AP Calculus Syllabus

AP Calculus Syllabus

AP^© Calculus AB

TEXT: CALCULUS: A New Horizon, 6^th Edition, Howard Anton

INSTRUCTOR: Mr. Roy Lanier, Jr.

MATERIALS REQUIRED: Textbook, graphing calculator, TI-83⁺, or TI-84⁺ recommended.

A. Course Goal

Since students tend to perform with more confidence and accuracy when they have an understanding of the concepts behind the subject at hand, this course focuses on the basic concepts as necessary for performing well in the course and on the AP test. The various concepts outlined in the College Board Calculus Course Description are presented and discussed, then applied to the process of solving problems, first from the class textbook, then from various sources focusing on advanced placement type problems. The primary source of these is problems from released tests and the course description. In addition a major effort is placed on understanding terms and conditions so that sound justifications for conclusions are reached in solving problems. The process of integrating technology in the form of graphing and analyzing calculators is also emphasized. This is accomplished by assigning activities for graphing calculators developed by Benita Albert and Phyllis Hillis of the Oak Ridge High School in Oak Ridge, Tennessee. This is supplementary to the requirement to use the calculators daily in the investigation and solution of problems. The goal, then, is to develop a student with a thorough understanding of the concepts of calculus and the tools available to assist in understanding and solving calculus problems.

B. Course Content

Chapter Time (approx.) Topics

1. Functions 2-3 weeks Review functions, graphs, trig, quadratic formula, distance,

absolute value, etc.

A review of basic functions and their graphs from algebra and trigonometry is accomplished during this time. The idea of modeling physical phenomena with functions is presented. Emphasis is placed on concepts and relating functions and their graphs using the graphing calculator. The concept of describing functions analytically, numerically, graphically and verbally is also introduced during this time.

2. Limits and 2-3 weeks Definitions and theorems on limits and continuity.

Continuity

Limits are presented as an intuitive concept and from a computational standpoint with examples and definitions. The idea of “close but not there” is emphasized, and calculator tables and graphs with zoom in are used to demonstrate this feature of limits. Specific examples are used to solidify the students understanding of the limit and one-sided limits. Continuity is defined with the idea of a limit in an effort to help the student understand the necessity for precision and clear verbal understanding of functions that are not continuous. Discussions of functions that have left or right hand limits and piecewise functions are used to solidify the students understanding of these key concepts. The Intermediate Value Theorem is introduced with continuity to show the necessity of a root occurring in a given interval and the zoom feature of calculator is used to approximate the root by approaching it. Certain key trigonometric limits such as are introduced, discussed and graphed at this time.

3. The Derivative 4-5 weeks Definition, tangent lines, rates, techniques.

The relationships of average and instantaneous rates of change are introduced as velocity concepts of which the student is more aware. The two definitions of derivative, for x approaching a point and for h approaching zero, are introduced and discussed as limits. The idea of a “rate of change at a point” is discussed in the context of the derivative and the tangent line at a point. Effort is directed toward having the student understand and verbalize this relationship. The definition of differentiability is then introduced and points of non-differentiability are discussed. The relationship of differentiability and continuity are discussed and investigated graphically. Techniques of differentiating polynomials and their products and quotients are derived and used in drills and simple application problems. The idea of higher derivatives is introduced and their potential applications are discussed. The derivatives of the trigonometric functions are then introduced and used in solving application problems. The chain rule is presented as the derivative of composition functions. The idea of local linear approximations as a tangent line approximation is introduced and investigated graphically. Differential notation is introduced at this point to stress the term or in differential approximations.

4. Logarithmic & 3-4 weeks Exponentials, log functions, inverses, implicit differentiation,

Exponential exponential growth and decay, related rates.

Functions

Exponential, logarithm and inverse trigonometric functions and their derivatives are introduced at this time to add these functions to the students “toolbox” of knowledge. The concept of implicit differentiation is introduced and practiced in conjunction with application problems such as exponential growth and decay, and related rate problems. The student is now expected to work problems from former AP tests including giving verbal justification for results and conclusions.

5&6. Derivatives 5-6 weeks Increasing, decreasing, concavity, absolute maximum and

in Analysis of minimum values, Newton’s Method, Mean Value Theorem.

Functions

The application of derivatives to locate relative extremes, absolute extremes and the criteria or justification for these is investigated. The process of developing a verbal justification based on limits, domains and function values are developed analytically, investigated with the graphing calculator and instilled in the student. The use of the derivative to determine concavity of curves and inflection points on a graph are also investigated. In addition Newton’s Method of locating roots of functions and the Mean Value Theorem are introduced, discussed and applied. All of these concepts are practiced in working AP type problems.

7. Integration 4-5 weeks The area problem, Riemann sums, indefinite and definite integrals.

The Fundamental Theorems of Calculus and the Mean Value Theorem. The Trapezoidal Method of approximation of integrals.

The problem of finding the area under a curve is investigated using Riemann sums with both analytical and calculator techniques to see the effect of evaluating curves above and crossing the x-axis as a basis for understanding antiderivatives for finding the value of an integral. Define antiderivatives and indefinite integrals and their relationship. Present integration techniques using substitution and integration by parts to assist in finding antiderivatives. Demonstrate and practice finding areas using the graphing calculator for analyzing functions where the antiderivatives are difficult to obtain. Introduce the definite integral and the Fundamental Theorem of Calculus, Part I and discuss the implication of the upper and lower limits. Present the Mean Value Theorem (MVT) for integrals and Part II of the Fundamental Theorem of Calculus. Revisit rectilinear motion with the application of the integral to develop the concept of distance traveled and change in position and use the MVT for integrals to find the average value of various functions.

8-10. Special Topics 3-4 weeks Volumes of solids, trigonometric integrals, separable

variable differential equations and slope fields.

Methods of determining the volumes of solids by means of summing known cross sectional areas, discs, washers and cylindrical shells are developed. The method of changing axes and limits to enhance the ability to find volumes is also investigated. Methods of solving integrals involving trigonometric functions are investigated. The solution of separable variable differential equations is analyzed and the process of using initial conditions to find a constant of integration is developed. The process of Euler’s Method to develop an approximate slope field from a differential equation and initial condition is discussed and compared to an actual solution to a differential equation.

C. Supplementary Materials

Materials required daily: Textbook, graphing calculator, TI-83⁺ or TI-84⁺ will be used in the classroom, but a list of allowed calculators is available.

Other:

· The College Board Calculus Course Description for May 2006, May 2007.

· Graphing Calculator Labs for Students by Benita Albert and Phyllis Hillis.

· Multiple Choice & Free Response Questions in Preparation for the AP Calculus (AB) Examination, eighth edition, by David Lederman.

D. Grading

There will be homework daily, quizzes frequently, calculator labs approximately three times each quarter outside the classroom (at lunch, before or after school or study hall), tests at appropriate times in the development of concepts and final exams. The purpose of the first two of these is to help you master the concepts. The calculator labs are designed to help you understand and use the calculator properly. The labs will each count as one quiz grade. The purpose of the remaining two of these is to help me know whether you are mastering the concepts and to prepare you for the AP test in May. The value of each toward your final grade is as follows:

homework--20%(full credit given only if submitted the class period following assignment), quizzes--20% (Calculator labs will count as quizzes), chapter tests--60% of daily grade; final exam--20% of overall grade.

E. Help!