We Cover Everything in the Calculus AB Topic Outline As It Appears in the AP

AP ® Calculus (AB)

Course Overview

We cover everything in the Calculus AB topic outline as it appears in the AP® Calculus Course Descriptions. The primary textbook is Calculus, Larson, Hostetler, et al., 8th ed. We study four major ideas: limits, derivatives, indefinite integrals, and definite integrals. As we develop the concepts, we explain how the mechanics go along with the topics, and stress application. Since concepts are so vital to AP® Calculus, there is an attempt to balance understanding, skills, and use of technology. The course provides students with the opportunity to work with functions represented graphically, numerically, analytically, and verbally—and emphasizes the connections among these representations. Chapter 1 - Limits

“Local Linearity” Day 1: Intro JT Sutcliffe Activity (available on AP Central) Day 2: JT Sutcliffe Activity (continued), Approximate slope for various x-values, data plot on calculator.

§ 1.2 "Finding Limits Graphically and Numerically" Objectives: Estimate a limit using a numerical or graphical approach. Learn different ways that a limit can fail to exist. Study and use a formal definition of limit. Day 1: p. 54 # 1-18, 63-66

§ 1.3 "Evaluating Limits Analytically" Objectives: Evaluate a limit using properties of limits. Develop and use a strategy for finding limits. Evaluate a limit using dividing out and rationalizing techniques. Evaluate a limit using the Squeeze Theorem. Day 1: p. 67 # 5-39 all, 41-44, 49-61 odd, 67- 77 odd Day 2: Review 1.2 -1.3

Test 1.3 (§1.2 - 1.3) § 1.4 "Continuity and One-Sided Limits" Objectives: Determine continuity at a point and continuity on an open interval. Determine one-sided limits and continuity on a closed interval. Use properties of continuity. Understand and use the Intermediate Value Theorem. Day 1: p. 78 # 1 – 28, 33-54, 57-60, 65-74 Day 2: p. 78 finish § 1.5 "Infinite Limits" Objectives: Determine infinite limits from the left and from the right. Find and sketch the vertical asymptotes if the graph of a function. Day 1: p. 88 # 1-52, 58 (9 – 28, find H.A., V.A., if any) Day 2: Review 1.5 Test 1.5 (§1.4 - 1.5) Chapter 2 - Differentiation

§ 2.1 "The Derivative and the Tangent Line Problem" Objectives: (1) Find the slope of a line to a curve at a point; (2) Use limit definition to find the derivative of a function; (3) Understand the relationship between differentiability and continuity Day 1: p. 103 # 2, 14 18, 19, 22, 24, 26, 33, 57

§ 2.2 "Basic Differentiation Rules and Rates of Change" Objectives: (1) Find derivatives using the Constant Rule, Power Rule, Constant Multiple Rule, and the Sum and Differences Rules; (2) Find the derivatives of the sine function and cosine function; (3) Use derivatives to find rates of change (velocity) Day 1: p. 115 # 1 -61 odd, 74

§ 2.3 "The Product and Quotient Rules and Higher-Order Derivatives" Objectives: (1) Find the derivative of a function using the Product Rule and Quotient Rule; (2) Find the derivative of a trigonometric function; (3) Find a higher-order derivative of a function (i.e., the second derivative, the third derivative, etc.) Day 1: p. 126 # 13, 15, 17 25-53 (every 4), 59-68 odd, 129-138

§ 2.4 "The Chain Rule" Objectives: (1) Find the derivative of a composite function using the Chain rule; (2) Find the derivative of a function using the General Power Rule; (3) Simplify the derivative of a function using algebra; (4) Find the derivative of a trigonometric function using the Chain Rule. Day 1: p. 143 # 1 - 32 odd, 39 - 58 odd., 59 – 65 odd Day 2: Review §2.4 and p. 117 #83-87, 89, 93, 95

Test 2.4 (§2.1 – 2.4)

§ 2.5 "Implicit Differentiation" Objectives: (1) Distinguish between functions written in implicit form and explicit form; (2) Use implicit differentiation to find the derivative of a function Day 1: p. 146 # 1 - 31 odd 45-49 odd

§ 2.6 "Related Rates" Objectives: (1) Find a related rate; (2) Use related rates to solve real-life problems Day 1: p. 154 # 1 – 8 all, 13 – 19 odd Day 2: p. 158 # 14-22 evens Day 3: Review §2.6

Test 2.6 (§2.5 – 2.6) Chapter 3 - Applications of Differentiation

§ 3.2 Rolle's Theorem and the Mean Value Theorem Objectives: (1) Definition of Rolle's Theorem and the Mean Value Theorem Day 1: p. 176 # 1, 2, 11 - 23 odd,35-36, 39-45 odd

§ 3.1 Extrema on an Interval Objectives: (1) Definition of relative and absolute extrema; (2) Definition of critical numbers; (3) Finding extrema on a closed interval Day 1: p. 169 # 3-7 odd, 13-17odd 19-39 odd Day 2: p. 169 # 2 – 36 even

§ 3.3 Increasing and Decreasing Functions and the First Derivative Test Objectives: (1) The first derivative test; (2) Sign charts Day 1: p. 186 # 3-7 odd, 17-45 odd Day 2: Review §3.3 Test 3.3: (§ 3.1 - 3.3)

§ 3.4 Concavity and the Second Derivative Test Objectives: (1) Connection between concavity and the second derivative; (2) Points of Inflection; (3) Second derivative test Day 1: p. 195 # 1 - 6 Day 2: Review, plus Refresh §3.5 + Intro Function Summaries

§ 3.5 Limits at Infinity Objectives: (1) Finding limits at infinity; (2) Horizontal asymptotes

§ 3.6 A Summary of Curve Sketching: Function Summaries Objectives: (1) Geometric connection between the tangent line and concavity; (2) Graphing a function based on its derivative Day 1: p. 215 # 57-64 all, 71-74 all Day 2: f( x )= x3 ( x 2 - 4)

TEST: (§3.4, 3.5, 3.6)

§ 3.7 Optimization Problems Objectives: (1) Applied minimum and maximum problems Day 1: p. 223 # 1,3, 7, 13 § 3.8 Newton’s Method Objectives: (1) Algebraic solutions (approximations) of polynomial equations; (2) Iterative processes Day 1: 233 # 1 - 17 odd Day 2: p. 233 # 2 –18 even Chapter 4 - Integration

§ 4.1 Antidifferentiation and Indefinite Integration Objectives: (1) Write the general solution of a differential equation (2) Use definite integral notation for antiderivatives; (3) Use basic integration rules to find antiderivatives; (4) Find a particular solution of a differential equation Day 1: p. 255 #9-48, 55-62 Day 2: p. 256 # 67,69,81

Test 4.1

Distance, Velocity, Acceleration Day 1: Worksheet 1 Day 2: Worksheet 2

§ 4.3 Riemann Sums and Definite Integrals Objectives: (1) Understand the definition of a Riemann Sum; (2) Evaluate a definite integral using limits; (3) Evaluate a definite integral using properties of definite integrals. Day 1: p. 278 #13-44 all, 55,56,63, plus 3 – 8 on calculator.

§ 4.4 The Fundamental Theorem of Calculus Objectives: (1) Evaluate a definite integral using the Fundamental Theorem of Calculus; (2) Understand and use the Mean Value Theorem for Integrals; (3) Find the average value of a function over a closed interval; (4) Understand and use the Second Fundamental Theorem of Calculus. Day 1: p. 291, #5 – 41, 79, 83 – 91 odds

§ 4.5 Integration by Substitution Objectives: (1) Use pattern recognition to evaluate an indefinite integral; (2) Use a change of variables to evaluate an indefinite integral; (3) Use the General Power Rule for Integration to evaluate an indefinite integral; (4) Use a change of variables to evaluate a definite integral; (5) Evaluate a definite integral involving an odd or even function. Day 1: p. 304 # 1 - 37 odd,43-55, 99-100. Day 2: Review Distance, Velocity, Acceleration and §4.3, 4.4, 4.5)

Test 4.5 (Distance, Velocity, Acceleration and §4.3, 4.4, 4.5) § 4.6 Numerical Integration Objectives: (1) Approximate a definite integral by the Trapezoidal Rule (Trapezium Rule). Day 1: p. 314, #1-10 on calculator. Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions

§ 5.1 The Natural Logarithmic Function and Differentiation Objectives: (1) Develop and use properties of the natural logarithmic function; (2) Understand the definition of the number e; (3) Use basic integration rules to find antiderivatives; (4) Find derivatives of functions involving the natural logarithmic function Day 1: p. 329 #7-10,11-17, 29-33,41-42,45-69,77-87 odd

§ 5.2 The Natural Logarithmic Function and Integration Objectives: (1) Use the Log rule for Integration to integrate a rational function; (2) Integrate trigonometric functions. Day 1: p. 338 # 1 -23, 29-35, 37-39, 47-57, 67-69, 83-85 odds, 91 Day 2: Review

Test 5.2 (§5.1 – 5.2)

§ 5.3 Inverse Functions Objectives: (1) Verify that one function is the inverse of another function; (2) Determine whether a function has an inverse function; (3) Find the derivative of an inverse function. Day 1: p. 347 # 1, 3, 15, 16, 23, 27 33, 43, 71-75, 101-105,

§ 5.4 Exponential Functions: Differentiation and Integration Objectives: (1) Develop properties of the natural exponential function; (2) Differentiate natural exponential functions; (3) Integrate natural exponential functions. Day 1: p. 356 # 1 – 28, 31, 35-47, 57, 61, 63, 65-71 odd, 85-97, 107, 109113-116 all Day 2: Review 5.4

Test 5.4: § 5.3-5.4

§ 5.5 Bases Other than e and Applications Objectives: (1) Define exponential functions that have bases other than e; (2) Differentiate and integrate exponential functions that have bases other than e; (3) Use exponential functions to model compound interest and exponential growth. Day 1: p. 366 # 1 – 19, 37-47, 53-55 odd, 61-69 odd § 5.6 Differential Equations: Growth and Decay Objectives: (1) Use separation of variables to solve a simple differential equation; (2) Use exponential functions to model growth and decay in applied problems. Day 1: p. 377 # 5-16, 21-27, 35, 36 Day 2: Review

§ 5.7 Differential Equations: Separation of Variables Objectives: (1) Use initial conditions to find particular solutions to differential equations; (2) Recognize and solve differential equations that can be solved by separation of variables; (3) Recognize and solve homogeneous differential equations; (4) Use a differential equation to model and solve an applied problem. Day 1: p. 385 # 1 – 19, 21-29, 31-41

Test 5.7 (§5.5, 5.6, 5.7) Chapter 6- Differential Equations

§ 6.1 Slope Fields Objectives: (1) Use initial conditions to find particular solutions of differential equations. (2) Use slope fields to approximate solutions of differential equations. Day 1: Slope Fields Packet #1 Day 2: Packet #2 Day 3: Packet #3

§ 6.2 Differential Equations: Growth and Decay Objectives: (1) Use separation of variables to solve a simple differential equation. (2) Use exponential functions to model growth and decay in applied problems. Day 1: p. 418 #1-15 odds, 41, 62

§ 6.3 Separation of Variables and the Logistic Equation Objectives: (1) Recognize and solve differential equations that can be solve by separation of variables. (2) Recognize and solve homogenous differential equations. (3) Use differential equations to model and solve applied problems. (4) Solve and analyze logistic differential equations. Day 1: p. 429 #1-15 odds Chapter 7 - Applications of Integration

§ 7.1 Area of a Region Between Two Curves Objectives: (1) Find the area of a region between two curves using integration; (2) Find the area of a region between intersecting curves using integration; (3) Describe integration as an accumulation process. Day 1: p. 452 # 1 –11, 15, 17-31, 43-47 odd Day 2: Review §6.1 § 7.2 Volume: The Disc Method Objectives: (1) Find the volume of a solid of revolution using the disc method; (2) Find the volume of a solid of revolution using the washer method; (3) Find the volume of a solid with known cross sections. Day 1: p. 463 # 1 - 30 Day 2: p. 463, 2 – 14 even

§ 7.3 Volume: The Shell Method Objectives: (1) Find the volume of a solid of revolution using the shell method; (2) Compare the uses of the disk method and the shell method. Day 1: p. 472 # 1 - 23 all

Test 7.3 (§7.1, 7.2, 7.3)

More on Differential Equations: Objectives: Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations. Days 1-3: Packet on slope fields

AP Review