AP® Calculus AB Course Overview
My main objective in teaching AP® Calculus is to enable students to receive a strong foundation that will give them the tools to succeed in future mathematics courses. We cover everything in the Calculus AB topic outline as it appears in the AP® Course Description. Students know that they will work harder than ever, and my expectation is that this hard work will enable them to succeed in the AP® Exam, this class and future math courses.
Course Planner Functions and models (Chapter 1) 1. Four ways to represent a function. (Numerically, Analytically, Graphically, and Verbally) 2. Mathematical models: a catalog of essential functions. 3. New functions from old functions. 4. Graphing calculators and computers. (With the use of a TI83 plus silver edition calculator they will be able to verify the endbehavior of a graph, find the zeros, the intercepts, the vertical asymptotes, the domain, the range, etc.) Limits and rates of change (Chapter 2) 1. The tangent and velocity problems. 2. The limit of a function. (Algebraically, graphically with the use of a calculator and by hand, and numerically from a table of values) 3. Calculating limits using the limit laws. 4. The precise definition of a limit. 5. Continuity. 6. Tangents, velocities, and other rates of change. Derivatives (Chapter 3) 1. Derivatives. 2. The derivative as a function. 3. Differentiation formulas. 4. Rates of change in the natural and social sciences. 5. Derivatives of trigonometric functions. 6. The chain rule.
7. Implicit differentiation. 8. Higher derivatives. 9. Related rates. 10. Linear approximations and differentials. 11. Using the calculator to obtain a numerical derivative and interpreting the result as the rate of change of a function. Applications of Differentiation (Chapter 4) 1. Maximum and minimum values by using the first and second derivative test. 2. The mean value theorem. 3. How derivatives affect the shape of a graph. 4. Limits at infinity: horizontal asymptotes, limits at points of discontinuity: vertical asymptotes. 5. Summary of curve sketching. 6. Graphing with calculus and calculators. (In this section students are asked to graph functions with a TI83 plus silver edition graphing calculator and to zero in on aspects such as finding the horizontal and vertical asymptotes of a function, how to find a local minimum and local maximum of a function, how to find intervals where a function is increasing and decreasing, finding points of inflection and concavity, etc.) 7. Optimization problems. 8. Applications to business and economics. 9. Newton’s method. 10. Antiderivatives. (An introduction to indefinite integrals) Integrals (Chapter 5) 1. Areas and distances. (Including right, left, and midpoint approximations to area under a function) 2. The definite integral. (Students will be asked do integrate by hand and also by using a graphing calculator. In this section students will be able to find the area under a curve and to interpret their results.) 3. The fundamental theorem of calculus. 4. Indefinite integrals and the net change theorem. 5. The substitution rule. (‘u’ substitution rule) Applications of integration (Chapter 6) 1. Areas between curves. 2. Volumes. (Disks, washers, and cross sections)
3. Volume by cylindrical shells. 4. Average value of a function and the mean value theorem for integrals. Inverse functions (Chapter 7) 1. Inverse functions. 2. The natural logarithmic function. 3. The natural exponential function. 4. General logarithmic and exponential functions. 5. Indeterminate forms and L’Hospital’s Rule. (optional) Techniques of integration (Chapter 8) 1. Trigonometric integrals. 2. Approximate integrals. (Trapezoidal rule)
Differential equations (Chapter 10) 1. Direction fields and Euler’s method. 2. Separable equations. 3. Exponential growth and decay. Review for AP® Calculus Exam Throughout the semester, students will be given sets of questions taken from: 1. AP® Calculus AB 2003 Released Exam questions. 2. AP® Calculus AB 2007 FreeResponse Questions. (Free Response questions and scoring guidelines are available on AP® Central) Students may work on these questions with one other person and come to me for extra help. Students will frequently be asked to present to the whole class their solutions to the given questions.
End of Semester Project At the end of each semester students will be asked to choose from their textbook one of the suggested project. They will have to return a written report, and orally present, explain, and defend their findings, results, and conclusions in front of the class. Sample Projects Topics include: Writing Projects: Early Methods for Finding Tangents;
Newton, Leibniz, and the Invention of Calculus; The Origins of L’Hospitals Rule; How Newton Discovered the Binomial Theorem. Applied Projects: Where Should a Pilot Start Descent? Building a Better Roller Coaster; The Calculus Rainbows; The Shape of a Can; Where to Sit at the Movies? How Fast Does a Tank Drain? Which is faster, Going up or coming down? Calculus and Baseball Radiation from the Stars. Discovery Projects: Area Functions Patterns in Integrals Arc Length Contest Rotating on a Slant
Technology and Computer software All students will be using the TI83 plus silver edition calculator to analyze functions and to compute various quantities such as definite integrals, derivatives, graphing functions, finding the zeros of a function, etc. With the aid of a calculator students will be able to obtain results and in turn they will be asked to interpret those results. Besides drawing conclusions from their results, students will be asked to use the calculator as a tool to help them provide evidence and to prove their conclusions.
Student Evaluation 90100=A, 8089=B, 7079=C, 6069=D, 059=D. The grade will be determined based on two categories: Homework/Projects will be 10% of the grade; and quizzes and exams will be 90% of the grade. Quizzes will be given on a weekly basis and Exams will be
given at the end of each chapter and a final cumulative exam will be given at the end of each semester. The exams will be structured in such a way that the students will be required to use a graphing calculator for half of each exam.
Textbook Stewart, James. Single Variable Calculus, 5th Edition. Brooks/Cole, a division of Thomson Learning, 2003.
Technology resources Students will be provided a TI83 plus silver edition calculator.
