AP Calculus AB
Course Requirements 2010-2011
Textbook: Calculus Third Edition by Hughes-Hallett, Gleason, McCallum, et al.
New York: John Wiley & Sons, 2002.
Because of the special nature of advanced placement classes, students need to be aware of specific commitments required. After reading the following expectations and responsibilities, your signature indicates your agreeing to comply with the requirements. Please note that four years of high school math are required before calculus and participation in Advanced Placement Calculus AB is subject to instructor approval. Your parent or guardian must sign also, indicating his/her understanding and support of the requirements of taking Advanced Placement Calculus AB, including the time commitment for the course and the financial commitment to take the test. Sign and return by Monday, August 21.
Topic Outline for Calculus AB
I. Functions, Graphs, and Limits
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.
Limits of functions (including one-sided limits).
An intuitive understanding of the limiting process.
Calculating limits using algebra.
Estimating limits from graphs or tables of data.
Asymptotic and unbounded behavior.
Understanding asymptotes in terms of graphical behavior.
Describing asymptotic behavior in terms of limits involving infinity.
Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)
Continuity as a property of functions.
An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.)
Understanding continuity in terms of limits.
Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).
II. Derivatives
Concept of the derivative.
Derivative presented graphically, numerically, and analytically.
Derivative interpreted as an instantaneous rate of change.
Derivative defined as the limit of the difference quotient.
Relationship between differentiability and continuity.
Derivative at a point.
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.
Tangent line to a curve at a point and local linear approximation.
Instantaneous rate of change as the limit of average rate of change.
Approximate rate of change from graphs and tables of values.
Derivative as a function.
Corresponding characteristics of 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.
Second derivatives.
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.
Applications of derivatives.
Analysis of curves, including the notions of monotonicity and concavity.
Optimization, both absolute (global) and relative (local) extrema.
Modeling rates of change, including related rates problems.
Use of implicit differentiation to find the derivative of an inverse function.
Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
Computation of derivatives.
Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
Basic rules for the derivative of sums, products, and quotients of functions.
Chain rule and implicit differentiation.
III. Integrals
Interpretations and properties of definite integrals.
Computation of Riemann sums using left, right, and midpoint evaluation points.
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:
f’(x)dx = f(b) - f(a)
Basic properties of definite integrals. (Examples include additivity and linearity.)
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.
Fundamental Theorem of Calculus.
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.
Techniques of antidifferentiation.
Antiderivatives following directly from derivatives of basic functions.
Antiderivatives by substitution of variables (including change of limits for definite integrals).
Applications of antidifferentiation.
Finding specific antiderivatives using initial conditions, including applications to motion along a line.
Solving separable differential equations and using them in modeling. In particular, studying the equation y’ = ky and exponential growth.
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.
This course is designed to prepare students to take the national college-level advanced placement examination to be administered in early May. Students successfully completing this exam may receive advanced placement and/or credit at many colleges and universities. Rifle High School participants in Advanced Placement Calculus AB are required to take the exam.
Materials:
Required materials: 2 inch 3 ring binder with pockets, 500 sheets college ruled paper, #2 pencils, Texas Instruments TI 83 Graphing Calculator, and a book cover.
Put everything in your notebook and don’t throw anything away!!
Suggested Materials: Texas Instruments TI 89 Graphing Calculator, a quality preparation guide for the 2006 advanced placement exam, any of which should be available at the major booksellers.
Grading Policy:
Exams and Quizzes = 60%
Homework = 40%
Letter Grades are assigned as follows:
A = 94 - 100 A- = 90 - 93
B = 84 - 89 B- = 80 - 83
C = 74 - 79 C- = 70 - 73
D = 64 - 69 D- = 60 - 63
F = 59 and below
A major project should be anticipated during Spring Break.
Student Responsibilities
1. It is the responsibility of the student to bring all required materials to class on a daily basis.
2. It is the student’s responsibility to keep up with note taking and homework assignments. Participating in homework assignments is essential to success in this class. Students planning to receive an “A” or “B” in this class (and that is all of you) and a 4 or 5 on the AP exam (and that is all of you) should plan on spending at least ONE HOUR on homework everyday. It is highly recommended that you begin your homework the same day that it is assigned
3. Exams and quizzes will be assigned as deemed necessary. However, students should plan on an exam about once a month. There will be a comprehensive final exam at the end of each semester.
4. All students who do not attain a perfect score will participate in test corrections, possibly Mastery Reform.
5. Students are responsible for all work missed because of absences. All make-ups must be arranged with the teacher as soon as possible. The make up policy is two days of absences allows for a two day make-up period. It is the responsibility of the student to find out what assignments were missed and to arrange for any make-up exams.
6. Students are expected to adhere to all school policies regarding absences, tardies, and academic honesty.
7. There will be no extra credit work provided.
8. Students who are accepted and enroll in the class must take the Advanced Placement Exam. Payment for the exam is expected on time.
9. Students should anticipate approximately 24 hours of required extra study sessions scheduled throughout the year. Extra study sessions will occur more frequently in April. Advanced Placement Calculus AB concludes when the exam is taken and the extra study sessions shall make up for the class sessions after the exam.