Unit 1: Pre-Calculus Review (2 weeks) A. Lines 1. Slope as rate of change 2. Parallel and perpendicular lines 3. Equations of lines
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1 Calculus AB Syllabus AB Course Outline The following is an outline of the topics we will cover and a typical sequence in which those topics will be covered. The time spent is only an estimate of the average number of days allotted to the topic because the actual time varies from year to year depending upon your and your classmates abilities and interest, and also upon the richness of the class discussion that is generated. I will be issuing you a TI-83Plus calculator or you may use your own. You are expected to bring a graphing calculator to class daily. Calculators may be used on some, but not all, assessments. Unit 1: Pre-Calculus Review (2 weeks) A. Lines 1. Slope as rate of change 2. Parallel and perpendicular lines 3. Equations of lines B. Functions and graphs 1. Functions(looking at both geometric and analytic information to predict and explain observed local and global behavior of a function) 2. Domain and range 3. Families of function (knowing the parent graphs and graphing with transformations) 4. Graphing functions using end behavior, symmetry and zeros 5. Piecewise functions 6. Composition of functions C. Exponential and logarithmic functions 1. Exponential growth and decay 2. Inverse functions 3. Logarithmic functions 4. Properties of logarithms 5. Comparing rates of growth D. Trigonometric functions 1. Unit circle values 2. Graphs of basic trigonometric functions a. Domain and range b. Transformations c. Inverse trigonometric functions
2 Unit 2: Limits and Continuity (2 weeks) A. Limits at a point 1. Properties of limits a. calculating limits with algebra b. calculating limits with tables 2. Two-sided 3. One-sided B. Continuous Functions 1. Definition of Continuity 2. Continuity Test 3. Discontinuities a. Removable discontinuity b. Jump discontinuity c. Infinite discontinuity d. Oscillating discontinuity 4. Properties of continuous functions 5. Max-Min Theorem for continuous functions 6. Intermediate Value Theorem for continuous functions C. Limits Involving Infinity 1. Asymptotic behavior 2. End behavior D. Slopes, Tangent lines and Derivative 1. Average rates of change and instantaneous rate of change 2. Local linearity 3. Definition of derivative a. derivative at a point b. approximate rates of change from graphs and tables 4. Differentiable and continuous functions B. Graphing Derivatives 1. Derivatives from Functions 2. Functions from Derivatives We study limits in a variety of ways including analytically, graphically and numerically. We use our graphing calculator to study the graph of functions with removable discontinuities as well as infinite discontinuities. We also use the table feature of the graphing calculator to investigate infinite discontinuities and limits as x goes to infinity. Students find the calculator very helpful in solving these problems. They are also required, however, to solve numerous problems analytically. We then use our calculator to confirm or support our analytical results. Doing this helps students see how the graph, table and symbolic representations relate. Students use the zoom feature of their graphing calculators to see that almost all functions can be made to look straight when a small enough window is used. We then discuss how this relates to finding slopes of non-linear curves. We also use our calculators to find the slope of secant lines around a point making the secant lines get smaller and smaller making them turn into a tangent line. We then discuss how this gives us the definition of derivative. Unit 3: The Derivative (5 weeks) (derivatives are seen geometrically, numerically and analytically) A. Numerical Derivatives 1. Numeric derivatives using the calculator 2. Differentiability B. Differentiation Rules C. Applications to speed, velocity and acceleration D. Derivatives of trigonometric functions
3 E. The Chain rule F. Implicit derivatives G. Related rates H. Linear Approximations and Differentials Before introducing the differentiation rules, students spend one to two days in the computer lab with an investigation using a template created with StudyWorks. They are given a function and a graph of its derivative. They are to guess the function s derivative rule and see if its graph matches the given derivative graph. Most students can discover the power rule as well as many trigonometric derivatives in this manner. After chapter 3 we begin our daily warm- up problems which consist of multiple choice questions based on the first 3 chapters (and then after chapter 4 based on the first 4 chapters, etc.) These problems are similar to AP multiple choice problems (some are actual released problems) and require students to apply and synthesize what they have previously learned. The problems are very difficult at first for the students, but eventually they become easier. These problems help keep the material fresh in their minds as well as helps students connect old material with the new material. This is also the time that I begin to assign weekly or biweekly free response questions with are graded on a scale of 0 to 9. In these problems students are expected to explain their work in written form. Unit 4: Applications of the Derivative (6 weeks) A. Extreme values 1. Local (relative) extrema 2. Global (absolute) extrema 3. Mean Value Theorem 4. Rolle s Theorem 5. Increasing and decreasing functions B. Predicting Hidden Behavior 1. Critical values 2. First derivative test for extrema 3. Concavity and points of inflection 4. Second derivative test for extrema C. Polynomial Functions and Optimization problems 1. Using the derivative tests to graph polynomials 2. Optimizing with polynomial functions D. Rational functions and Economics Applications 1. Using the derivative tests to graph rational functions 2. Optimizing with rational functions in economics E. Radical and Transcendental Functions 1. Using the derivative tests to graph radical and transcendental functions 2. Optimizing with radical and transcendental functions G. Antiderivatives and Initial Value Problems Unit 5: Integration (5 weeks) Students are taught how to use their calculators to graph first and second derivatives and to use these graphs to solve problems dealing with maximums, minimums, intervals of increasing, decreasing and concavity and inflection points. Students interpret these graphs so that they can solve problems. Many days are spent on optimization and related rates problems. Students work in groups where they discuss the problems and determine the solutions together. They are then expected to present their answers to the rest of the class verbally.
4 A. Calculus and Area At least one week is spent working on 1. RAM released AP free response problems 2. Sigma notation dealing with integrals. Students work B. Definite Integrals together on a packet of 16 problems. 1. Riemann sums They discuss the problems and then 2. Area write their solutions as if they were C. The Fundamental Theorem of Calculus going to be graded by the AP readers. 1. Part 1 2. Part 2 D. Indefinite Integrals E. Accumulation Problems 1. Numerical calculation of areas geometrically and with tables 2. Motion along a line 3. Fundamental theorem of calculus 4. Application problems of integrals (lots and lots) F. Integration by Substitution Unit 6: Applications of Definite Integrals (5 weeks) A. Areas between curves B. Volumes 1. Volumes of solids of revolution a. Disk method b. Shell method 2. Volumes of solids with known cross-sections. C. Average Value of a Function D. Trapezoidal rule Manipulatives are used to help students visualize these solids. Presentations using Geometer s Sketchpad are used to show revolutions. Unit 9: Differential Equations (1 week) A. Modeling with Differential Equations B. Direction Fields C. Separable Equations Unit 7: Techniques of Integration (3 weeks-some items done after the exam) A. Integration by Parts B. Trig Integrals C. Trig Substitutions D. Rational Functions and Partial Fractions E. Strategy for Integration This schedule leaves 3 weeks to review for the AP Exam.
5 Assessment Students are assessed through homework, class work, AP free response problems, quizzes and tests. Students are expected to present problems where they verbally explain the solution a number of times throughout each marking period. They earn participation points through these presentations. Calculators Students who do not already own a graphing calculator are issued a TI- 83 Plus calculator for in class and at home use for the entire school year. Textbook James Stewart. Single Variable Calculus with Vector Functions. Thomson Brooks/Cole.
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