 Syllabus

Fall 2008       Instructor: Mrs. Regina Ramey School: Boone County Honors Academy E-mail See Edline

TEXT:

Calculus Graphical, Numerical, Algebraic, 3rd edition;  Finney, Demana, Waits, Kennedy

REQUIREMENTS:

3-ring binder, Pencil, Notebook, Graph Paper, and Graphing Calculator

### PREREQUISITES:

Before studying calculus, all students should complete four courses of secondary mathematics designed for college-bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include those that are linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and so on) and know the values of the trigonometric functions of the numbers 0, pi/6, pi/4, pi/3, pi/2, and their multiples.

DESCRIPTION:

Advanced Placement (AP) Calculus AB is a college-level mathematics course for students that have previously demonstrated mastery of' algebra, geometry, coordinate geometry, and trigonometry. This course will develop the students' understanding of the concepts of calculus and provide experience with its methods and applications. The course will emphasize a multirepresentational approach to calculus, with concepts, results, and problems being expressed geometrically, numerically, analytically, and verbally. The connections among these representations also are important. Technology is used regularly to reinforce the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results.

### COURSE GOALS

Students should be able to:

§  work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations.

§  understand the meaning of the derivative in terms of a rate of change and local linear approximation and they should be able to use derivatives to solve a variety of problems.

§  understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems.

§  understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.

§  communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.

§  model a written description of a physical situation with a function, a differential equation, or an integral.

§  use technology to help solve problems, experiment, interpret results, and verify conclusions.

§  determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.

§  develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

TOPICAL OUTLINE:

I.    Limits

1. Use graphical and numerical evidence to estimate limits and identify situations where limits fail to exist.

2. Apply rules to calculate limits.

3. Use the limit concept to determine where a function is continuous.

II.   Derivatives

1. Use the limit definition to calculate a derivative, or to determine when a derivative fails to exist.

2. Calculate derivatives (of first and higher orders) with pencil and paper, without calculator or computer algebra software, using:

(a) Linearity of the derivative;

(b) Rules for products and quotients and the Chain Rule;

(c) Rules for powers, trigonometric and inverse trigonometric functions, and for logarithms and exponentials.

3. Use the derivative to find tangent lines to curves.

4. Calculate derivatives of functions defined implicitly.

5. Interpret the derivative as a rate of change.

6. Solve problems involving rates of change of variables subject to a functional relationship.

7. Approximate functions by using linearization (differentials).

III.   Applications of the Derivative

1. Find critical points, and use them to locate maxima and minima.

2. Use critical points and signs of first and second derivatives to sketch graphs of functions:

(a) Use the first derivative to find intervals where a function is increasing or decreasing.

(b) Use the second derivative to determine concavity and find inflection points.

(c) Apply the first and second derivative tests to classify critical points.

3. Use Differential Calculus to solve optimization problems.

4. Apply the Mean Value Theorem.

5. Use Newton's method to improve approximate roots of equations.

IV. The Integral

1. Find antiderivatives of functions; apply antiderivatives to solve separable first-order differential equations.

2. Use the definition to calculate a definite integral as a limit.

3. Apply the Fundamental Theorem of Calculus to evaluate definite integrals and to differentiate functions defined as integrals.

4. Calculate elementary integrals with pencil and paper, without calculator or computer algebra software, using:

(a) Linearity of the integral;

(b) Rules for powers (including exponent -1) and exponentials, the six trigonometric functions and the inverse sine, tangent and secant;

(c) Simple substitution.

V.   Transcendental Functions

1. Use the relation between the derivative of a one to one function and the derivative of its inverse.

2. Calculate with exponentials and logarithms to any base.

3. Calculate derivatives of logarithmic, exponential and inverse trigonometric functions; interpret and apply such derivatives as usual.

4. Use logarithmic differentiation.

5. Use models describing exponential growth and decay.

TENTATIVE PACING GUIDE

Chapter 1 Prerequisites for Calculus                                                        11 days

Chapter 2 Limits and Continuity                                                               10 days

Chapter 3 Derivatives                                                                              30 days

Chapter 4 Applications of Derivatives                                                        28 days

Chapter 5 The Definite Integral                                                                 26 days

Chapter 6 Differential Equations and Mathematical Modeling                      22 days

Chapter 7 Applications of Definite Integrals                                               21 days

This timeline gives approximately 15 days to “review” the course before the exam.

# THE EXAM

Put your knowledge to the test: The AP Calculus AB Exam assesses your mastery of Calculus AB concepts and techniques. It also gives you the chance to earn college credit while in high school.

### About the Exam

The AP Calculus AB Exam is 3 hours and 15 minutes. The 105-minute, 45-question multiple-choice section tests your proficiency on a wide variety of topics. The 90-minute, six-problem free-response section gives you the chance to demonstrate your ability to solve problems using an extended chain of reasoning.

#### Section I: Multiple-Choice

The multiple-choice section of the exam has two parts. For Part A, you'll have 55 minutes to complete 28 questions without a calculator. For Part B, you'll have 50 minutes to answer 17 questions using a graphing calculator. For more information, see the calculator policy for the AP Calculus Exams.

Unlike other multiple-choice tests, random guessing can hurt your final score. While you don't lose anything for leaving a question blank, one quarter of a point is subtracted for each incorrect answer on the test. But if you have some knowledge of the question and can eliminate one or more answers, it's usually to your advantage to choose what you believe is the best answer from the remaining choices.

#### Section II: Free-Response

The free-response section tests your ability to solve problems using an extended chain of reasoning. You'll have 45 minutes for each of the two parts of the free-response section. In Part A, you'll answer three questions using a graphing calculator. In Part B, you'll answer three questions without a calculator. During the second timed portion of the free-response section (Part B), you are permitted to continue work on problems in Part A, but you are not permitted to use a calculator during this time. For more information, see the calculator policy for the AP Calculus Exams.

### Scoring the Exam

The multiple-choice and free-response sections each account for one-half of your final exam grade. Since the exams are designed for full coverage of the subject matter, it is not expected that all students will be able to answer all the questions.

# Exam Grades

AP Grade Reports are sent in July to the college you designated on your answer sheet, to you, and to your high school. Each report is cumulative and includes grades for all the AP Exams you have ever taken, unless you have requested that one or more grades be withheld from a college or canceled.

RESOURCES:

Karl's Calculus Tutor                    http://www.karlscalculus.org/

Ask Dr. Math                               http://mathforum.org/dr.math/dr-math.html

TI Graphing Calculators                http://education.ti.com/educationportal/

Tutorials for the Calculus Phobe

Mr. Calculus                                http://users.adelphia.net/~sismondo/index.html

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