AP® Calculus AB Course Overview This course is designed so that students who successfully complete it will do well on the AP Calculus AB Exam and be prepared for subsequent college level math classes. Every topic outlined in the AP® Calculus AB Course Description is covered. Additional topics such as L’Hopital’s Rule and integration by parts are covered as time permits during the course and after the AP Exam. Students are required to take the AP Calculus AB Exam in May.
Course Planner The specific topics covered in this course are as follows. Unit 1: Review of prerequisite topics (2 weeks) Equations of lines Functions and relationships more than meets the eye Exponential functions Inverse functions and logarithms Trigonometric functions Functions and their graph families Functions on the calculator Finding zeros and intersections with the calculator Unit 2: Limits and Continuity (34 weeks) The limit as introduced by the rate of change Limits Limits involving infinity Continuity The tangent line and applications
Unit 3: Differential Calculus (6 weeks) Definition of Derivative Derivative Shortcuts Differentiability Computing derivatives on the calculator
Position/velocity and other applications Derivatives of trig functions The Chain Rule Implicit Differentiation The relationship between derivatives of functions and their inverses Derivatives of inverse trigonometric functions Derivatives of exponential and logarithmic functions
Unit 4: Applications of the Derivative (4 weeks) Extreme Values of Functions The Mean Value Theorem for derivatives The behavior of functions: increasing, decreasing, and concavity Relationship between functions, derivatives, and second derivatives Oblique asymptotes Particle Motion problems displacement and total distance traveled L'Hôpital's Rule Unit 5: More applications of derivatives (3 to 4 weeks) Modeling and Optimization Linearization Related rates Unit 6: Integral Calculus (5 weeks) Estimating by adding up rectangular areas The Riemann sum and the definite integral Definite integrals and antiderivatives Definite integrals on the calculator The Fundamental Theorem of Calculus The Trapezoid Rule and Simpson's Rule Antiderivatives and Slope Fields Integration by substitution Separable differential equations Exponential Growth and Decay Unit 7: Applications of Integrals (3 to 4 weeks) The integral as a net change Areas in the plane Volumes Science and statistics applications
Unit 8: AP Review (remaining time prior to AP exam) Practice MultipleChoice Questions Practice Free Response Questions Unit 9: [(Time Permitting) or after AP Exam (Time Permitting)] Integrating More Complicated Functions Integration by Parts Trigonometric Integrals Trigonometric Substitution Integrals Partial Fraction Integrals
Teaching Strategies The students are exposed to topics and questions throughout the course graphically, numerically, analytically, and verbally. This forces the students to think about what is going on instead of memorizing a laundry list of formulas and equations. Students are encourage to work and study together when they are learning new material as this interaction helps them to understand the concepts more than through just individual effort. Emphasis is placed on communicating with proper mathematical language and syntax. Most of the students entering the course are already familiar with graphing calculators through their precalculus class taken the previous year. It is constantly stressed that calculator answers are never sufficient to justify a solution and that it takes an analytical method to do that. However, many students find that their graphing calculator can help them to investigate and come up with the analytical solution for the problem. The students are reminded that for AP Calculus their calculator can do five things (1) arithmetic, (2) graphing, (3) find zeros/solutions to equations, (4) numerical derivatives, and (5) definite integrals. TISmartView is used extensively so the students can see exactly how a calculator is used to solve problems and how it can be used more efficiently to solve more complicated and/or lengthy multistep problems. During the spring semester the students spend a substantial amount of time reviewing and preparing for the AP Exam in May in addition to learning the new material that still needs to be covered. They are assigned 20 multiple choice questions from previous exams every week to be completed at home and graded the next week. Approximately one month (April) is spent reviewing previous AP free response questions and additional multiple choice questions. The
students take an actual previous AP test in April. At the end of this review students are comfortable with the style and complexity of an actual AP Exam and they are able to perform to the best of their ability.
Student Activities Students participate in activities, labs, and/or projects throughout the year. For example, during the fall semester we have a project to calculate the distance driven in an RC car by recording the velocity of the vehicle every 30 seconds. This gives the students an opportunity to numerically integrate the velocity to obtain distance traveled. They then write up a report to show their findings. During the spring semester we have a project to construct 3D solids using disks and washers, or known cross sections so that they can better visualize these volume integrals.
Student Evaluation Students are graded on a semester basis with 80% of this score coming from tests, quizzes, homework, and other activities. The final 20% of their grade is composed of a comprehensive final exam for the fall semester. During the spring semester, a practice AP Exam is considered their final exam and as a result the final test of the year (material covered after the AP Exam) is treated like a normal test. The exams in the class check to see understanding at a range of levels with all tests having some questions comparable in difficulty to actual AP exams. On these written tests, some questions are asked analytically while others are asked graphically or numerically as is done on actual AP exams.
Primary Textbook Finney, Demana, Waits and Kennedy. Calculus: Graphical, Numerical, Algebraic. New Jersey: Pearson Prentice Hall, 2003.
Technology Resources Students are required to have and use a graphing calculator throughout the course. Most students use one of the TI83/TI84/TI89 version graphing calculators. Using a TISmartView, a program that models a TI84 calculator, the students can see and follow along as a graphing calculator is used to solve problems and explore concepts. In addition, MAPLE software is used to model and demonstrate various topics.
