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  • Rebecca Holliman

     
    Department:Math

    Period 1 Math Analysis Honors/Room 5116

    Period 2 Math AnalysisHonors/Room 5116

    Period 3 AP Calculus/Room 5116

    Period 4 AP Calculus/Room 5116

    Period 5 Math Analysis Honors/Room 5116

    Period 6 Conference/Room 5116

    Tutoring is available every weekday from 7:15 a.m. to 7:45, and a "required" after school AP Calculus workshop every Wednesday from 3:05 p.m. to 4:00pm. 

    AP

    ® Calculus AB

    Syllabus

    Course Design and Philosophy

    Students do best when they have an understanding of the conceptual underpinnings of

    calculus. Rather than making the course a long laundry list of skills that students have to

    memorize, we stress the “why” behind the major ideas. If students can grasp the reasons

    for an idea or theorem, they can usually figure out how to apply it to the problem at hand.

    We explain to them that they will study four major ideas during the year: limits,

    derivatives, integrals, and modeling/ applications. As we develop the concepts, we

    explain how the mechanics go along with the topics. Thus, although facility with

    manipulation and computational competence are important outcomes, they are not the

    core of this 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.

    Students should understand the meaning of the derivative in terms of a rate of

    change and local linear approximation and should be able to use derivatives to

    solve a variety of problems.

    Students should 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.

    Students should understand the relationship between the derivative and the

    definite integral as expressed in both parts of the Fundamental Theorem of

    Calculus.

    Students should be able to communicate mathematics both orally and in wellwritten

    sentences and should be able to explain solutions to problems.

    Students should be able to model a written description of a physical situation with

    a function, a differential equation, or an integral.

    Students should be able to use technology to help solve problems, experiment,

    interpret results, and verify conclusions.

    Students should be able to determine the reasonableness of solutions, including

    sign, size, relative accuracy, and units of measurement.

    Students should develop an appreciation of calculus as a coherent body of

    knowledge and a human accomplishment.

    AP Calculus AB Course Outline /Timeline

    Unit 1: Precalculus Review (Summer Workshops)

    A. Lines

    1. Slope as rate of change

    2. Parallel and perpendicular lines

    3. Equations of lines

    B. Functions and graphs

    1. Functions

    2. Domain and range

    3. Families of function

    4. Piecewise functions

    5. Composition of functions

    C. Exponential and logarithmic functions

    1. Exponential growth and decay

    2. Inverse functions

    3. Logarithmic functions

    4. Properties of logarithms

    D. Trigonometric functions

    1. Graphs of basic trigonometric functions

    a. Domain and range

    b. Transformations

    c. Inverse trigonometric functions

    2. Applications

    Unit 2: Limits and Continuity (3 weeks)

    A. Rates of change

    B. Limits at a point

    1. Properties of limits

    2. Two-sided

    3. One-sided

    C. Limits involving infinity

    1. Asymptotic behavior

    2. End behavior

    3. Properties of limits

    4. Visualizing limits

    D. Continuity

    1. Continuous functions

    2. Discontinuous functions

    a. Removable discontinuity

    b. Jump discontinuity

    c. Infinite discontinuity

    E. Instantaneous rates of change

    F. Limit Lab Activity

    Unit 3: The Derivative (5 weeks)

    A. Definition of the derivative

    B. Differentiability

    1. Local linearity

    2. Numeric derivatives using the calculator

    3. Differentiability and continuity

    C. Derivatives of algebraic functions

    D. Derivative rules when combining functions

    E. Applications to velocity and acceleration

    F. Derivatives of trigonometric functions

    G. The chain rule

    H. Implicit derivatives

    1. Differential method

    2. y' method

    I. Derivatives of inverse trigonometric functions

    J. Derivatives of logarithmic and exponential functions

    Unit 4: Applications of the Derivative (4 weeks)

    A. Extreme values

    1. Local (relative) extrema

    2. Global (absolute) extrema

    B. Using the derivative

    1. Mean value theorem

    2. Rolle’s theorem

    3. Increasing and decreasing functions

    C. Analysis of graphs using the first and second derivatives

    1. Critical values

    2. First derivative test for extrema

    3. Concavity and points of inflection

    4. Second derivative test for extrema

    D. Optimization problems

    1. Max/Min Project

    E. Linearization models

    F. Related rates

    1. Tootsie Roll Pop Lab

    Unit 5: The Definite Integral (3 weeks)

    A. Approximating areas

    1. Riemann sums

    2. Trapezoidal rule

    3. Definite integrals

    B. The Fundamental Theorem of Calculus

    (part 1)

    C. Definite integrals and antiderivatives

    1. The Average Value Theorem

    D. The Fundamental Theorem of Calculus (part 2)

    Unit 6: Differential Equations and Mathematical Modeling (3-4 weeks)

    A. Antiderivatives

    B. Integration using u-substitution

    C. Separable differential equations

    1. Growth and decay

    2. Slope fields

    3. General differential equations

    Unit 7: Applications of Definite Integrals (3 weeks)

    A. Summing rates of change

    B. Particle motion

    C. Areas in the plane

    D. Volumes

    1. Volumes of solids with known cross sections, and project

    2. Volumes of solids of revolution

    a. Disk method

    b. Shell method

    This schedule leaves 4–6 weeks for flexibility with teaching and learning time

    management.

    Teaching Strategies

    Students better understand the concepts of calculus when they see concrete applications.

    Students are encouraged to participate in ACE (architecture, construction, and

    engineering program). The students are exposed to real world applications of calculus.

    During the first few weeks, we spend extra time familiarizing students with their

    graphing calculators. Students are taught the rule of three: Ideas can be investigated

    analytically, graphically, and numerically. Students are expected to relate the various

    representations to each other.

    It is important for them to understand that graphs and tables are not sufficient to prove an

    idea. Verification always requires an analytic argument. Each chapter exam includes one

    or two questions that involve only graphs or numerical data.

    I believe it is important to maintain a high level of student expectation. I have found that

    students will rise to the level that I expect of them. A teacher needs to have more

    confidence in the students than they have in themselves.

    We also stress communication as a major goal of the course. Students are expected to

    explain problems using proper vocabulary and terms. Like many teachers, I have students

    explain solutions on the board to their classmates. This lets me know which students need

    extra help and which topics need additional reinforcement.

    Students better understand the concepts of calculus when they see concrete applications.

    Much of calculus depends on an understanding of a concept taught in a previous lesson.

    Students are encouraged to form study groups and tutor themselves.

    Calculator Ideas

    The graphing calculator is used to help students develop an intuitive feel for concepts

    before they are approached through typical algebraic techniques.

    I use the calculator as a tool to illustrate ideas and topics. I stress the four required

    functionalities of graphing technology:

    1. Finding a root

    2. Sketching a function in a specified window

    3. Approximating the derivative at a point using numerical methods

    4. Approximating the value of a definite integral using numerical methods

    Activities

    The following sample activities demonstrate ways to help students gain an increased

    understanding of calculus.

    Limits

    If your calculator has a “table” feature, it can be used to zoom in on a

    limit numerically. For example, to find

    we view the values of the function from x-values from 1.5 to 2.5 with an increment step

    of 0.1. At x = 2 the table records “error” or “not defined.” Students should see that the yvalues

    seem to follow a pattern. Redo the process beginning at 1.9 with a step size of

    0.01, and observe that the y-values are converging to 0.25. The process can be repeated

    with smaller and smaller steps.

    The limit can also be shown visually by graphing the function in a window that has a

    pixel step of 0.1. Trace the function beginning at x = 1. Each step shows the

    corresponding x- and y-coordinates, but at x = 2, the y-coordinate disappears. It

    “reappears” when the tracing continues at x = 2.1. Students can see graphically that the ycoordinates

    cluster at about 0.25 as x is near 2.

    For comparison, do the same exploration with

    This function is also undefined at x = 2, but the y-values do not converge as x approaches

    2. Instead, the values explode, giving students a numerical look at asymptotic behavior.

    The Derivative of the Sine Function (This activity works well on an overhead display.)

    Graph the function y = sin x in a standard trigonometric viewing window. Estimate the

    slope of the tangent line at various x-values and plot the slope values as a function of x on

    the overhead screen. (The slope values are clearly zero at the turning points and can be

    estimated to be +1 or -1 at the x-intercepts. A few more estimates will enable students to

    guess the curve.) Students should see that the slope curve follows the path of the cosine

    function. To test this conjecture, graph the numerical derivative of the sine in the same

    window. Then graph the cosine function and note that the two graphs are superimposed.

    Tracing gives the same values on both curves. From this point it is easy to proceed to an

    analytic proof of

    Major Text

    Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus—

    Graphical, Numerical, Algebraic. 1st ed. Menlo Park: Scott-Forseman Addison-

    Wesley, 1999.