AP Calculus AB
2012-2013
Syllabus
Overview of Advanced placement Calculus AB Most students take Calculus in either their junior or senior year. Most AP Calculus students are easily identified early in their high school career by their algebra teachers. These teachers encourage the students to plan a course sequence that will carry them through calculus. The majority of students follow a standard track of Geometry, Algebra 2 and Pre-Calculus. 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 student 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, indefinite integrals, and definite integrals. As we develop the concepts, we explain how the mechanics go along with the topics. Teaching Strategies We believe it is important to maintain a high level of student expectation. We have found that students will rise to the level that we 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. The students work in cooperative groups which allows students to communicate mathematically with their peers. Students are expected to explain problems using proper vocabulary and terms. Like many teachers, we have students explain and present their solutions on the board to their classmates. This lets us know which students need extra help and which topics need additional reinforcement. During the daily starter students are expected to come to the board to present the problems. All students are given a chance. Individual white boards make presentation of quick problems easy. Students are expected to be able to write about mathematics. In section 2.3 I assign #60 as journal writing. x 1 Let f x 1 x a) Find the domain b) Draw the graph of f c) Writing to Learn Explain why x=-1 and x=0 are points of discontinuity of f. d) Writing to Learn Are either of the discontinuities in part c removable? Explain. e) Use calculators, graphs and tables to estimate lim f x x
The most important key for students’ success is for them to do homework assignments each day. It is much easier to stay current than to catch up. Much of calculus depends on an understanding of a concept taught in a previous lesson. Fortunately, AP students are highly motivated and are willing to put forth the needed effort. They are encouraged to form study groups and tutor themselves. I let students correct their own homework assignments. This allows them to make notes on their work for later review.
Our textbook has many exploration activities that allow students to explore concepts graphically, numerically, and analytically. Calculator Use We will be using graphing calculators in class. I encourage every student to have a TI 83 or 84 PLUS. The graphing calculator is used to help students develop an intuitive feel for concepts before they are approached through typical algebraic techniques. As an example, at the beginning of the unit on slope fields students explore the SLOPESL program on their calculator. The calculator plots the slope field so we can talk about the family of solutions. While I expect students to understand how the mathematics relates to a problem, we do learn to use a calculator to solve problems. I stress the four required functionalities of graphing technology as defined by the College Board for AP testing: 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 Review and Preparation for the Exam Our text does a nice job of preparing students for the AP Exam. In addition we will practice using released items from previous exams throughout the year and the second semester final will be an exam made up of past AP questions.
AP Calculus AB Course Outline Unit 1: Prerequisites for Calculus (2 weeks). During this chapter students are instructed on basic calculator use. 1.1
Lines a. b. c.
1.2
Slope as rate of change Parallel and perpendicular lines Equations of lines
Functions and Graphs
a. b. c. d. e. f. g. 1.3
Exponential Functions (Oral group presentation to the class) a. b.
1.4
Relations Circles and ellipses Lines and other curves
Functions and Logarithms a. b. c.
1.6
Exponential growth and decay Inverse functions
Parametric Equations (Concepts worksheet 1.4 for graphic representation of parametric equations) a. b. c.
1.5
Functions Domain and range Families of function Piecewise functions Composition of functions Absolute value functions Even and odd functions
One-to-one functions Logarithmic functions Properties of logarithms
Trigonometric Functions a. b.
Graphs of basic trigonometric functions Applications
Unit 2: Limits and Continuity (2 weeks) 2.1
Rates of Change and Limits – Using the graphing calculator on the overhead projector we develop the intuitive meaning of limit. a. b. c. d.
2.2
Average and instantaneous speed Definition of limit – Bob Barefoot’s worksheet “Exploring the Concept of the Limit of a Function Intuitively” One-sided and two –sided limits Sandwich theorem
Limits Involving Infinity a.
Asymptotic behavior
b. c. d. 2.3
Continuity a. b. c.
2.4
End behavior Properties of limits Visualizing limits
Continuity at a point Continuous functions Discontinuous functions
Rates of change and Tangent Lines a. b. c.
Average rate of change Tangent to a curve Slope of a curve
Unit 3: Derivatives (6 weeks) The roller coaster project will be due at the end of the chapter. Students design a “roller coaster” ride. They need to come up with a piecewise function of 5 parts, so that the function is both continuous and differentiable at the endpoints of each interval. A complete piecewise function and correct graph is required. The project is to be presented to the class and the expectation is that the presentation will be creative and colorful. 3.1
Derivative of a Function a. b. c.
3.2
Differentiability a. b. c.
3.3
Local linearity Numeric derivatives using the calculator Differentiability and continuity
Rules for Differentiation a. b. c. d.
3.4
Definition of derivative Graphing the derivative from data One-sided derivatives
Positive integer powers, multiples, sums and differences Products and quotients Negative integer powers of x Second and higher order derivatives
Velocity and Other Rates of Change
a. b. 3.5
Derivatives of Trigonometric Functions a. b.
3.6
Derivative of the sine and cosine functions Simple harmonic motion
Chain Rule a. b. c. d.
3.7
Instantaneous rates of change Motion along a line
Derivative of a composite function Outside-inside rule Repeated use of the chain rule Power chain rule
Implicit Differentiation a. Differential method b. y ' Method
3.8
Derivatives of Inverse Trigonometric Functions a. b.
3.9
Derivatives of inverse functions Derivative of the arcsine, arctangent, and arcsecant
Derivative so Exponential and Logarithmic Functions a. b. c. d.
Derivative of Derivative of Derivative of Derivative of
ex ax ln x log a x
Unit 4: Applications of Derivatives (4 weeks) 4.1
Extreme Values of Functions a. b.
4.2
Mean Value Theorem a. b.
4.3
Local (relative) extrema Global (absolute) extrema
Physical interpretation Increasing and decreasing functions
Connecting f ' and f '' with the graph of f.
a. b. c. d. 4.4
First derivative test for local extrema Concavity Points of inflection Second derivative test for local extrema
Modeling and Optimization a.
Examples from business and industry, mathematics, economics and discrete phenomena with differentiable functions
4.5
Linearization and Newton’s Method
4.6
Related Rates a.
Related rate equations
Unit 5: The Definite Integral (4 weeks) 5.1
Estimating with Finite Sums a. b. c.
5.2
Definite Integrals a. b. c. d. e.
5.3
The Average Value Theorem Mean Value Theorem for definite integrals
Fundamental Theorem of Calculus a. b.
5.5
Riemann sums Terminology and notation of integration Definite integral and area Constant functions Integrals on a calculator
Definite Integrals and Antiderivatives a. b.
5.4
Distance traveled Rectangular approximation method (RAM) Volume of a sphere
Fundamental Theorem parts 1 and 2 Graphing
Trapezoidal Rule
Unit 6: Differential Equations and Mathematical Modeling (4 weeks)
6.1
Antiderivatives and Slope Fields a. b. c.
6.2
Integration by Substitution a. b. c. d.
6.3
Law of exponential change Newton’s law of cooling
Population Growth a. b. c.
6.6
Product rule in integral form Solving for the unknown integral Tabular integration
Exponential Growth and Decay a. b.
6.5
Power rule in integral form Trigonometric integrands Substitution in indefinite integrals Separable differential equations
Integration by Parts a. b. c.
6.4
Solving initial value problems Antiderivatives and indefinite integrals Properties of indefinite integrals
Exponential model Logistic growth model Logistic regression
Numerical Methods a. b. c. d.
Euler’s method Numerical solutions Improved Euler’s method Graphical solutions
Unit 7: Applications of Definite Integrals (3 weeks) 7.1
Integral as Net Change a. b.
7.2
Consumption over time Net change from data
Areas in the Plane
a. b. c. 7.3
Volumes a. b.
7.4
Volumes of solids with known cross-sections. Volumes of solids of revolution
Lengths of Curves a. b. c.
7.5
Area between curves Area enclosed by intersection curves Integrating with respect to y
Length of a smooth curve A sine wave Vertical tangents, corners, and cusps
Applications from Science and Statistics
End of course project. To allow for flexibility of teaching and learning and to prepare for the AP Exam there are 3 to 4 weeks left before the end of the semester.
References and Materials Major Text Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus - Graphical, Numerical, Algebraic. 3rd edition. Boston, Massachusetts: Pearson Educations, Inc., 2007. Reference Books Best, George, Sally Fischbeck. AP Calculus with the TI-83 Graphing Calculator. Andover, MA: Venture Publishing, 1998. Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus - Graphical, Numerical, Algebraic. Boston: Pearson Prentice Hall, 2003. Hughes-Hallett, Deborah, Andrew M. Gleason, William G. McCallum, et al. Calculus – Single Variable. 4th edition. Hoboken, New Jersey: John Wiley and Sons, Inc., 2005. Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable. Boston: Houghton Mifflin Company, 2006. Stewart, James. Calculus. 5th edition. Belmont, CA: Thomson Learning, Inc., 2003.
