AP Calculus AB Syllabus Course Description: This is a college level course covering derivatives, integrals, limits, applications and modeling of these topics. Course Objectives: The objectives of this course are: To work with functions represented in a variety of ways: graphically, analytically, numerically. They should understand the relationship of each way. To understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve several problems To understand the meaning of a definite integral as both a limit of a Riemann sums and the net accumulation of a rate of change and use integrals to solve several problems To know the relationship between a derivative and a definite integral dealing with the Fundamental Theorem of Calculus. To be able to use a graphing calculator to plot a graph of a function within an arbitrary viewing window, find zeros of a function, numerical calculate the derivative of a function, and numerically calculate the value of a definite integral. Course Expectations: To inquire about material and concepts that may not be clear To strive to apply knowledge from previous courses in a new way To conceptualize the calculus curriculum and apply knowledge To communicate mathematics both orally and in well written sentences. Teaching Strategies Some examples are imbedded in the curriculum using parenthesis. Students must write out solutions to application problems with logically explanations to this. They must interpret graphs and tables. Use mirroring techniques that are given in the AP released calculus exams to show how to communicate mathematically and what an acceptable answer looks like. Test and quizzes are given over skills and applications. Students are assigned problems in their text, pre-made handouts, calculator labs, board problems, and released items. Overheard projector attached to a computer is used to show graphical representation, especially in area of revolution to create volume and solids of a known cross section. Course Outline: I. Functions and Graphs: A Precalculus Review (approximately 3 weeks) a. Functions & function notation i. Definition of function ii. Vertical line test iii. Function notation
iv. Finding input and output values v. Domain and range b. Absolute Value & Piecewise Defined Functions i. Piecewise defined functions ii. The absolute value function iii. Solving equations involving absolute value c. Inequalities i. Solving inequalities of all types ii. Writing situations and applying inequalities d. Composition & Combination of Functions i. Composition of functions ii. Inverse functions iii. Non-invertible functions iv. Combinations of functions e. Transformation of Functions i. Vertical and horizontal shifts ii. Reflections and symmetry iii. Vertical stretches and compressions iv. Horizontal stretches and compressions v. Combining transformations f. Trigonometric Functions i. The graph of cosx, sinx, tranx, cotx, secx, cscx ii. Shifts and distortions of trigonometric functions iii. Trigonometric identities iv. Inverse trigonometric functions v. Solving trigonometric equations g. Culminating Presentation from Precalculus review II. Limits and Continuity (approximately 3 weeks) a. Intuitive definition of a limit i. Definition ii. Using table and graphs to find limits (students use a table in their calculator to approach a limit numerically) b. Algebraic techniques for finding limits i. Calculating limits using limit laws ii. Direct substitution properly iii. Indeterminate forms
c. One-Sided limits i. Definition ii. Finding one-sided limits iii. Existence of limits d. Infinite limits i. Definition ii. Vertical asymptotes e. Limits at Infinity i. End-behavior of functions and horizontal asymptotes ii. Limit laws for limits at infinity iii. Oblique asymptotes f. Limits of special trigonometric functions i. Special limits involving the sine function ( evaluating sine x as is it approaches zero graphically and comparing it to y=x at that time students can see the limit) ii. Special limits involving the cosine function g. Continuity i. Continuity at a point ii. Continuity on a closed interval iii. Continuity on an open interval iv. Discontinuous functions a. removable discontinuity b. Jump discontinuity c. Infinite discontinuity v. Intermediate Value Theorem III. Derivatives (approximately 5 weeks) a. Definition of the Derivative i. The derivative as the slope of a tangent (limit of difference quotient) ii. The derivative as the rate of change iii. The derivative as a function b. Differentiation Rules i. Constant rule, constant multiple rule ii. Power rule iii. Sum rule, difference rule iv. Product rule v. Quotient rule vi. Trigonometric rules vii. Chain rule viii. Higher order derivatives
c. Differentiability & Continuity i. Differentiability implies continuity ii. Non-differentiable functions iii. Local linearity iv. Numerical derivatives with a calculator d. Application as applied to certain concepts i. Position function ii. Velocity function iii. Acceleration function e. rectilinear motion i. position, velocity, acceleration (analyzing signs of velocity and acceleration) ii. turning points, coasting, speeding up slowing down iii. graphing in parametric mode IV. Application of the Derivative (approximately 5 weeks) a. Tangent and Normal Lines b. Related Rates i. Sample Problems c. Relative Extrema and the First Derivative Test i. Theorem on increasing and decreasing ii. First derivative test for extrema: critical points d. Concavity and the Second Derivative Test i. Definition of concavity ii. Theorem: test for concavity iii. Definition of point of reflection iv. Second derivative test for extrema v. Curve sketching ( students apply what they know and then use written sentences to explain how to back them selves up that their curve is correct, students also present this to the class verbally, they evaluate the functions analytically, demonstrate them graphically and test some components numerically) e. Absolute Extrema and Optimization i. Extreme value theorem ii. Sample problems with applications (students form conclusions and use support such as sign change with derivatives) f. Rolle s and the Mean Value Theorems i. Rolle s Theorem
ii. Mean Value Theorem g. Differentials V. Antidervatives and Definite Integrals (approximately 5 weeks) a. Differential equations and slope fields ( students discover the parent functions and see the beginnings of an antiderivative. b. Antiderivaties i. Definitions of an antiderivative ii. Theorem on antiderivative iii. Notation iv. Vocabulary v. Constant multiple rule vi. Sum and difference rule vii. Trigonometric rules viii. Power rule c. Substitution of Variable i. Simple substitution ii. Trigonometric integrals d. Antiderivatives on Inverse Trigonometric Functions e. Trigonometric Substitutions VI. Application of Antderivatives and Definite Integrals a. Definitions of the definite integral i. Riemann sums: left, right and midpoint ii. Definition of the definite integral; notation iii. Theorem: Trapezoidal Rule iv. Theorem: Area between curves ( the students explain what this represents both verbally and written form) b. Fundamental theorem of calculus i. Integral defined functions ii. Four properties of the definite integral iii. Property of even and odd functions iv. Variable bounds property v. Average-value property vi. Integrating rate of change (use a calculator to find a definite integral, getting the number of people in an amusement park at a given time, or amount of sand on a beach, here students use written sentences to explain the solution. They also use their calculator to get their values in which to integrate from.
c. Analyzing curves with Antiderivatives i. Position, velocity and acceleration d. Volume i. Vertical axis of rotation (disc/washer) ii. Horizontal axis of rotation (disc/washer) iii. Solids with known cross sections (students find volumes of irregular shapes both analytically and with a calculator) VII. Exponential and Logarithmic Functions (approximately 5 weeks) a. The family of exponential functions b. The number e c. Logarithmic differentiation d. Antiderivatives and the Natural log e. Antiderivative of other Exponentials and Logs f. Differential equations with dy/dt=ky VIII. AP exam review (use released items and ABCD cd ROM) IX. Final project ( take different vases, plot curvature, store in calculator, use regression, find volume, fill vase with water, measure volume, find percent error, analyze error, and write a conclusion) X. Teacher resourses a. primary text book - Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus I with Precalculus. Boston: Houghton Mifflin. b. Graphing calculator labs for students ---Benita Albert and Phyllis Hillis c. AP Calculus Resource Notebook from AP Calculus Summer Institute with Phyllis Hillis d. Web sites-- AP Central, Visual Calculus, and Calculus graphics e. ABCD Calculus Review CD ROM f. The College Board AP advanced placement program course description Calculus AB, Calculus BC g. Released exams 1998 AP Calculus AB and Calculus BC h. All released exams on AP Central Calculus