AP CALCULUS AB SYLLABUS TEXT: Larson, R., Hostetler, R., Edwards, B., and Heyd, D. (2002) Calculus of a Single Variable. Boston: Houghton Mifflin. MAIN REFERENCE MATERIALS: Albert, B. & P. Hillis. Calculus Calculator Labs. Andover: Skylight Publishing. College Board Professional Development Workshop Materials Finney, Demana, Waits, Kennedy. Calculus, Graphical, Numerical, Algebraic. (2003) Prentice Hall, ISBN: 0-13-063131-0 Foerster, P. Calculus Concepts and Applications. Key Curriculum Press. ISBN: 0-471-48481-4 Kamischke, E.A Watched Cup Never Cools. Key Curriculum Press. ISBN: 1-55953-318-8 Lederman, D. Multiple Choice & Free Response Questions in Preparation for the AP Calculus (AB) or (BC) Examination. 8th Edition. D & S Marketing Systems, Inc. Lipp, Alan. Little Books of Big Ideas. The Peoples Publishing Group, Inc. ISBN:1-4138-1322-4 Course Description: 3113A AP Calculus Pre-requisites and/or Requirements: Calculus Honors This course is concerned with developing the students’ understanding of the concepts of differential and integral single variable calculus providing experience with its methods and applications. The course emphasizes a multipresentational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The course represents a college-level mathematics. The students will be expected to participate in the corresponding advanced placement exam. COURSE PLANNER: Unit 1: Prerequisites for Calculus (3 weeks) Review solving equations and using the graphing calculator o Intercepts, symmetry, domain, range Linear models and rates of change Functions and graphs o Linear, power, inverses, exponential, logarithmic, trigonometric, inverse trigonometric, piecewise and composite functions o Transformations of functions and corresponding graphs o Analysis of graphs Unit 2: Limits and Continuity (3 weeks) Finding limits graphically and numerically Evaluating limits analytically Continuity and one-sided limits Infinite limits AP - Limits of functions (including one-sided limits). o An intuitive understanding of the limiting process o Calculating limits using algebra o Estimating limits from graphs or tables of data AP - Asymptotic and unbounded behavior o Understanding asymptotes in terms of graphical behavior o Describing asymptotic behavior in terms of limits involving infinity o Comparing rates of change of exponential, polynomial and logarithmic functions AP - Continuity as a property of functions o An intuitive understanding of continuity o Understanding continuity in terms of limits o Geometric understanding of graphs of continuous functions o Intermediate Value Theorem and Extreme Value Theorem Unit 3: Differentiation (6 weeks) The Derivative and the Tangent Line Problem Discover the Derivative Function from a Table of Values and by Exploration Basic Differentiation Rules and Rates of Change The Product and Quotient Rules and Higher-Order Derivatives The Chain Rule Implicit Differentiation Related Rates Derivatives of Exponential and Logarithmic Functions AP – Concept of the Derivative o Derivative presented graphically, numerically, and analytically o Derivative interpreted as an instantaneous rate of change o Derivative defined as the limit of the difference quotient o Relationship between differentiability and continuity AP - Derivative at a point o Slope of a curve at a point o Tangent line to a curve at a point o Instantaneous rate of change as the limit of average rate of change o Approximate rate of change from graphs and tables of values AP - Computation of derivatives o Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric and inverse trigonometric functions o Basic rules for the derivatives of sums, products, and quotients of functions o Chain rule and implicit differentiation Unit 4: Applications of Derivatives (4 weeks) Extrema on an Interval Rolle’s Theorem and the Mean Value Theorem Increasing and Decreasing Functions and the First Derivative Test Concavity and the Second Derivative Test Limits at Infinity A Summary of Curve Sketching Optimization Problems Newton’s Method Differentials o Linear Approximation AP – Derivative of a function o Corresponding characteristics of f, f’, and f’’ o Relationship between the increasing and decreasing behavior of f and the sign of f’ o The Mean Value Theorem o Equations involving derivatives. Verbal descriptions translated in equations involving derivatives and vice versa. AP – Derivative at a point o Local linear approximation AP – Second derivatives o Corresponding characteristics of the graphs f, f’, and f’’ o Relationship between concavity of f and the sign of f’’ o Points of inflection as places where concavity changes AP – Applications of Derivatives o Analysis of curves, including the notions of monotonicity and concavity o Optimization, both absolute (global) and relative (local) extrema o Modeling rates of change, including related rates problems o Use of implicit differentiation to find the derivative of an inverse function o Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration Unit 5: Integration (4 weeks) Antiderivatives and Indefinite Integration Area Riemann Sums and Definite Integrals The Fundamental Theorem of Calculus Integration by Substitution AP - Interpretations and properties of definite integrals o Computation of Riemann Sums using left, right, and midpoint evaluation points o Definite integral as a limit of Riemann Sums over equal subdivisions o Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: b f ( x)dx f (b) f (a) a o Basic properties of definite integrals AP – Applications of Integrals AP – Fundamental Theorem of Calculus o Use of the Fundamental Theorem to evaluate indefinite integrals o Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined AP – Techniques of antidfferentiation o Antiderivatives following directly from derivatives of basic functions Unit 6: Differential Equations and Mathematical Models (4 weeks) Numerical Integration; Trapezoidal Rule Antiderivatives and Slope Fields Integration of Logarithmic and Exponential Functions Exponential Growth and Decay Population Growth and Numerical Methods AP – Numerical Approximations to definite integrals o Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values AP – Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations AP – Applications of Integrals AP - Techniques of antidifferentiation o Antiderivatives by substitution of variables (including change of limits for definite integrals) AP – Applications of antidifferentiation o Solving separable differential equations and using them in modeling (i.e. studying the equation and exponential growth) Unit 7: Applications of Integration (3 weeks) Area of a Region Between Two Curves Volume: The Disk Method Volume: The Shell Method Arc Length and Surfaces of Revolution AP – Applications of integrals o Finding specific antiderivatives using initial conditions, including applications to motion along a line o Solving separable differential equations and using them in modeling (i.e. studying the equation and exponential growth) REVIEW FOR THE AP EXAM Beginning in January, I assign an AP free-response question for students to work on every week. Students may work on these questions with one other person and come to me for extra help. I randomly select from these questions an AP free-response question on every test. Students are graded as they would be graded on an AP Exam. Every test beginning in January will consist of a multiple-choice section of AP Calculus Released Exams. Times allotments should allow ample weeks of review for the AP exam, which is scheduled for the 2nd week of May. The review will consist of sessions with parts of previous exams. This will vary between calculator and non-calculator problems from the multiple choice and free response sections. The review period will also include a mock scenario using the 2003 exam in order to simulate the actual conditions of the exam. STUDENT EVALUATION Grades for this course will be based on two categories: 1. Quizzes, homework, classwork, and projects (40% of each grading period average). Daily grades will be taken from homework, classwork, and bell ringers. Short quizzes (25 points) cover the previous day’s homework; longer quizzes (50 points) cover 1 or 2 concepts or book sections. Labs (examples follow) usually count 50 points; projects (examples follow) count 100 points. 2. Tests and 9-week examinations (60% of each grading period average). Chapter tests count 100 points and are given at the end of each chapter. The 9-week exam counts the same as a chapter test. The exams given at the end of the first and second 9-week grading periods will test the material covered during that grading period. The exam given at the end of the third grading period will be comprehensive and use AP style questions. The fourth grading period exam will be given after the AP exam and will cover any additional topics covered that grading period. Quizzes, tests and exams will feature both calculator and non-calculator sections. LABS/PROJECTS The following is a brief description of several of the homework/projects assigned: 1. To review functional transformations a group project in the form of bingo is used. A list of functions is provided and the class chooses 25 at random to add to their bingo sheet. The corresponding graphs are then shown using the projector. The class is expected to match the graphs appropriately based on their knowledge of functions, graphs, and transformations. 2. Activity #2 is repeated using only graphs of functions and their corresponding graphs of the first derivative and second derivative. 3. Class activity involving rules of differentiation. Again, bingo is used, but in this case, the boxes are already completed with the derivatives and the students are provided a list of functions. 4. Discover the derivative by exploration. ln( x h) ln( x) Exploration 1. Let Y1 = . Set up the Table: Tbl Start 0; ΔTbl = 1 h Looking at the Table, can you guess the derivative of y = ln(x)? x e x 0.001 e x Exploration 2. To discover the derivative of y =e , graph Y1 = . Use 0.001 ZOOM4. Can you determine the derivative function? Exploration 3. We want to investigate the derivative of y = sin x, using one of the f ( x h) f ( x ) sin( x L1 ) sin x definitions of the derivative, lim . Enter Y1 = . h 0 h L1 Press STAT, EDIT, and enter 5, 1, 0.1,and 0.001 in L1. Use ZOOMTRIG to graph Y1. Each value in L1 is the “h” value in the derivative definition, so there will be four graphs displayed. As the “h” values approach 0, what does the graph of Y 1 resemble? You can graph Y2 = cos x simultaneously with Y1. Does this confirm your guess? 5. Students explore the concept of average rate of change and discover the concept of instantaneous rate of change in a ball toss lab. We use the Vernier computer interface, Logger Pro, Vernier Motion Detector, or the Calculator-Based Laboratory. The students examine the distance-verses-time graph. They fit the data with a quadratic equation for the position function to determine how high the ball traveled and time traveled. The students calculate the average velocity of the ball over a period of time. Students are then asked to investigate the exact velocity at different points in time. Depending on the use of either the Vernier computer interface and Logger Pro or CBL the students either trace or zooms in on the position functions graph to find the slope of the line and compare it to their estimation of the instantaneous velocity. 6. During the study of related rates, students suck on Tootsie Roll Pops to determine the rate of change of their radius; they then calculate the rate of change of the Pops’ volume. Students measure the initial radius of a Pop with dental floss. They then suck on the Pop for 30 seconds, record its radius, suck for another 30 seconds, etc. They model the rate of change of the radius with some function of time. Students then use this rate of change to estimate the rate of change of the volume of the Pop when its radius is three-fourths of its original radius. This lab, “How Many Licks?,” can be found in Ellen Kamischke’s book, A Watched Cup Never Cools. 7. Optimization Lab - Working with a variety of materials, some students will form prisms and pyramids of maximum volume. First, the maximum volume possible will be calculated, then the solid will be formed. Other students will calculate and form solids of minimum surface area which would contain a given volume. A short paper will be written describing the process. 8. As an exploration problem and introduction to optimization problems, the students are asked to form a 5-sided box from a standard sheet of poster board with a maximum volume. 9. Volume Lab -- Working with a partner, the student will build a model of a solid formed by rotating a function around an axis. The model will be formed by shells or disks. The volume of the model will be calculated. A short paper will be written discussing the project and findings. Technology Each of my students must have a graphing calculator of his or her own. We use graphing calculators daily in class to introduce, explore, discover, and reinforce the concepts of calculus; therefore, the students are expected to have it in class each day. The calculator is also used to analyze data, analyze graphs, and compare derivatives. The majority of students either use a TI-83 Plus, TI-84, or TI-89. We begin using the graphing calculators with the review of many functions. We investigate roots, domain, range, minimums, maximums, and continuity using the different features of the graphing calculator. We continue the use of the calculator to investigate limits. We use the table and trace features to view the graphs and make the connection of the limit graphically, numerically, analytically, and verbally. The students use the calculator when guessing the derivative function from a table of values. Before the students learn the rules for differentiating the elementary functions, the students take some simple functions and create a small table of values for f(x), say at x = 0, 1, 2, 3, then uses the difference quotient to approximate f’(x). The students look at the values in the table, try to see the pattern, and discover the equation for f’(x). The students use the graphing calculator to explore and discover for themselves certain derivatives. (lab/project #4) Students explore the concept of average rate of change and discover the concept of instantaneous rate of change as described in the ball toss lab (#5 lab/project). We explore the position function verses velocity function verses acceleration function in a simple pendulum lab similar to the ball toss lab in the investigation of the sine and cosine functions. We continuously use the graphing calculator in making connections between f, f’, and f’’ in tables and in the graphs. Investigating functions differentiable at x = a as well as not differentiable at x = a will be included algebraically and graphically. Algebraically and graphically the students will investigate and confirm increasing/decreasing behavior using f’. Students use graphing technology exploring Riemann sums. Although the investigation of area is the basic context we include regions below the axis as well as those above. Students discover that Riemann sums are not always good predictors of the answer if area is the question.