Mr. Olson AP Calculus (BC) 2014 Syllabus Course Overview

Transcription

1 Mr. Olson AP Calculus (BC) 2014 Syllabus Course Overview The objective of this course is to enable students to understand and apply the concepts of calculus in a variety of situations. All students should achieve an appreciation of the logical, sequential, development of calculus in theory and in practice. All students should feel comfortable using this knowledge in situations that are graphical, algebraic, practical, and theoretical. All students should be able to use the correct terminology and notation to develop and articulate their responses to various prompts. Written forms for work are expected to be expressed using full descriptive sentences as opposed to merely symbolic, abbreviated, or fragmented statements. Problem-solving, individual and group work, and labs are an essential part of their learning experience. Activities will often involve cooperative learning, small groups activities. A significant component of class work will require the formal presentation of work in written form as well as verbal form to the class. A component of the course will require students to watch video tutorials outside of class, take notes, and answer simple comprehension questions in order to be prepared for class. Students are required to have a TI-83 (Plus) or TI-84 (Plus) calculator for this course. Our text will be Stewart, James. Calculus Concepts & Context. 4 th ed. Thomson Brooks/Cole, Unit 1 Functions and Models. Introduction to Limits (7 days) 1. Four ways to represent a function 2. Mathematical models essential functions 3. New functions from old translations, algebraic formation, composite functions * 4. Using the graphing calculator and computer graphing software *, ** 5. Exponential functions 6. Inverse functions and logarithms 7. Parametric curves 8. Polar and vector functions 9. Tangent and Velocity Problems 10.The Limit of a Function * Student Activities for Unit 1 Exploration #1: Door closer finding average and instantaneous rates of change. Graphing models sketch a graph of power consumption at home M-Fr, and on the weekend. Given a graph provide a verbal description *** Given a graph write a piecewise definition Exploration #2: Sketch graph of function and describe rate of change at specific x value. Which is the original? Given set of translated graphs find the original f(x) Given graphs label the translation

2 Determine values of f(g(x)) given the graphs of f(x) and g(x) Introduction to limits of functions. Geometrically looking at tangent line using graphing calculator, estimating slope by zooming, discuss meaning of instantaneous velocity in different situations, analyze f(x) = x 4-3x 2 - function increasing, decreasing, concave up and down, what is occurring with slopes of tangent lines What is the pattern? - find slopes of secant lines, moving points closer and then estimate the slope of the tangent line - using f (x) = x + 1 Unit 2 - Limits and Derivatives (10-12 days) 1. Calculating limits using limit laws, squeeze theorem, graphs, and tables *, ** 2. Continuity Definition - composite functions - Intermediate Value Theorem 3. Limits involving infinity - horizontal and vertical asymptotes as limits, limits of different types of functions as x infinity *, ** 4. Tangents, instantaneous velocity and other rates of change 5. Derivatives at a point, tangent lines, derivative as a rate of change 6. Derivatives as a function, notation, differentiability and continuity, definition as the limit of a difference quotient, higher derivatives 7. Relation between the derivative and function, increasing and decreasing, maximum and minimum, concavity, inflection points, anti-derivatives Student Activities for Unit 2 x+16 4 Class work find limit of f (x) = x graphically, numerically, and algebraically. Exploring limits - looking at limits of sum and product of functions that are not continuous Limits of functions with holes and asymptotes. Investigating the Intermediate Value Theorem - understanding why continuity is required. Making a piecewise defined function continuous Limits involving Infinity To Infinity and Beyond - develops students intuition about infinite limits Order from chaos - from a graph ordering the values of tangent slopes and secant slopes Derivatives and Inverses - meaning of derivatives in real world *** Tangent lines and the derivative function The derivative function - given graphs of functions students required to sketch the graph of the derivative Sorting them out - given three graphs determine which the function, the derivative, and the second derivative The major curve pieces- what can be concluded about the function, its derivative, and its second derivative.

3 Unit 3 Differentiation Rules. (12-15 days) 1. Derivatives of polynomial and exponential functions 2. Product and quotient rules product rule derived algebraically and geometrically 3. Rates of change in natural and social sciences *** 4. Derivatives of trigonometric functions 5. Chain rule tangents to parametric curves 6. Implicit differentiation - derivatives of inverse trig functions and inverse function in general. 7. Derivatives of logarithmic functions 8. Linear approximations and differentials Student Activities for Unit 3 Group Work: Doing a lot with a little - practice with power rule and foreshadowing of the chain rule. Major project and presentation for the students. Students divide into groups and research applications physics, chemistry, biology, and economics and make graded presentations to class explaining applications. *** Group Work: Back and Forth - students find derivatives of different functions and have their partner try to determine the original functions. Group Work: Students find derivatives involving product and quotient rules with only information from a table. Group Work: Follow that particle - students analyze the motion of a typical f (x) = e t (5 t) 5 particle. Group Work: The magnificent six - having found the derivatives of sine and cosine as a class, the students determine the derivatives of the other four trig functions. Group Work: Using our new knowledge - tangent lines, vertical asymptotes Group Work: Unbroken chain - introduction to the mechanics of finding derivatives using the chain rule. Group Work: Chain rule without formulas - determining derivatives using the chain rule with only the graphs of f(x) and g(x) Lab: Linking up with the chain rule - further explorations of the chain rule Group Work: Looking for the minimum - practice with derivative of arcsine * Group Work: Logarithmic differentiation - differentiation of functions of the form: f(x) g(x) Group Work: e as a limit * Group Work: Practice with linear approximation - allows students to see how well they understand the geometry of linear approximation ** Lab - Taylor polynomial - students find Taylor approximations for the function and graph results using calculators

4 Unit 4 Applications and Computation of Derivatives. (10-12 days) 1. Related rates 2. Absolute and local maximum and minimum problems - Extreme Value Theorem, Fermat s Theorem, closed intervals 3. Derivatives and the shapes of curves - Mean Value Theorem, first and second derivative tests 4. Indeterminate form and L Hospital s Rule 5. Optimization problems - absolute extreme values 6. Applications to Economics *** 7. Newton s method ** 8. Anti-derivatives, slope fields, position, velocity and accelerations 9. Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration 10.Numerical solution of differential equations using Euler s method 11.Derivatives of parametric, polar and vector functions. 12.Use the calculator to find numerical derivative, graph a derivative, find zeros, maximum and minimum points, and estimate inflection points of a function *, ** Student Activities for Unit 4 Group Work: Nobody escapes the cube - basic related rate problem Group Work: The swimming pool - related rate filling a swimming pool Group Work: What does the derivative say - use graph of derivative (first & second) to gain information about a function Group Work: The little dip -analyzing a function whose graph does not clearly indicate all properties. Group Work: Graphing with the derivative - graphing equations using the derivatives. Group Work: Finding the error - using L Hospital s Rule Group Work: The waste free box - open top box optimization problem Group Work: A shortcut - maximizing area of triangle with two equal sides. Calculus in England -maximizing the volume of a cone Group Work: An unusual cost function - minimum marginal cost and minimum average cost. Group Work: The problem of Archimedes - cutting a sphere with a plane in order to have one volume half the other. Group Work: Direction fields - students sketch a graph for the direction field of a given function and then given direction fields, graph at least two different anti-derivatives. Group Work: Velocity Vectors (from Chapter 10) Group Work: Anti-differentiation formulas - activity to introduce students to anti-differentiation. Group Work: A strange antiderivative - students produce an anti-derivative of a natural piecewise defined continuous function. AP EXAM PRACTICE (Differential Calculus Only) 3 to 5 days

5 Unit 5 Integrals. (10-12 days) 1. Area and distance Riemann sums * 2. The definite integral limit of Riemann sum, properties, use Riemann sum to approximate definite integral algebraically, geometrically, and by table of values 3. Evaluating definite integrals ** 4. The fundamental theorem of calculus, use to represent a particular anti-derivative, analytical and graphical analysis of function 5. Integral of a rate of change represents the accumulated change 6. The substitution rule - changing limits 7. Integration by parts, choosing u and dv 8. Simple partial fraction integration (only non-repeating linear factors) 9. Approximate integration, midpoint rule, trapezoidal rule, and Simpson s rule 10.Using L Hospital s Rule to determine limits and convergence of improper integrals 11.Using technology to find area under a curve and evaluate definite integrals *, ** Student Activities for Unit 5 Group Work: Choose your weapon - groups estimate the area under a function using left endpoints, right endpoints, midpoints and any point in interval * Group Work: Two easy pieces - use left and right endpoints Riemann sum approximations for a piece-wise defined function Group Work: The area function - foreshadowing for the Fundamental Theorem of Calculus Group Work: Exploring definite integrals -finding definite integrals when given some definite integral values and when given a graph. Group Work: Clearing the hill - a car climbing and descending a hill Group Work: Some different integrals - integrals of some interesting functions Group Work: Integration potpourri - function defined by a definite integral Group Work: What is the fundamental theorem of calculus Group Work: Finding the error - deeper understanding of the FTC Group Work: Where do these areas go in the limit - computing a limit using FTC and L Hospital s Rule Group Work: An application - another application of FTC Group Work: Unexpected equality - definite integrals and substitution Group Work: Some surprising areas - examining area under f (x) = 1 Group Work: Loki s dilemma Group Work: Guess the method - integrations involving parts and substitution. Group Work: Find the error and sequel - finding errors in definite and indefinite integrations. Group Work: Comparison of methods - left, right endpoints and trapezoidal methods xln(x)

6 Unit 6 Applications of Integration. (6-8 days) 1. More about areas between curves - using Simpson s rule with tables and graphical situations - parametric curves - variety of examples of applications 2. Finding areas using polar curves 3. Volumes, known cross section, revolution disk, washer variety of examples of applications 4. Average value of a function - Mean Value Theorem for Integrals 5. Applications to Economics and Biology *** Student Activities for Uint 6 Group Work: Practice with area - sketch regions and setting up definite integrals. Group Work: The revolution will not be televised - practice with regions rotated about different lines Group Work: Geometric volume - cross sectional areas of familiar shapes. Group Work: Average value theory and practice Group Work: Choosing a bank - average value and banking Group Work: Picture Pages (Chapter 10) Group Work: Calculus by the Light of the Moon (Chapter 10) Group Work: The Weighty Chain, Homer s Blood, Foretelling the Future Unit 7 Differential Equations. (6-8 days) 1. Modeling with differential equations, initial values 2. Direction fields and Euler s method, calculator program for Euler s ** 3. Logistic differential equations and modelling 4. Separable equations 5. Exponential growth and decay Student Activities for Unit 7 Group Work: How do solutions behave - a look at solutions to differential equations. Group Work: Solutions to differential equations - stresses idea of family of solutions. Group Work: Fun with differential equations - exploration of differential equations. Group Work: Direction fields - practice drawing directions fields. Group Work: Euler with care - applying Euler s method, seeing some of the dangers. Group Work: Determining whether differential equations are separable. Group Work: The coffee window - a look at Newton s laws Group Work: We want information - spread of information Group Work: The rule of 72 - doubling an investment Group Work: More of Homer s blood - rate of change of sedative in the blood. Group Work: Find the error Group Work: Growing at Half Capacity Group Work: The Modified Logistic Model

7 Unit 8 Polynomial Approximation and Series. (6-8 days) 1. Sequences and Series *, ** 2. The Integral and Comparison Tests: Estimating Sums 3. Alternating Series Tests 4. Absolute Convergence 5. Ratio Test 6. Power Series 7. Taylor and Maclaurin Series 8. Lagrange Error Student Activities for Unit 8 Group Work: Practice with Convergence Group Work: Made in the Shade Group Work: How Do I Compare? Group Work: The Three Conditions Group Work: From Power Series to Polynomials Group Work: The Secret Function In sections marked with an *, students are expected to use their graphing calculators to experiment in order to come to an appropriate solution. In sections marked with an **, students are expected to use their graphing calculators to interpret results and support conclusions. In sections marked with an ***, students are expected to explain solutions to problems in written sentences. Review for AP (5-8 days) Students review past AP problems and grading rubrics to become familiar with types of questions they will encounter on the exam. Students also take a timed practice AP and then a full timed AP, which is graded.

8 Student Evaluations: Grades are determined using tests and quizzes worth 60%, homework and projects worth 20% of the semester grade and 10% of the grade for class participation. Homework is graded on completeness and quality. Tests and quizzes can involve both multiple choice questions and free-response questions. Calculators may be used all tests and quizzes. Students will also present solutions to homework problems to help reinforce their ability to explain mathematical concepts. A complete timed AP is used as the student s final graded activity in the course. Major Text: Stewart, James. Calculus Concepts & Context. 4 th ed. Thomson Brooks/Cole, Supplementary Materials: Barron s: AP Calculus (9 th Edition), Barron s Educational Series, Inc Shaw, Douglas, Keynes, Harvey, Stewart, James, Hall, John. Instructor s Guide for AP Calculus for Stewart s Calculus Concepts and Context, 3 rd,ed. Thomson Brooks/Cole, 2005 Solow, Anita, Editor, Learning By Discovery A Lab Manuel for Calculus, Mathematical Association of America, Straffin, Philip, Editor, Applications of Calculus, Mathematical Association of America, Fraga, Robert, Editor, Calculus Problems For A New Century, Mathematical Association of America, Foerster, Paul, Calculus Explorations, Key Curriculum Press, Hockettt, Shirley, Bock, David, Barron s How to Prepare for AP Calculus. Baron s Educational Series Lipp, Alan, Visualizing Differential Equations with Slope Fields. People s Publishing Group, Antinone, Linda, Dick, Thomas, Fitzpatrick, Kevin, Grasse, Michael, Howell, Mark, TI-84 Plus / TI-83 Plus Explorations. Texas Instruments, 2004.