Advanced Placement Calculus AB. South Texas ISD. Scope and Sequence with Learning Objectives

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1 Advanced Placement Calculus AB South Texas ISD Scope and Sequence with Learning Objectives

2 Advanced Placement Calculus AB Scope and Sequence - Year at a Glance AP Calculus AB - First Semester Three Weeks 1 st 3 weeks 2 nd 3 weeks 3 rd 3 weeks 4 th 3 weeks 5 th 3 weeks 6 th 3 weeks Topics/ Concepts Analysis of graphs Functions Trigonometric equations and graphs Limits of functions Asymptotic behavior Continuity Concept of the derivative Derivative at a point Computation of derivatives Modeling Rates of Change Derivative as a Function Second Derivatives Optimization Rate of change in varied contexts Thematic applications Techniques of antidifferentiation Area Riemann sums Curve Sketching Resource: Calculus of a Single Variable; Larson/Hostetler Chapter P Chapter 1 Chapter 2 Chapter 3 Chapter 3 Chapter 4 Advanced Placement Calculus AB Scope and Sequence - Year at a Glance AP Calculus AB - Second Semester Three Weeks 7 th 3 weeks 8 th 3 weeks 9 th 3 weeks 10 th 3 weeks 11 th 3 weeks 12 th 3 weeks Topics/ Concepts Properties of definite integrals Fundamental theorem of calculus Mean Value Theorems Techniques of antidifferentiation Numerical Integration Inverse Functions Derivatives of Transcendental Functions Integrals of Transcendental Functions Relative Rates of Change Applications of antidifferentiation Differential Equations Applications of integrals Area and Volume Review for AP Examination Further applications of integration, derivatives and limits Resource: Calculus of a Single Variable; Larson/Hostetler Chapter 4 Chapter 4 Chapter 5 Chapter 6 Chapter 7 All Chapters

3 Prerequisites: A Review of Precalculus (Approximate Time: 3 weeks) 1 st 3 weeks Solve and graph linear and quadratic inequalities Define and apply the concept of absolute value Relate absolute value to distance Apply the distance formula Write equations of circles Find x and y intercepts of functions Find points of intersection for various functions and evaluate the feasibility of these intersections Test the symmetry of a graph with respect to an axis and the origin Write and graph linear equations using the general form Interpret slope as a rate in a real-life application Write equations of lines parallel and perpendicular to given lines Use function notation to represent and evaluate a function Find the domain and range of a function Sketch the graph of a function Analysis of graphs Functions Trigonometric equations and graphs IA IA IA IA PC PC PC PC PC PC PC PC The graph of an equation. Intercepts. Symmetry. Points of Intersection. Function notation Domain and range Transformations Composite functions Inverse functions Evaluating trigonometric functions Solving trigonometric equations Graphs of trigonometric equations Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

4 Prerequisites: A Review of Precalculus (Approximate Time: 3 weeks) 1 st 3 weeks Identify different types of transformations of functions Find expressions for composite functions Test for even and odd functions Evaluate trigonometric functions using special triangles and the unit circle Graph the basic trigonometric functions Solve trigonometric equations Fit linear, quadratic and trigonometric models to reallife data sets

5 Limits and Their Properties (Approximate Time: 3 weeks) 2 nd 3 weeks Estimate limits using graphical and numerical approaches Learn different ways a limit can fail to exist Study and use the ε δ definition of a limit Evaluate limits using properties of limits Develop and use a strategy for finding limits Evaluate a limit using division and rationalization techniques Evaluate a limit using the Squeeze Theorem Determine continuity at a point and on an open interval Determine one-sided limits and continuity on a closed interval Use properties of continuity Understand and apply the Intermediate Value Theorem Limits of functions Asymptotic behavior Continuity IB1 IB2 IB3 IB3 IC1 IC2 ID1 ID2 ID3 Intuitive understanding of limiting process Calculating limits using algebra Estimating limits using graphs Estimating limits using tables of data Understanding asymptotes in terms of graphical behavior Asymptotic behavior in terms of limits involving infinity Intuitive understanding of continuity Continuity in terms of limits Geometric understanding of graphs of continuous functions Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

6 Limits and Their Properties (Approximate Time: 3 weeks) 2 nd 3 weeks Determine infinite limits from the left and right Find and sketch the vertical asymptotes of the graph of a function Use limits to justify vertical asymptotes Find limits as x approaches infinity Find and sketch the horizontal asymptotes of the graph of a function

7 Derivatives (Approximate Time: 3 weeks) 3 rd 3 weeks Find the slope of the tangent line to a curve at a point Use the limit definition to find the derivative of a function Understand the relationship between differentiability and continuity Find the derivative of a function using the basic differentiation rules Find the derivative of sine and cosine functions Use derivatives to find rates of change Find the derivative of a function using the Product Rule Find the derivative of a function using the Quotient Rule Find the derivative of the six basic trigonometric functions Find higher order derivatives Find the derivative of a composite function using the Chain Rule Concept of the derivative Derivative at a point Computation of derivatives IIA1 IIA2 IIA3 IIA4 IIB1 IIB2 IIB2 IIB4 IIB3 IIF1 IIF2 IIF3 IIF3 Derivative presented graphically, numerically and analytically Derivative interpreted as instantaneous rate of change Derivative defined as the limit of the difference quotient Relationship between continuity and differentiability Slope of a curve at a point Tangent line to a curve at a point Local linear approximation Approximate rate of change from graphs and tables of values Instantaneous rate of change as the limit of average rate of change Derivatives of basic functions Basic rules for the derivative of sums, products and quotients of functions Chain rule Implicit differentiation Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

8 Derivatives (Approximate Time: 3 weeks) 3 rd 3 weeks Find the derivative of a function using the General Power Rule Simplify the derivative of a function using algebra Distinguish between functions written in implicit and explicit forms Use implicit differentiation to find the derivative of a function Find a related rate Use related rates to solve reallife problems

9 Applications of Differentiation (part 1) (Approximate Time: 3 weeks) 4 th 3 weeks Understand the definition of extrema of a function on an interval Understand the definition of relative extrema of a function on an open interval Find extrema on a closed interval Understand and apply the Mean Value Theorem including Rolle s Theorem Determine intervals on which a function is increasing or decreasing Apply the First Derivative Test to find relative extrema Determine intervals on which a function is concave upward or concave downward Find and justify points of inflection Modeling Rates of Change Derivative as a Function Second Derivatives IIE5 IIE5 IIE3 IIE3 IIC1 IIC2 IIC3 IIC4 IID1 IID2 Using derivatives to find velocity, speed and acceleration Average rates of change Finding related rates Problem solving with related rates Corresponding characteristics of f and f. Relationship between the increasing and decreasing behavior of f and the sign of f. The Mean Value Theorem and its geometric consequences. Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations IID3 Corresponding characteristics of the graphs of f, f, and f. Relationship between the concavity of f and the sign of f. Points of inflection as places where concavity changes.

10 Applications of Differentiation (part 1 continued) (Approximate Time: 3 weeks) 4 th 3 weeks Apply the Second Derivative Test to find relative extrema Determine limits at infinity Determine horizontal asymptotes Analyze and sketch the graph of a function Curve Sketching IIC2 IIC1 IID1 IID2 IID3 IIE1 Relationship between the increasing and decreasing behavior of f and the sign of f. Corresponding characteristics of the graphs of f and f, and f. Relationship between the concavity of f and the sign of f. Points of inflection as places where concavity changes Analysis of curves, including the notions of monotonocity and concavity. Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

11 Applications of Differentiation (part 2) (Approximate Time: 3 weeks) 5 th 3 weeks Solve applied minimum and maximum problems Approximate a zero of a function using Newton s Method Understand the concept of a tangent line approximation Compare the value of the differential with the actual change Estimate a propagated error using a differential Find the differential of a function Optimization Rate of change in varied contexts Thematic applications IIE2 IIE2 IIE2 IIE5 IIE5 IIE6 IIE2 IIE2 IIE2 Absolute (global) extrema Relative (local) extrema Applied maximum and minimum problems Newton s method Differentials Slope fields with polynomials and circles Business and Economic applications Engineering applications Medical applications Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

12 Introduction to Integration (Approximate Time: 3 weeks) 6 th 3 weeks Write the general solution of a differential equation Use indefinite integral notation for antiderivatives Use basic integration rules to find antiderivatives Find a particular solution of a differential equation Use sigma notation to write and evaluate a sum Understand the concept of an area Approximate the area of a plane region Find the area of a plane region using limits Understand the definition of a Riemann Sum Techniques of antidifferentiation Area Riemann sums IIID1 IIIB IIIB IIIA1 IIIA2 Antiderivatives following directly from derivatives of basic functions Finding accumulated area using rectangles Computation of Riemann sums using left, right and midpoint evaluation points. Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

13 Integration (Approximate Time: 3 weeks) 7 th 3 weeks Evaluate a definite integral using limits Evaluate a definite integral using the properties of definite integrals Evaluate a definite integral using the Fundamental Theorem of Calculus Understand and use the Mean Value Theorem for Integrals Find the average value of a function over a closed interval Understand and use the Second Fundamental Theorem of Calculus Evaluate definite integrals of even and odd functions Properties of definite integrals Fundamental theorem of calculus IIIA2 IIIA3 IIIA4 IIIC1 IIIC2 Definite integral as a limit of Riemann sums over equal subdivisions. Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: b a f ( x) dx = f ( b) f ( a) Basic properties of definite integrals. (Examples include additivity and linearity.) Use of the Fundamental Theorem to evaluate definite integrals. Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.

14 Integration(continued) (Approximate Time: 3 weeks) 8 th 3 weeks Use a change of variable to evaluate definite and indefinite integrals Approximate a definite integral using Trapezoidal and Simpson s Rules Techniques of antidifferentiation Numerical Integration IIID2 IIIF IIIF IIIF Antiderivatives by substitution of variables (including change of limits for definite integrals). Use of Riemann sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values Use of trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values Other numerical integration techniques Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

15 Exponential and Logarithmic Functions (Approximate Time: 3 weeks) 9 th 3 weeks Develop and use properties of the natural logarithmic function Understand the definition of the number e Find derivatives of functions involving the natural logarithmic function Integrate a rational function using the log rule Integrate trigonometric functions Verify that one function is the inverse function of another Determine whether a function has an inverse Find the derivative of an inverse function Develop properties of the natural exponential function Differentiate natural exponential functions Integrate natural exponential functions Define exponential functions that have other bases than e Differentiate and integrate exponential functions that have other bases than e Develop properties of inverse trigonometric functions Differentiate inverse trigonometric functions Inverse Functions Derivatives Integrals IIE4 IIE4 IIB1 IIB2 IIB3 IIF1 IIF2 IIF3 IIIA3 IIIA4 IIIC1 IIIC2 IIID2 Review inverse relationships Use of implicit differentiation to find the derivative of an inverse function Slope and equation of a tangent line to a curve at a point Instantaneous rate of change Derivatives of basic functions Basic rules for the derivative of sums, products and quotients of functions Chain rule and implicit differentiation Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: b a f ( x) dx = f ( b) f ( a) Basic properties of definite integrals. Use of the Fundamental Theorem to evaluate definite integrals. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Antiderivatives by substitution of variables (including change of limits for definite integrals). Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

16 Exponential and Logarithmic Functions (Approximate Time: 3 weeks) 10 th 3 weeks Integrate functions whose antiderivatives involve inverse trigonometric functions Use completing the square to integrate a function Use exponential functions to model compound interest and exponential growth Relative Rates of Change Applications of antidifferentiation IC3 IIIE1 IIIE2 IIE6 Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth. Finding specific antiderivatives using initial conditions. Solving separable differential equations and using them in modeling. In particular, studying the equation y' = ky and exponential growth. Geometric interpretation of differential equation via slope fields and the relationship between slope fields and solution curves for differential equations. Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

17 Applications of Integration (Approximate Time: 3 weeks) 10 th 3 weeks Use initial conditions to find particular solutions of differential equations Use slope fields to approximate solutions of differential equations Use Euler s Method to approximate solutions of differential equations Use separation of variables to solve simple differential equations Use exponential functions to model growth and decay in applied problems Applications of integrals IIIB IIIB IIIB IIIB CC CC CC CC Finding the area of a region. Finding the volume of a solid with known cross sections. Finding the average value of a function. Finding the distance traveled by a particle along a line. Finding arc length Work including springs, pumping and lifting Fluid forces Moments and centers of mass Larson Calculus of a Single Variable 8th Edition Finney/Weir/Giordano Calculus 10 th Edition Technology Lab Guide (Larson Teacher Resources) Forester Calculus Explorations

18 Course Review and Final Exam (Approximate Time: 3 weeks) 11th 3 weeks Write the general solution of a differential equation Use basic integration rules to find antiderivatives Evaluate a definite integral using limits Use a change of variable to evaluate definite and indefinite integrals Find the derivative of a composite function using the Chain Rule Find the derivative of a function using the General Power Rule Find the derivative of sine and cosine functions Use derivatives to find rates of change Find the derivative of a function using the Product Rule Find the derivative of a function using the Quotient Rule Find the derivative of the six basic trigonometric functions Evaluate limits using properties of limits Develop and use a strategy for finding limits Determine one-sided limits and continuity on a closed interval Find limits as x approaches infinity Integration Derivatives Limits III II I Overview of integration General survey of application problems Overview of differentiation General survey of application problems Limits as related to differentiation and integration Multiple Choice and Free- Response Questions in Preparation for the AP Calculus (AB) Examination (D&S Marketing) Calculus Problems for a New Century (MAA,Volume 2)

19 Review for the AP Exam (Approximate Time: 3 weeks) 11 th 3 weeks Evaluate limits using properties of limits Determine continuity at a point and on an open interval Determine one-sided limits and continuity on a closed interval Use properties of continuity Understand the relationship between differentiability and continuity Use derivatives to find rates of change Find a related rate Use related rates to solve reallife problems Evaluate limits using properties of limits Determine continuity at a point and on an open interval Determine one-sided limits and continuity on a closed interval Use properties of continuity Understand the relationship between differentiability and continuity Use derivatives to find rates of change Find a related rate Use related rates to solve reallife problems Find derivatives of functions involving the natural logarithmic function Use exponential functions to model compound interest and exponential growth Past year AP questions District-wide final exam Selected problems from suggested resources in preparation for districtwide, AP and concurrent exams. Multiple Choice and Free- Response Questions in Preparation for the AP Calculus (AB) Examination (D&S Marketing)

20 Extension Topics (Approximate Time: 3 weeks) 12 th 3 weeks ELO Correlation Objectives/concepts TEKS Topics (not in sequential order) Suggested Resources Find a related rate Use related rates to solve reallife problems Use exponential functions to model compound interest and exponential growth Use initial conditions to find particular solutions of differential equations Use exponential functions to model growth and decay in applied problems Find the volume of a solid Computer applications Further applications of calculus EA EA Animations of various calculus concepts Learning by discovery Problems for student investigations Readings for calculus Learning by discovery

21 Topic Outline for Calculus AB (TEKS) I. Functions, Graphs, and Limits A. Analysis of graphs. With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function. B. Limits of functions (including one-sided limits). 1. An intuitive understanding of the limiting process. 2. Calculating limits using algebra. 3. Estimating limits from graphs or tables of data. C. Asymptotic and unbounded behavior. 1. Understanding asymptotes in terms of graphical behavior. 2. Describing asymptotic behavior in terms of limits involving infinity. 3. Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.) D. Continuity as a property of functions. 1. An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.) 2. Understanding continuity in terms of limits. 3. Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem). II. Derivatives

22 A. Concept of the derivative. 1. Derivative presented grnphical1y, numerically, and analytically. 2. Derivative interpreted as an instantaneous rate of change. 3. Derivative defined as the limit of the difference quotient. 4. Relationship between differentiability and continuity. B. Derivative at a point. 1. Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. 2. Tangent line to a curve at a point and local linear approximation. 3. Instantaneous rate of change as the limit of average rate of change. 4. Approximate rate of change from graphs and tables of values. C. Derivative as a function. 1. Corresponding characteristics of graphs of f and f, and f. 2. Relationship between the increasing and decreasing behavior of f and the sign of f. 3. The Mean Value Theorem and its geometric consequences. 4. Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. D. Second derivatives. 1. Corresponding characteristics of the graphs of f and f, and f.

23 2. Relationship between the concavity of f and the sign of f. 3. Points of inflection as places where concavity changes. E. Applications of derivatives. 1. Analysis of curves, including the notions of monotonicity and concavity. 2. Optimization, both absolute (global) and relative (local) extrema. 3. Modeling rates of change, including related rates problems. 4. Use of implicit differentiation to find the derivative of an inverse function. 5. Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration. F. Computation of derivatives. 1. Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions. 2. Basic rules for the derivative of sums, products, and quotients of functions. 3. Chain rule and implicit differentiation.

24 III. Integrals A. Interpretations and properties of definite integrals. 1. Computation of Riemann sums using left, right, and midpoint evaluation points. 2. Definite integral as a limit of Riemann sums over equal subdivisions. 3. Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: b a f ( x) dx = f ( b) f ( a) 4. Basic properties of definite integrals. (Examples include additivity and linearity.) B. Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line. C. Fundamental Theorem of Calculus. 1. Use of the Fundamental Theorem to evaluate definite integrals. 2. Use of the Fundamental Theorem to represent a particular Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. D. Techniques of antidifferentiation. 1. Antiderivatives following directly from derivatives of basic functions.

25 2. Antiderivatives by substitution of variables (including change of limits for definite integrals). E. Applications of antidifferentiation. 1. Finding specific antiderivatives using initial conditions, including applications to motion along a line. 2. Solving separable differential equations and using them in modeling. In particular, studying tile equation y' = ky and exponential growth. F. Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values. PC: Precalculus Review CC: College Course EA: Extra Assignments