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 . h0N 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.