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AP Calculus AB - Syllabus Teacher: Eduardo Ramirez School: Jefferson HS District: EPISD Course Requirements 1) All students are expected to learn the material detailed in the Course Outline so that they are well-prepared for college-level mathematics and the AP Exam in May a) Students are expected to participate in preparatory mock-exams as well as the actual AP exam b) A variety of resources exist at the district and school level to insure that all students have the opportunity to take the exam 2) All students are expected to understand and explore the "Rule of 4" which provides for the representation of functions in various ways including graphically, numerically, analytically, and verbally a) Students will be given a wide range of problems demonstrating their knowledge of how to work with each type of representation as well as the connections that exist between the different representations b) Students will use TI-83/84 type calculators on a daily basis to explore the relationships between numeric data, formulas, and their corresponding graphs 3) All students are expected to be able to communicate mathematical concepts and describe solutions to problems both orally and in writing a) Students will work extensively in small groups where they will be expected to collaborate with their peers as they develop their understanding of key mathematical concepts b) Students will be expected to keep a journal of notes for key concepts taken in Cornell note form where students will be expected to write a summary using complete sentences of the math concept being explored c) Students will be given homework assignments beginning early in the year where they will be expected to develop the necessary skills to successfully answer freeresponse questions from previous AP tests by explaining and justifying their answers with a written response 4) The graphing calculator will be used to aid the student in developing a perceptive feel for the concepts before they are approached through typical algebraic methods. All students are expected to use a TI-83/84 type calculator on a daily basis for use in solving problems, experimenting with graphs and solutions, exploring the reasonableness of results, and verifying answers. Students will also be expected to identify relations between the graphs of functions and their analysis, and conclusions about the behavior of functions when using a graphing calculator. Conclusions can be in written form and in mathematics vernacular. a) Students will have access to school supplied TI-83/84 calculators for use in class and at home b) Students are free to use their own graphing calculator provided that it meets the requirements for use on the actual AP Calculus exam c) Students will be assessed in a manner similar to that of the AP exam with students being expected to answer questions which are calculator-active and questions which are calculator-inactive d) Students will be able to use the calculator to regularly perform the following: i) Plot the graph of a function using an appropriate window, including the ability to zoom in on a point to demonstrate properties such as local linearity ii) Find the zeros of functions iii) Numerically calculate the derivative of a function iv) Numerically calculate the value of a definite integral 5) It is strongly desired that all students will develop an appreciation of calculus as a truly amazing human accomplishment as it provides a cohesive mathematical approach to solving problems Textbook Anton, Bivens, and Davis. Calculus early trancedentals-single variable: Eighth Edition. Danvers, MA: John Wiley and Sons, 2005. Students will be given a copy of this textbook for use in the course. The Course Outline below references specific sections in the textbook. While this textbook is our primary source of material for this course, it will be supplemented as necessary and appropriate to enhance the experience of learning calculus. Course Outline Unit I: Functions, Graphs, and Limits (20 days) A) Analysis of Graphs (4 Days) i) Interplay between the geometric and analytic information ii) Use of calculus to predict and explain observed local and global behavior B) Limits of Functions (Including one-sided limits)(9 Days) i) An intuitive understanding of the limiting process ii) Calculating limits algebraically iii) Estimating limits graphically and numerically C) Continuity as a Property of Functions (4 Days) i) An intuitive understanding of continuity using sufficiently close domain values ii) Understanding continuity in terms of limits iii) Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem) D) Asymptotic and Unbounded Behavior (3 Days) i) Understanding asymptotes graphically ii) Asymptotes and limits at infinity Unit II: Derivatives (60 days) A) Concept of Derivative (8 Days) i) Derivative presented graphically, numerically, and analytically ii) Relationship between differentiability and continuity iii) Derivative as the limit of the difference quotient iv) Derivative as an instantaneous rate of change B)Derivative at a Point (6 Days) i) Slope of a curve at a point including vertical tangents ii) Instantaneous rate of change as the limit of average rate of change iii) Approximate rate of change from graphs and tables iv) Tangent Line to a curve at a point and local linear approximation C)Computation of Derivatives (10 Days) i) Derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions ii) Basic rules for the derivative of sums, products, and quotients iii) Chain Rule and implicit differentiation D) Derivative as a Function (10 Days) i) Curve sketching and corresponding characteristics between f and f by finding critical values ii) Mean Value Theorem and its graphical and analytic implications iii) Relationship between the increasing and decreasing behavior of f and f iv) Equations involving derivatives E) Second Derivative (4 Days) i) Corresponding characteristics of f , f , and f ii) Relationship between sign of second derivative and concavity iii) Points of inflection as a location where concavity changes with many examples and counterexamples (no sign change) F) Applications of Derivatives (22 Days) i) Analysis of curves, including the notions of monotocity and concavity ii) Optimization, absolute and relative extrema iii) Modeling rates of change including related rates problems iv) Use of implicit differentiation to find the derivative of an inverse v) The derivative as a rate of change in applied context including velocity, speed, and acceleration vi) Geometric interpretation of differential equations via slope fields Unit III: Integrals (50 days) A) Numerical Approximations to Definite Integrals (6 Days) i) Use of Riemann sums (left, right, midpoint, and trapezoidal) B) Interpretations and Properties of Definite Integrals (6 Days) i) Definite integral as a limit of Riemann sums ii) Properties of definite integrals iii) Definite integral of the rate of change of a quantity over an interval 𝑏 ∫𝑎 𝑓 ′ (𝑥) = 𝑓(𝑏) − 𝑓(𝑎) C) Techniques of Anti-differentiation (6 Days) i) Anti-derivatives following directly from derivatives of basic functions ii) Anti-derivatives by substitution of variables D) Fundamental Theorem of Calculus (6 Days) i) Use of the Fundamental Theorem to evaluate definite integrals ii) Use of the Fundamental Theorem to represent a particular anti-derivative and the analytical and graphical analysis of functions so defined E) Applications of Anti-differentiation (6 Days) i) Finding specific anti-derivatives using initial conditions, including applications to motion along a line ii) Solving separable differential equations and using them in modeling with exponential growth and the equation y ky F) Applications of Integrals (20 Days) i) Physical, biological, and/or economic situations ii) Using the integral of a rate of change, to give accumulated change, area of a region, volume of a solid with known cross-sections, the average value of a function, and the distance traveled by a particle along a line Unit IV: Comprehensive Review (15+ days) A) Mock Exam B) Comprehensive Review of Limits, Derivatives, and Integrals using previously released tests Grading This class will utilize the approved grading scale for the district for all AP courses. A minimum of 6 daily grades will be given each grading period that constitutes 40% of a student's grade. Daily grades include daily assignments, homework, quizzes, written summaries and free-response questions, and presentations. A minimum of 3 test grades will be given each grading period that constitutes 60% of a student's grade. Test grades include major tests given at appropriate points during the Course Outline and any major projects given. Sample of Additional Materials AP Calculus Multiple Choice and Free Response Questions 2003 by D&S Additional materials collected during Summer AP Institute Computer generated images for visual representations of rotated solids using GeoGebra