Mathematics – AP/IB Calculus AB Mathematics Curriculum Map AP/IB Calculus AB 2016 The College Board Chandler Unified School District #80 Page 1 of 17 Revised: September 2017 Mathematics – AP/IB Calculus AB AP/IB Calculus AB – At a Glance NOTE: Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among mathematical topics. AP/IB Calculus AB Curriculum Map Semester 1 Flex Semester 2 Unit 1 Unit 2 Unit 3 Flex Unit 4 Unit 5 Unit 5 Limits and Continuity Derivative Applications of the Integration Application of Differential Derivative Integrals Equations 1.1A Limit Notation 2.1A Definition of the 2.3A Interpreting the 3.1A Antiderivatives 3.4A Applications of 2.3D Solving Problems Derivative Meaning of the the Definite Integral Involving Rates of 1.1B Estimating Limits Derivative 3.2A Riemann Sums Change 2.1B Estimating 3.4B Average Value of 2.3B Linear 3.2B Approximating 2.3E Verifying Solutions 1.1C Evaluating Limits Derivatives a Function Approximation Definite Integrals to Differential Equations 1.1D Function Behavior 2.1C Calculating 2.3C Related Rates, 3.2C Properties of 3.4C Integrals and 2.3F Estimating and Limits Derivatives Optimization, Definite Integrals Rectilinear Motion Solutions to Differential Rectilinear motion Equations 1.2A Continuity 2.1D Higher Order 3.3A The Fundamental 3.4D Area and Solids Derivatives 2.4A The Mean Value Theorem of Calculus 3.5A General and 1.2B Applications of of Integration Theorem Specific Solutions of Continuity 2.2A Functions and 3.3B Calculating Differential Equations Derivatives Indefinite and Definite Integrals 3.5B Differential 2.2 B Differentiability Equations from Context Mathematical Practices for AP Calculus (MPACs) All units will include the Mathematical Practices MPAC 1: Reasoning with definitions and theorems MPAC 4: Connecting multiple representations MPAC 2: Connecting concepts MPAC 5: Building notational fluency MPAC 3: Implementing algebraic/computational processes MPAC 6: Communicating 2016 The College Board Chandler Unified School District #80 Page 2 of 17 Revised: September 2017 Mathematics – AP/IB Calculus AB AP/IB Calculus AB Overview Big Idea 1: Limits Big Idea 3: Integrals and the Fundamental Theorem of The concept of a limit can be used to understand the behavior Calculus of functions. Antidifferentiation is the inverse process of differentiation. Continuity is a key property of functions that is defined using The definite integral of a function over an interval is the limit of limits. a Riemann sum over that interval and can be calculated using a variety of strategies. Big Idea 2: Derivatives The Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and integration. The derivative of a function is defined as the limit of difference quotient and can be determined using a variety of strategies. The definite integral of a function over an interval is a mathematical tool with many interpretations and application A function’s derivative, which is itself a function, can be used involving accumulation. to understand the behavior of a function. Antidifferentiation is an underlying concept involved in solving The derivative has multiple interpretations and application separable differential equations. Solving separable differential including those that involve instantaneous rates of change. equations involves determining a function or relation given its The Mean Value Theorem connects the behavior of a rate of change. differentiable function over an interval to the behavior of the derivatives of that function at a particular point in the interval. Mathematical Practices for AP Calculus (MPACs) MPAC 1: Reasoning with definitions and theorems MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes MPAC 4: Connecting multiple representations MPAC 5: Building notation fluency MPAC 6: Communicating Source: AP Calculus AB/BC Course and Exam Description 2016 The College Board Chandler Unified School District #80 Page 3 of 17 Revised: September 2017 Mathematics – AP/IB Calculus AB Semester 1 Unit 1 – Limits and Continuity Essential Question(s): How can the concept of limits be used to understand the behavior of functions? How can you define continuity as a key property of functions using limits? Topic College Board Standard Mathematical Resources Practices Express limits 1.1A1: Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be MPAC 1 Finney, Demanna, Waits, symbolically made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit Kennedy, and Bressoud using correct exists and is a real number then the common notation is lim 푓(푥) = 푅. MPAC 2 2.1, 2.2 notation 푥→푐 1.1A2: The concept of a limit can be extended to include one-sided limits, limits at infinity, and Larson Edwards 2.1, 2.2, infinite limits. 2.4, 2.5 1.1A3: A limit might not exist for some function at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this Smith & Minton 1.1, 1.2 value, or if the limit from the left does not equal the limit from the right. Estimating 1.1B1: Numerical and graphical information can be used to estimate limits. MPAC 3 Finney et al. 2.1, 2.2 Limits of MPAC 4 Functions Larson 2.2 Smith & Minton 1.1, 1.2 Determining 1.1C1: Limits of sums, differences, products, quotients, and composite functions can be found MPAC 1 Finney et al. 2.1, 2.2, limits of using the basic theorems of limits and algebraic rules. Functions MPAC 2 Larson 2.3 1.1C2: The limit of a function may be found by using algebraic manipulation, alternate forms of trigonometric functions or the squeeze theorem Smith & Minton 1.1, 1.2 MPAC 3 MPAC 4 Function 1.1D1: Asymptotic and unbounded behavior of functions can be explained and described MPAC 2 Finney et al. 2.1, 2.2 Behavior and using limits. Limits Larson 2.5, 3.2, 3.4, 4.5 1.1D2: Relative magnitudes of functions and their rates of change can be compared using limits. Smith & Minton 1.4 2016 The College Board Chandler Unified School District #80 Page 4 of 17 Revised: September 2017 Mathematics – AP/IB Calculus AB Continuity 1.2A1: A function f is continuous at x = c provided that f(x) exists, lim f (x) exists, and MPAC 1 Finney et al. 2.3 xc MPAC 2 lim f (x) f (c) . Larson 2.1, 2.2, 2.3, 2.4, xc MPAC 3 2.5 MPAC 4 1.2A2: Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are Smith & Minton 1.3 continuous at all points in their domains. 1.2A3: Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes. Applications of 1.2B1: Continuity is an essential condition for theorems such as the Intermediate Value MPAC 1 Finney et al. 2.3 Theorem, the Extreme Value Theorem, and the Mean Value Theorem. Continuity MPAC 2 Larson 2.4, 4.1 MPAC 3 MPAC 4 Smith & Minton 1.3 2016 The College Board Chandler Unified School District #80 Page 5 of 17 Revised: September 2017 Mathematics – AP/IB Calculus AB Semester 1 Unit 2 - Derivatives How do we use derivatives to describe the rate of change of one variable with respect to another variable, and how do we use multiple representations to understand change in a variety of contexts? Topic College Board Standard Mathematical Resources Practices Definition of the 2.1A2: The instantaneous rate of change of a function at a point can be expressed by MPAC 1 Finney et al. 3.1, 3.2 푓(푎+ℎ)−푓(푎) 푓(푥)−푓(푎) MPAC 2 Derivative lim or lim , provided the limit exists. These are common forms of the 푥→ℎ ℎ 푥→푥 푥−푎 MPAC 3 Larson 3.1 definition of the derivative and are denoted f’(a). MPAC 4 MPAC 5 푓(푎+ℎ)−푓(푎) MPAC 6 2.1A3: The derivative of f is the function whose value at x is lim provided this limit Smith & Minton 2.1, 2.2 푥→0 ℎ exists. 2.1A4: For y = f(x), notations for the derivative include 푑푦, f’(x), and y’. 푑푥 2.1A5: The derivative can be represented graphically, numerically, analytically, and verbally. Estimating 2.1B1: The derivative at a point can be estimated from information given in tables or graphs. MPAC 1 Finney et al. 3.1 Derivatives MPAC 2 MPAC 3 Larson 3.1, 3.2, .3.3 MPAC 4 Smith & Minton 2.1, 2.2 MPAC 5 MPAC 6 Continuity and 2.2B1: A continuous function may fail to be differentiable at a point in its domain. MPAC 1 Finney et al. 3.2 Differentiability MPAC 2 2.2B2: If a function is differentiable at a point, then it is continuous at that point. MPAC 3 MPAC 4 Larson 3.1 MPAC 5 MPAC 6 Smith & Minton 2.2 Derivative 2.1C1: Direction application of the derivative of the definition of the derivative can be used to MPAC 1 Finney et al. 3.3, 3.5, 4.3, 4.4 Rules find the derivative for selected functions, including polynomial, power, sine, cosine, MPAC 2 Larson 3.2, 3.3, 3.6 exponential, and logarithmic functions. MPAC 3 MPAC 4 Smith & Minton 2.3. 2.4, 2.5, 2.1C2: Specific Rules can be used to calculate derivatives for classes of functions, including MPAC 5 2.6 polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse MPAC 6 trigonometric.