AP Calculus Mrs. Stansbery Course Syllabus BHS Room 309 (360) 473-0875 sandra.stansbery@bremertonschools.org Classroom Expectations 1. Come to class on time and prepared to learn. Take care of locker visits, bathroom breaks, and personal business before class begins. Bring all materials to class each day.. Be respectful to yourself, other students, the teacher, and the classroom. Use appropriate language. Be courteous to others. Use classroom materials and equipment appropriately. 3. Follow all school rules and policies that are included in the student handbook. Materials Textbook Calculus of a Single Variable, to be checked out from the library. Notebook You will need either a spiral bound or composition notebook. A graph paper notebook is ideal, but not required. Pencil Graphing Calculator Preferred model is the TI-84 Plus. Grading Policy Unit tests will be given at the end of each chapter and account for 75% of your grade. o Unit tests may be re-taken once if proper arrangements are made with the teacher. o Re-takes will not be given during class time. o All tests must be completed in one sitting and within the allotted time. Quizzes will be given approximately once a week and account for 15% of your grade. o Quiz corrections may be submitted up until the scheduled date of the unit test. Skills for Success will be assessed regularly and account for 10% of your grade. **Students may make up any assignment that is missed due to an excused absence. It is the student s responsibility to make arrangements with the teacher in a timely manner.
Course Description 1 This course is designed to prepare the mathematically talented and advanced student for university work in mathematics. In addition, it is designed to prepare those who wish to take the Calculus AB Advanced Placement exam. Students will study functions, analytic geometry, differentiation, integration, and problem solving applications of these topics. Text Calculus of a Single Variable, sixth edition, by Roland E. Larson, Robert P. Hostetler, and Bruce H. Edwards. Copyright 1998 by Houghton Mifflin Company. (A graphing calculator is also required for this course. The suggested model is the TI-83 Plus.) Goals Students should be able to work with functions represented in a variety of ways: graphically, numerically, analytically, or verbally. They should understand the connections among these representations. Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems. Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems. Students should understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus. Students should be able to communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems. Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral. Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions. Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment. Course Topics Chapter 1 Limits and Their Properties (3 weeks) A Preview of Calculus: Students are introduced to the idea of the slope of a tangent line as the limit of the slope of a secant line, and the area underneath a curve as the limit of the sum of the areas of rectangles. Finding Limits Graphically and Numerically: Students explore limits of functions by examining tables of values and graphs of functions. They learn the definition of a limit and identify circumstances in which a limit does not exist. Evaluating Limits Analytically: Students use properties of limits and algebraic techniques to determine limits analytically. The Squeeze Theorem is used to find limits of some trigonometric functions. 1 From the Knight Planning Guide, Bremerton High School s course catalog, page 33. From AP Calculus: 005-006 Workshop Materials, page 133.
Continuity and One-Sided Limits: Students explore one-sided limits and their role in the definition of continuity. They use graphing calculators to examine continuous and discontinuous functions. Students also discuss the Intermediate Value Theorem and its dependence on continuity. Infinite Limits: Students explore unbounded behavior numerically, graphically, and analytically. There is a focus on the relationship between infinite limits and vertical asymptotes on the graphs of functions. 3 x 4x, x 1 1. Consider the function f ( x) x x, x 1 a) Find the right-hand and left-hand limits as x approaches 1. b) Is f(x) continuous at x 1? Explain how you know.. Use the Intermediate Value Theorem to show that the function ( x) 3x 6x 5 0,1 g has a zero on the closed interval Chapter Differentiation (4 weeks) The Derivative and the Tangent Line Problem: Students learn the definition of the derivative of a function and find derivatives using the limit process. They also determine differentiability of functions based on the properties of their graphs. Basic Differentiation Rules and Rates of Change: Students are introduced to basic rules of differentiation the constant rule, power rule, constant multiple rule, sum and difference rules, and the derivatives of the sine and cosine functions. These rules are used to solve problems involving position and velocity. The Product and Quotient Rules and Higher-Order Derivatives: Students use the product and quotient rules to find more complex derivatives and to derive the derivatives of the remaining trigonometric functions. First and second derivatives are used to solve problems involving position, velocity, and acceleration. The Chain Rule: Students use the chain rule to find derivatives of composite functions. Emphasis is placed on deciding when to use the different derivative rules and practice using combinations of rules in a single problem. Implicit Differentiation: Students use implicit differentiation to find derivatives of functions. Related Rates: Students practice creating a mathematical model from a verbal description and use implicit differentiation to solve problems involving related rates. 1. If a ball is shot vertically upward with a velocity of 00 ft/sec, then its height after t seconds is given by the position function s( t) 00t 10t. a) Find the velocity and acceleration functions. b) Find the instantaneous velocity of the ball when t = 6 seconds. Is the ball still on its way up or is it coming down at that time? Explain how you know. c) How high does the ball get? (Find the position when the velocity is 0.). A spherical balloon is being inflated and the radius is increasing at a rate of.5 in/sec. Find the rate of increase of the surface area of the balloon when the radius is 7 inches. (The surface area of a sphere is given by the formula S 4r.)
Chapter 3 Applications of Differentiation (6 weeks) Extrema on an Interval: Students learn the definitions of a critical number and relative and absolute extrema. They find critical numbers and extrema both by examining graphs and by using derivatives. The Mean Value Theorem: Students use the Mean Value Theorem to solve problems involving average and instantaneous rates of change. Increasing and Decreasing Functions and the First Derivative Test: Students use the sign of the first derivative to determine on what intervals a function is increasing or decreasing. Sign charts are introduced as a valuable tool in applying the first derivative test to determine relative maxima and minima of functions. Concavity and the Second Derivative Test: Students use the sign of the second derivative to determine concavity and points of inflection. They learn to use the second derivative test as another method of finding relative extrema. Limits at Infinity: Students first use tables of values to examine end behavior of functions numerically, and then use guidelines for rational functions to determine end behavior analytically. They use infinite limits to find horizontal asymptotes. A Summary of Curve Sketching: Students use properties such as domain and range, symmetry, intercepts, asymptotes, relative extrema, and concavity to sketch graphs of functions. Optimization Problems: Students use techniques they have learned to solve problems involving maximizing or minimizing quantities such as area, volume, and time. There is an emphasis on creating a mathematical model from a verbal description. Differentials: Students use differentials to find linear approximations and propagated and percent error. Business and Economics Applications: Students use differentials to solve problems involving marginals such as revenue, cost, and profit. f ( b) f ( a) 1. Find the value of c such that f ( c) for the function b a f ( x) x x on the interval 0 x 4.. Suppose R 8 x represents revenue and C x represents cost, with x measured in thousands of units produced. How many units should be produced in order to maximize profit? Chapter 4 Integration (5 weeks) Antiderivatives and Indefinite Integrals: Students explore the concept of an antiderivative and learn the vocabulary and notation associated with indefinite integrals. They use basic integration rules to find general and particular solutions to simple differential equations. Area: Students approximate the areas of plane regions using left-hand, right-hand, and midpoint Riemann sums. They then use the limit process to find precise areas of plane regions. Riemann Sums and Definite Integrals: Students are introduced to the concept of a definite integral as the limit of a Riemann sum. There is a focus on representing a definite integral as the area of a region on a graph. Properties of definite integrals are discussed using sums, differences, and multiples of areas of regions. The Fundamental Theorem of Calculus: Students connect the concepts of definite integrals and antiderivatives using the Fundamental Theorem of Calculus. After practicing evaluating
definite integrals analytically, students solve problems involving the Mean Value Theorem and average values of functions on given intervals. The Second Fundamental Theorem of Calculus is used to differentiate functions involving definite integrals. Integration by Substitution: Students focus on recognizing patterns and using change of variables to integrate composite functions. Numerical Integration: Students use the Trapezoid Rule and Simpson s Rule to approximate the values of definite integrals, and analyze the propagated and percent error associated with each. 1. A baseball is thrown upward from ground level with an initial velocity of 10 meters per second. Determine its maximum height. (Use a( t) 9.8m / s as the acceleration due to gravity.). Use Simpson s Rule with n 4 to approximate the definite integral. 0 4 3x dx Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions (6 weeks) The Natural Logarithmic Function and Differentiation: Students review properties of logarithmic functions and are introduced to the derivative of the natural log function. They use tables and graphs to better understand where this derivative comes from in reference to the slope of a tangent line. The Natural Logarithmic Function and Integration: Students use the derivative of the natural log function to develop the log rule for integration. They then derive the integration rules for tan x, cot x, sec x, and csc x. There is a focus on recognizing different forms of functions and choosing the integration technique to use accordingly. Inverse Functions: Students review properties of inverse functions and examine graphs to explore the relationship between a function s derivative and its inverse s derivative. Exponential Functions Differentiation and Integration: Students learn about the derivative and antiderivative of the natural exponential function. They review methods of solving logarithmic and exponential equations for the purpose of solving optimization problems.] Bases Other than e and Applications: Students learn derivative rules for logarithmic and exponential functions with various bases. They use the skills and techniques they have available to solve application problems involving compound interest, depreciation, population growth, etc. Differential Equations Growth and Decay: Students use growth and decay models to solve differential equations of the form y ky when initial conditions are given. Differential Equations Separation of Variables: Students find general and particular solutions to separable differential equations. They use slope fields to examine behavior and aid in sketching solution curves. Inverse Trigonometric Functions and Differentiation: Students learn derivative rules for the six inverse trigonometric functions. At this point, the students should be able to differentiate any elementary function. Inverse Trigonometric Functions and Integration: Students learn the arcsine, arctangent, and arcsecant rules for integration. There is a continued focus on recognizing different forms of functions and choosing the integration technique to use accordingly.
1. 1 Find an equation of the tangent line to the graph of y 4 x ln x 1 at the point (0, 4).. dy 3 Find the particular solution to the differential equation dt 4 t whose graph passes through the point (0, 10). Chapter 6 Applications of Integration ( weeks) Area of a Region Between Two Curves: Students use integration to solve problems involving areas between curves. They find limits of integration and evaluate integrals both by hand and using technology. There is a focus on what the regions between curves might represent. Volume The Disc Method: Students use the disc method to find the volumes of solids of revolution and solids with known cross sections. There is an emphasis on setting up the integrals and using technology to evaluate them. 1 x 1. In the figure below [figure omitted], f ( x) sin( x) and g( x) 4. 4 a) Find the area of R. b) Find the area of S. c) Find the volume of the solid generated when S is revolved about the horizontal line y 1. A sphere of radius r is generated by revolving the region bounded by the graph of y r x and the x-axis about the x-axis. Use the disc method to verify that the volume of a sphere is given by 4 r 3 3. Review for AP Test (5 weeks) Students use released exams to prepare for the Advanced Placement exam. They become very familiar with the format of the test and the sort of questions that appear on it. Students practice scoring their own free-response items to learn where points are earned and deducted. After the AP Test (5 weeks) Students complete and present projects covering various topics from calculus. They also prepare for and complete their final in-class exam.