Maywood Academy High School Maywood, California 90270
AP Calculus AB Course Overview and Philosophy This syllabus has been written to provide evidence of meeting two basic requirements that this course must satisfy for the success of my calculus students: first, that the scope of this course covers all of the topics on the AP exam and; second, that this AP course consists of a full high school academic year of work that is comparable to calculus courses in colleges and universities. Further, the course I offer emphasizes a multirepresentational approach to calculus by employing the notion that problems, concepts, and results can often be expressed in a variety of ways or a combination of those ways. I expect my students to learn to express both questions and answers graphically, numerically, analytically, and verbally so that clear and useful connections can be made between what is needed and what is available when working with calculus.
Use of Technology Our school has purchased and placed on order enough TI-83 and TI-89 calculators so that these calculators are available to all of our calculus students both for class and home use. Additionally, I use the Texas Instrument projection adaptors, along with an overhead projector and a wire connected calculator, to project TI-83 and TI-89 screen demonstrations to the class. Because I incorporate classroom activities and exercises that require using these calculators, I expect my students to improve not only their level of calculator competency but also their ability to solve problems, experiment, interpret results, and support their conclusions. For example, my students learn through classroom activities that with the aid of these calculators that functions can be more easily and accurately viewed, especially for the purpose of graphical and numeric analysis. Further, because these calculators are programmable, I have a classroom activity in which students use a downloadable program that is used to observe and analyze slope fields. Nevertheless, my admonition to my students is that they keep in mind that these calculators can provide useful geometrical and analytic information but they cannot substitute for a basic understanding of calculus or fundamental computational abilities.
Student Activities with calculators A specific activity I employ to teach students the usefulness of the TI-83 in learning calculus is centered on slope fields. This activity was demonstrated to me at a summer AP week-long workshop. First, I have the students download the program that I received from my AP instructor. Then I demonstrate how to use the program using the overhead projector. My students then have the opportunity to follow the program and view the resultant change in the slope fields with each change they make with various differential equations. Or they can experiment by manipulating the original program to see different viewing possibilities while working with the 1
same differential equation. Finally, the students work with a partner to practice testing the other’s ability to identify a type of differential equation or the change to a differential equation. In this way, not only do students have the opportunity to view various slope fields using the TI-83 but also students see the usefulness of a program that they downloaded into the calculator.
Student Activities with computers Although facility with mathematical manipulation and computational competence both with and without calculators are important outcomes for all beginning calculus students, I also expect my students to make use of additional available technology. Our students are encouraged to use the computer and the Internet for research. In the first semester of calculus when students are introduced to the history and development of the calculus, I assign research projects that are meant to develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment. These projects are self-assigned as individual or group efforts, with a corresponding rubric of expectations and grading. Further, I recognize the computer as a special tutorial aid and means of obtaining information. Our school is currently trying to purchase a site license for the use of Hot Math, which is a program that will be used at our school for all levels of high school math. Nevertheless, Hot Math will be especially useful for our calculus students because of the on-line availability of written solutions to some of the problems from the calculus textbook we are currently using.
Teaching Strategies I teach and encourage my students to improve their communication of solutions to calculus problems both verbally and in written sentences. I have found that sometimes the ability to adequately communicate a solution is as important as arriving at the correct solution. One way I use to encourage verbal skills is to hold a classroom discussion over possible strategies in solving an open-ended calculus problem. I tell my students to put away their pens and pencils but encourage them to use a calculator if they desire. They are then encouraged to present their own individual strategies as I act as a facilitator, allowing everyone who cares to speak to share their opinion on alternate strategies. The intent is to promote critical thinking and mathematical understanding through verbal interpretation and response. But the best way I have discovered to encourage my students to write complete and literate explanations is to ask them to silently write their individual answers to a mathematical question that I have written on the board. When a reasonable amount of time has passed, I ask each one in turn to read their response aloud. Then as a class we review and discuss the merits of each response. I also use the above mentioned strategies with the students alternately working in groups of two, three or fours. Finally, I teach my students that communication of solutions must be correct and reasonable in terms of sign, size, relative accuracy, and units of measure. As a direct result class and small group discussions, along with practice writing sessions, my students are expected to be able to model a written description of a physical situation with a function, a differential equation, or an integral. 2
AP Calculus AB Course Outline Unit I. Precalculus Review 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 functions 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 functions D. Trigonometric functions 1. Graphs of basic trigonometric functions a. Domain and range b. Transformations c. Inverse trigonometric functions 2. Applications Unit II. Limits and Continuity A. Rates of change B. Limits at a point 1. Properties of limits 2. Two-sided 3. One-side 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 Unit III. The Derivative A. Definition of the derivative B. Differentiability 1. Local linearity 3
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. The derivative as function method I. Derivatives of logarithmic and exponential functions J. Second derivatives Unit IV. Applications of the Derivative A. Extreme values 1. Local and relative extrema 2. Global and absolute extrema B. Using the derivative 1. Mean value theorem 2. Rolle’s theorem 3. Increasing and decreasing theorems 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 E. Linearization models F. Related rates Unit V. The Definite Integral A. Approximating areas 1. Riemann sums 2. Trapezoidal rule 3. Definite integrals B. The Fundamental Theorem of Calculus, parts I and II C. Definite integrals and antiderivatives 1. The Average Value Theorem Unit VI. Differential Equations and Mathematical Modeling A. Antiderivatives B. Integration using u substitution C. Separable differential equations 1. Growth and decay 2. Slope fields 3. General differential equations
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Unit VII. Applications of Definite Integrals A. Summing rates of change B. Particle motion C. Areas in the plane D. Volumes 1. Volumes of solids with known cross sections 2. Volumes of solids of revolution a. Disk method b. Shell method
Major Textbook Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus eighth edition. Boston: Houghton Mifflin, 2006.
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