AP_Calculus_AB_Outline

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					AP Calculus AB Outline
Course Description
It is the goal of this course to develop a solid understanding of what calculus is, and to
equip the students with the tools to successfully pursue higher-level mathematics courses.
At the conclusion of this course the students will: understand the fundamental concepts of
calculus (functions, limits, derivatives, integrals) and be able to apply the methods of
calculus to solve real problems.


Teaching Style
I begin the year by stating that Calculus is about the relation between a quantity and its
rate of change. We calculate several constant rates of change over different time intervals
           d
using v  and then I give the students the task of finding the rate of a particle that
           t
moves along the curve dbg t 2 where 0  t  3 . The students realize quickly the
                           t

problems posed by this situation and that the rate changes as time changes. This leads to
a discussion about instantaneous rates and average rates. I continually develop new
concepts in this manner for the purpose of emphasizing importance and application.
Throughout the course, the text provides data in a variety of formats, so that the students
get a rich exposure to performing symbolic manipulations, graphical analysis, and
estimations from tabular data.

Assessments
Throughout the year I quiz and test them in a variety of ways. Section quizzes and
chapter tests may be given as multiple choice, free-response, or a combination of the two.
Some of the quizzes are taken in groups. Some assessments are completely without a
calculator, completely with a calculator, or a mix. The students also regularly present
solutions for homework problems to the class and must answer, verbally, all questions
posed by their peers and myself.

Writing
I also have the students occasionally answer writing prompts. They are usually a one-
page essay, and should include diagrams with labels when appropriate. An example of
one such prompt is:
        You are going to explain to a friend of yours in Pre-Calculus what the formal
        definition of derivative means. Your discussion should include, but is not limited
        to, how secant lines and tangent lines are part of the development of the definition
        of derivative. Just to make sure, here is the definition:
                                                         L
                                                         M
                                                             f
                The derivative of a function is the Lim ( x h )
                                                                      O
                                                                    f ( x)
                                                                      P     .
                                                      h0N        h   Q
        Keep in mind that your friend is in the early part of Pre-Calculus, and does not
        know the concepts and terminology that you have picked up in this course. Thus,
        you need to define or explain any Calculus specific terms that you use, as you use
        them.
Graphing Calculators
Students may checkout a TI-83+ from the Library, although it has been my experience
that they buy their own. The calculator is used as a tool to aid the students in their quest
for a numeric, graphical, and analytic solution. It also helps them arrive at conjecture that
would otherwise not be quickly discovered.
Chapter 1: Prerequisites for Calculus (Pre-Calculus Review)
(2-3 weeks)
1.1: Lines
       A. Slope as a rate of change
       B. Parallel and perpendicular lines
       C. Equations of lines ( Point-slope, Slope-intercept, & Standard form)
1.2: Functions and Graphs
       A. Functions
       B. Domain and range
       C. Viewing and interpreting Graphs
       D. Piecewise Functions
       E. Composition of functions
1.3: Exponential Functions
       A. Exponential growth and decay
       B. Applications
       C. The number e
1.5: Functions and Logarithms
       A. One-to-one functions
       B. Inverses
       C. Logarithmic functions
       D. Properties of logarithms
1.6: Trigonometric functions
       A. Graphs of trigonometric functions
              1. Period
              2. Domain/range
              3. Transformations
       B. Inverse Trigonometric functions
              1. Restricting domain to make one-to-one function
              2. Finding angle measures
       C. Applications
Chapter 2: Limits and continuity
(3 weeks)
2.1: Rates of change and limits
       A. Average and instantaneous speed
       B. Definition of limit
       C. Properties of limits
       D. Two sided limits
       E. One sided limits
2.2: Limits involving infinity
       A. Horizontal asymptotes
       B. Vertical asymptotes
       C. Properties of limits as x  
       D. End behavior models
               1. Left end behavior
               2. Right end behavior
2.3: Continuity
       A. Continuous functions
       B. Discontinuous functions
               1. Removable
               2. Jump
               3. Infinite
2.4: Rates of change and Tangent lines
       A. Tangent to a curve
       B. Slope of a curve
       C. Instantaneous rates of change
Chapter3: Derivatives
(5 weeks)
3.1: Derivative of a function
       A. Definition of derivative
       B. One-sided derivative


3.2: Differentiability
        A. Local linearity
        B. Numeric derivatives using the calculator
        C. Differentiability implies continuity
3.3: Rules for Differentiation
        A. Power rule
        B. Sum and Difference
        C. Products and Quotients
        D. Second and higher derivatives
3.4: Velocity and other rates of change
        A. Instantaneous rates of change
        B. Motion along a line
        C. Sensitivity to change
3.5: Trigonometric functions
3.6: Chain rule
3.7: Implicit Differentiation
3.8: Inverse Trigonometric functions
3.9: Exponentials and Logarithms

Chapter 4: Applications of Derivatives
(4 weeks)
4.1: Extreme Values of Functions
4.2: Mean Value Theorem
       A. Physical Interpretation
       B. Increasing and Decreasing Functions
4.3: Connecting f’ and f” with the Graph of f
       A. First Derivative Test for Local Extrema
       B. Concavity
       C. Points of Inflection
       D. Second Derivative Test for Local Extrema
       E. Learning about Functions from Derivatives
4.4: Modeling and Optimization
4.5: Linearization and Newton’s Method
       A. Differentials
       B. Estimating Change
4.6: Related Rates

Chapter 5: The Definite Integral
(3 weeks)
5.1: Estimating with Finite Sums
       A. Distance Traveled
       B. Rectangular Approximation
5.2: Definite Integrals
       A. Riemann Sums
       B. Definite Integral and Area
       C. Integrals on a Calculator
       D. Discontinuous Integrable Functions

5.3: Definite Integrals and Antiderivatives
       A. Properties of Definite Integrals
       B. Average Value of a Function
       C. Mean Value Theorem for Definite Integrals
       D. Connecting Differential and Integral Calculus
5.4: Fundamental Theorem of Calculus
       A. Part 1
       B. Part 2
5.5 Trapezoidal rule

Chapter 6: Differential Equations and Mathematical Modeling
(3-4 weeks)
6.1: Slope Fields
6.2: Integration with U-Substitution
6.3: Integration by Parts
6.4: Separable Differential Equations
6.5: Partial Fractions

Chapter 7: Applications of Definite Integrals
(3 weeks)
7.1: Summing Rates of change
7.2: Areas in the plane
       A. Areas between curves
       B. Integrating with respect to y
7.3: Volumes
       A. Volumes of solids with known cross sections
       B. Volumes of revolution
              1. Disk Method
              2. Shell Method

This leaves 4-6 weeks for review

Course Textbook
Ross L. Finney, Franklin D. Demana, Bert K. Waits, and Daniel Kennedy.
       Calculus-Graphical, Numeric, Algebraic. 3rd ed. Pearson Prentice Hall, 2007.

AP Exam Review
Along with released question I use Pearson Education AP test prep series, Fast track to A
“5”, and Kaplan AP Calculus AB 2004 edition. I have the students take practice exams
and give the same amount of time that they will have on each section of the AP exam.
We then grade the exams using the grading guides that accompany each practice test.