.2-3.3, 3.5-3.6  Derivative rules for sums, products, and AB BC quotients of functions 3.2-3.3, 3.6 3.2-3.3, 3.6 AB BC  Chain rule and implicit differentiation 3.7, 3.10 3.7-3.8  Derivatives of parametric, polar, and vector BC functions 11.1, 11.3, 11.7 12.1, 12.3, 12.7 AB BC III. Integrals Interpretations and properties of definite AB BC integrals 5.1-5.2 5.1-5.2 AB BC  Definite integral as a limit of Riemann sums 5.2 5.2  Definite integral of the rate of change of a quantity over an interval interpreted as the AB BC change of the quantity over the interval 5.5 5.5 AB BC  Basic properties of definite integrals 5.2 5.2 6.1-6.3, 8.1, 6.1-6.3, 9.1, AB BC Applications of integrals 11.2, 11.4 12.2, 12.4 AB BC Fundamental Theorem of Calculus 5.7 7.1, 7.3, 7.8 © W. H. Freeman & Company Rogawski’s Calculus for AP* , Early Transcendentals ix  Use of the Fundamental Theorem to evaluate AB BC definite integrals 5.3-5.4 5.3-5.4  Use of the Fundamental Theorem to represent a AB BC particular antiderivative 5.3-5.4 5.3-5.4 AB BC Techniques of antidifferentiation 3.5, 5.7 3.5, 7.1, 7.3, 7.8  Antiderivatives following directly from AB BC derivatives of basic functions 4.9, 5.3-5.4, 5.6 4.8, 5.3-5.4, 5.6 AB BC  Antiderivatives by substitution of variables 5.6 5.6  Antiderivatives by substitution of variables, BC parts, and simple partial fractions 5.6, 7.1, 7.5 5.6, 8.1, 8.5 BC  Improper integrals as limits of definite integrals 7.6 8.6 AB BC Applications of antidifferentiation 4.8 4.7  Finding specific antiderivatives using initial AB BC conditions 4.9, 9.1, 9.3 4.8, 10.1-10.2  Solving separable differential equations and AB BC using them in modeling 5.8, 9.1, 9.3 7.4, 10.1-10.2  Solving logistic differential equations and using BC them in modeling 9.4 10.3 AB BC Numerical approximations to definite integrals 7.8 8.8  Use of Riemann sums and trapezoidal sums to AB BC approximate definite integrals of functions 8.1 9.1 IV. Polynomial approximations to BC definite integrals BC Concept of series 10.1-10.2 11.1-11.2 BC Series of constants 10.2-10.5 11.2-11.5  Motivating examples, including decimal BC expansion 10.2-10.3 11.2-11.3 BC  Geometric series with applications 10.2-10.3 11.2-11.3 BC  The harmonic series 10.2-10.3 11.2-11.3 BC  Alternating series with error bound 10.4 11.4  Terms of series as areas of rectangles and their BC relationship to improper integrals 10.3 11.3 BC  The ratio test for convergence and divergence 10.5 11.5  Comparing series to test for convergence or BC divergence 10.3, 10.5 11.3, 11.5 BC Taylor series 10.6-10.7 11.6-11.7  Taylor polynomial approximation with BC graphical demonstration of convergence 8.4, 10.6-10.7 9.4, 11.6-11.7  Maclaurin series and the general Taylor series BC centered at x = a 8.4, 10.6-10.7 9.4, 11.6-11.7  Maclaurin series for the functions ex, sin x, cos BC x and 1/1-x 8.4, 10.6-10.7 9.4, 11.6-11.7  Formal manipulation of Taylor series and BC shortcuts to computing Taylor series 10.6-10.7 11.6-11.7 BC  Functions defined by power series 10.6-10.7 11.6-11.7 x Preface © W. H. Freeman & Company