AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

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1 AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods and applications. The five major topics of this course are limits, differential calculus, integral calculus, polynomial approximations & series, and their applications. Parametric, polar, and vector functions will be studied in this course. All students will be required to analyze problems graphically, numerically, analytically and verbally in this course. The course content will follow the outlines set forth by the College Board and the state. All students will be required to TI-89 series graphing calculator for this course. Course Prerequisites: All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB. Textbook: Larson, Hostetler, Edwards. Calculus, 6 th Edition, Houghton Mifflin, 1998 AP Review Supplemental Workbook: Lederman, David. Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus (BC) Examination, 7 th Edition, D&S Marketing Systems, 2005 Student Evaluation: 1 st, 2 nd & 3 rd Nine Weeks: Exams 70% Quizzes 20% Assignments 10% 4 th Nine Weeks Exam 40% AP Free-Response 25% AP Multiple Choice 25% Assignments 10% AP Calculus BC Page 1 of 17

2 Student Assessments: Exams will be given at the completion of a unit (typically at the end of a chapter). Most exams will span one 57 minute class period. Quizzes will be given during the coverage of the chapter, usually covering 2 4 sections of textbook material. These quizzes will be completed during one class period and some will be with the use of a graphing calculator and others will be without. AP Free Response will be assigned during the review period for the AP examination. All AP free response questions will be assigned to students to complete from the past five examination cycles. AP Multiple Choice will be review assignments from the D&S Marketing review workbook. Students will practice the multiple choice questions in these workbooks to help them prepare for the AP examination. Graded homework will usually consist of challenging problems that are assigned at the completion of a unit. Old AP questions will be given throughout the year as appropriate on quizzes and exams, as well as homework assignments. AP Calculus BC Page 2 of 17

3 AP Calculus BC: Course Outline All students take a full year of AP Calculus AB before enrolling in AP Calculus BC. Students will receive the complete AP Calculus BC course as well as several topics that are not covered in the course description. Students also have more time to explore the concepts and receive a better understanding of the concepts. For the purpose of this syllabus I have included my outline for Calculus AB first as a reference to the material covered during the first year of this two year course. AP Calculus AB (First Semester): Chapter 1: Limits and Their Properties Section 1.1: Section 1.2: Section 1.3: Section 1.4: Section 1.5: A Preview of Calculus Finding Limits Graphically and Numerically Evaluating Limits Analytically Continuity and One-Sided Limits Infinite Limits Chapter 2: Differentiation Section 2.1: Section 2.2: Section 2.3: Section 2.4: Section 2.5: The Derivative and the Tangent Line Problem Basic Differentiation Rules and Rates of Change The Product and Quotient Rules and Higher-Order Derivatives The Chain Rule Implicit Differentiation Supplemental Worksheet of The Chain Rule & Implicit Differentiation Section 2.6: Related Rates Supplemental Worksheet on Related Rates Section 2.A: Supplemental Unit on Particle Motion AP Calculus BC Page 3 of 17

4 Chapter 3: Applications of Differentiation Section 3.1: Section 3.2: Section 3.3: Section 3.4: Section 3.5: Section 3.6: Section 3.7: 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 Supplemental Worksheet on Optimization Problems Section 3.8: Section 3.9: Newton s Method Differentials Section 3.10: Business and Economic Applications AP Calculus AB (Second Semester): Chapter 4: Integration Section 4.1: Section 4.2: Section 4.3: Section 4.4: Section 4.5: Section 4.6: Antiderivatives and Indefinite Integration Area Riemann Sums and Definite Integrals The Fundamental Theorem of Calculus Integration by Substitution Numerical Integration (Trapezoidal Rule and Simpson s Rule) AP Calculus BC Page 4 of 17

5 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions Section 5.1: Section 5.2: Section 5.3: Section 5.4: Section 5.5: Section 5.6: Section 5.7: Section 5.A: Section 5.8: Section 5.9: Section 7.1: The Natural Logarithmic Function: Differentiation The Natural Logarithmic Function: Integration Inverse Functions Exponential Functions: Differentiation and Integration Bases Other than e and Applications Differential Equations: Growth and Decay Differential Equations: Separation of Variables Supplemental Unit on Slope Fields Inverse Trigonometric Functions: Differentiation Inverse Trigonometric Functions: Integration Basic Integration Rules Chapter 6: Applications of Integration Section 6.1: Section 6.2: Section 6.3: Section 7.7: Area of a Region Between Two Curves Volume: The Disk Method Volume: The Shell Method Indeterminate Forms and L Hôpital s Rule Review: Review for AP Examination AP Calculus BC Page 5 of 17

6 The following is the order of topics for AP Calculus BC. During the five weeks of school, I review the material that the students learned in AP Calculus AB. Usually one week for each of the first five chapters in the textbook. The first five chapters and their topics can be found on the previous pages. On the sixth week of the school year, I begin to follow the outline listed below. AP Calculus BC (First Semester): Chapter 6: Applications of Integration Section 6.1: Section 6.2: Section 6.3: Section 6.4: Area of a Region between Two Curves Volume: The Disk Method Volume: The Shell Method Arc Length and Surfaces of Revolution Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions Section 5.10: Hyperbolic Functions Chapter 7: Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 7.1: Section 7.2: Section 7.3: Section 7.4: Section 7.5: Section 7.6: Section 7.7: Section 7.8: Basic Integration Rules Integration by Parts Trigonometric Integrals Trigonometric Substitution Partial Fractions Integration by Tables and Other Integration Techniques Indeterminate Forms and L Hôpital s Rule Improper Integrals Supplement: Differential Equations Section A.1: Section A.2: Section A.3: Slope Fields and Euler s Method Separation of Variables and the Logistic Equation First-Order Linear Differential Equations AP Calculus BC Page 6 of 17

7 Chapter 9: Conics, Parametric Equations, and Polar Coordinates Section 9.1: Section 9.2: Section 9.3: Section 9.4: Section 9.5: Section 9.6: Conics and Calculus Plane Curves and Parametric Equations Parametric Equations and Calculus Polar Coordinates and Polar Graphs Area and Arc Length in Polar Coordinates Polar Equations of Conics and Kepler s Laws Chapter 10: Vectors and the Geometry of Space Section 10.1: Vectors in the Plane Section 10.2: Space Coordinates and Vectors in Space Section 10.3: The Dot Product of Two Vectors Section 10.4: The Cross Product of Two Vectors in Space Section 10.5: Lines and Planes in Space Section 10.6: Surfaces in Space Section 10.7: Cylindrical and Spherical Coordinates AP Calculus BC (Second Semester): Chapter 11: Vector-Valued Functions Section 11.1: Vector-Valued Functions Section 11.2: Differentiation and Integration of Vector-Valued Functions Section 11.3: Velocity and Acceleration Section 11.4: Tangent Vectors and Normal Vectors Section 11.5: Arc Length and Curvature AP Calculus BC Page 7 of 17

8 Chapter 8: Infinite Series Section 8.1: Section 8.2: Section 8.3: Section 8.4: Section 8.5: Section 8.6: Section 8.7: Section 8.8: Section 8.9: Sequences Series and Convergence The Integral Test and p-series Comparisons of Series Alternating Series The Ratio and Root Tests Taylor Polynomials and Approximations Power Series Representation of Functions by Power Series Section 8.10: Taylor and Maclaurin Series Review: Review for AP Examination AP Calculus BC Page 8 of 17

9 AP Calculus BC: College Board s Topic Outline & Correlation to Textbook The sections listed next to each topic, lists a section (or supplemental unit) of the textbook which corresponds to each topic. Although several of these topics appear throughout a calculus course, the section where it is first introduced is included for reference. TOPIC TEXTBOOK SECTION I. Functions, Graphs, and Limits Analysis of Graphs With the aid of technology, graphs of functions are often easy to produce. (1.2, 1.3) The emphasis is on the interplay between the geometric and analytic information and on the use of Calculus both to predict and to explain the observed local and global behavior of a function. Limits of Functions (Including one-sided limits) An intuitive understanding of the limiting process (1.2) Calculating limits using algebra (1.3) Estimating limits from graphs or tables of data (1.2) Asymptotic and Unbounded Behavior Understanding asymptotes in terms of graphical behavior (1,4, 1.5, 3.5) Describing asymptotic behavior in terms of limits involving infinity (3.5) Comparing relative magnitudes of functions and their rates of change (5.6) (for example, contrasting exponential growth, polynomial growth, and logarithmic growth) Continuity as a Property of Functions An intuitive understanding of continuity. (The function values can be made (1.4) as close as desired by taking sufficiently close values of the domain.) Understanding continuity in terms of limits (1.4) Geometric understanding of graphs of continuous functions (Intermediate (1.4) Value Theorem and Extreme Value Theorem) Parametric, Polar and Vector Functions The analysis of planar curves includes those given in parametric form, (9.3, 9.5, 11.1) polar form, and vector form. AP Calculus BC Page 9 of 17

10 II. Derivatives Concept of the Derivative Derivative presented graphically, numerically, and analytically (2.1, 2.2) Derivative interpreted as an instantaneous rate of change (2.2, 2.A) Derivative defined as the limit of the difference quotient (2.1) Relationship between differentiability and continuity (2.1) Derivative at a Point Slope of a curve at a point. Examples are emphasized, including points (2.1) at which there are vertical tangents and points at which there are no tangents. Tangent line to a curve at a point and local linear approximation (2.1) Instantaneous rate of change as the limit of average rate of change (2.2) Approximate rate of change from graphs and tables of values (2.2) Derivative as a Function Corresponding characteristics of graphs of f and f (3.3) Relationship between the increasing and decreasing behavior of f and the (3.3) sign of f The Mean Value Theorem and its geometric consequences (3.2) Equations involving derivatives. Verbal descriptions are translated into (5.6) equations involving derivatives or vice versa. Second Derivatives Corresponding characteristics of the graphs of f, f, f (3.3, 3.4, 3.6,) Relationship between the concavity of f and the sign of f (3.4, 3.6) Points of inflection as places where concavity changes (3.4, 3.6) AP Calculus BC Page 10 of 17

11 Applications of Derivatives Analysis of curves, including the notations of monotonicity and concavity (3.6) Analysis of planar curves given in parametric form, polar form, and vector (9.3, 9.5, 11.3) form, including velocity and acceleration Optimization, both absolute (global) and relative (local) extrema (3.1, 3.7) Modeling rates of change, including related rates problems (2.6) Use of implicit differentiation to find the derivative of an inverse function (5.3) Interpretation of the derivate as a rate of change in varied applied contexts, (2.2, 2.A, 4.4) including velocity, speed, and acceleration Geometric interpretation of differential equations via slope fields and the (5.A) Relationship between slope fields and solution curves for differential Equations Numerical solution of differential equations using Euler s method (A.1) L Hôpital s Rule, including its use in determining limits and convergence (7.7, 7.8, Of improper integrals and series Chapter 8) Computation of Derivatives Knowledge of derivatives of basic functions, including power, exponential (2.2, 2.3, 5.1 logarithmic, trigonometric, and inverse trigonometric functions 5.3, 5.4, 5.8, 5.9) Basic rules for the derivative of sums, products, and quotients of functions (2.2, 2.3) Chain rule and implicit differentiation (2.4, 2.5) Derivatives of parametric, polar, and vector functions (9.3, 9.4, 11.2) III. Integrals Interpretation and Properties of Definite Integrals Definite integral as a limit of Riemann sums (4.2) Definite integral of the rate of change of a quantity over an interval (4.4) interpreted as the change of the quantity over the interval: b f x dx f b f a a Basic properties of definite integrals (examples include additivity and (4.4) linearity) AP Calculus BC Page 11 of 17

12 Applications of Integrals Appropriate integrals are used in a variety of application to model physical, (3.10, 4.1, 4.2, biological, or economic situations. Although only a sampling of applications 4.3, 4.4, 6.1, can be included in any specific course, students should be able to adapt 6.2, 9.5) their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line, and the length of a curve (including a curve given in parametric form). Fundamental Theorem of Calculus Use of the Fundamental Theorem to evaluate definite intergrals (4.4) Use of the Fundamental Theorem to represent a particular antiderivative, (4.4) and the analytical and graphical analysis of functions so defined Techniques of Antidifferentiation Antiderivatives following directly from derivatives of basic functions (4.1) Antiderivatives by substitution of variables (including change of (4.5, 7.2, 7.5) limits for definite integrals), parts, and simple partial fractions (nonrepeating factors only) Improper integrals (as limits of definite integrals) (7.8) Applications of Antidifferentiation Finding specific antiderivatives using initial conditions, including (4.5) applications to motion along a line Solving separable differential equations and using them in modeling (5.6, 5.7) (in particular, studying the equation y ky and exponential growth) Solving logistic differential equations and using them in modeling (A.2) Numerical Approximations to Definite Integrals Use of Riemann sums (using left, right, and midpoint evaluation points) (4.2, 4.6) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values. AP Calculus BC Page 12 of 17

13 IV. Polynomial Approximations and Series Concept of Series A series is defined as a sequence of partial sums, and convergence is (Chapter 8) defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence. Series of Constants Motivating examples, including decimal expansion (8.1) Geometric series with applications (8.2) The harmonic series (8.2) Alternating series with error bound (8.5) Terms of series as areas of rectangles and their relationship to improper (8.3) integrals, including the integral test and its use in testing the convergence of p-series The ratio test for convergence and divergence (8.6) Comparing series to test for convergence or divergence (8.4) Taylor Series Taylor polynomial approximation with graphical demonstration of (8.7) convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve) Maclaurin series and the general Taylor series centered at x a (8.7) x 1 Maclaurin series for the functions e, sin x, cos x, and 1 x (8.10) Formal manipulation of Taylor series and shortcuts to computing Taylor (8.7, 8.10) series, including substitution, differentiation, antidifferentiation, and the formulation of new series from known series Functions defined by power series (8.8) Radius and interval of convergence of power series (8.9) Lagrange error bound for Taylor polynomials (8.7) AP Calculus BC Page 13 of 17

14 Curricular Requirement #1: AP Calculus BC: Evidence of Curricular Requirements The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. This curricular requirement is met and on the five previous pages the entire Calculus BC topic outline was reproduced and the section or supplemental unit where it is covered is denoted. In addition, some topics that are not in the Calculus BC Topic Outline are also covered in this course. Examples of these topics include but is not limited to three dimensional vectors, partial derivatives and double integration. Curricular Requirement #2 This course provides students with the opportunity to work with functions represented in a variety of ways graphically, numerically, analytically, and verbally and emphasizes the connections among these representations. Example #1: Students are first introduced to limits numerically, then graphically, then analytically. Students will construct tables of values, then graph the function to see if the two answers agree. During the next class the analytical method is introduced and the three methods are compared. Students are encouraged to verbalize their explanations and answers in class on a daily basis. Example #2: To find the area under the curve the students are exposed to several approximation techniques prior to learning how to evaluate the definite integral algebraically. Examples of this would include problems of the following nature: Problem #1: (Numerical) Suppose a volcano is erupting and readings of the rate r t at which solid materials are spewed into the atmosphere are given in the table. The time t is measured in seconds and the units for r t are tones (metric tons) per second. t r t Use the Trapezoidal Rule with 3 subdivisions of equal length to approximate 6 0 r t dt. AP Calculus BC Page 14 of 17

15 Problem #2: (Graphical) Use the midpoint rule to approximate the area of the region bounded by the graph of the function and the x-axis over the indicated interval. Use n 4 subintervals. 2 f x x 3 ; 0,2 y x Problem #3: (Analytical) A particle moves along the x-axis so that its acceleration at any time t is given by a t 6t 18. At time t 0 the velocity of the particle v 0 24, and at time t 1 its position is x (A) Write an expression for the velocity vt of the particle at any time t. (B) For what value(s) of t is the particle at rest? (C) For what value(s) of t is the particle moving to the right on the time interval 0,8? (D) Write an expression for the position xt of the particle at any time t. At the conclusion of this students can present their finding to class by writing their solutions on the board and verbally explaining the steps to their classmates. AP Calculus BC Page 15 of 17

16 Curricular Requirement #3 The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences. Students are encouraged to verbally express their suggestions on solving problems in class on a daily basis. Students will also come to board and demonstrate their solutions to the class. Examples of questions that have been used on examinations for students to express the mathematics in written sentences appear below. Example #1: If oil leaks from a tank at a rate of 120 r t gallons per minute at time t, what does r 0 t dt represent? Example #2: During the first 40 seconds of a flight, the rocket is propelled straight up so that in t seconds it reaches a 3 height of s 5t feet. It can be shown that ds dt 15t 2. Interpret ds dt in the context of the problem. AP Calculus BC Page 16 of 17

17 Curricular Requirement #4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. Example for Solving Problems: Each student enrolled in this course will have a TI-89 series graphing calculator to use. The students are expected to bring their graphing calculators to class with them each day. The calculators are used frequently in class to explore, discover and reinforce the concepts that we are learning. Example for Experiment: When introducing the limit definition of a derivative, the students will sketch a function using their graphing calculator and graph various secant lines to the curve which pass through a specific point x, f x on the curve. Students will calculate the slope of their secant lines. To graph the other secant lines students will choose a second point on the curve closer to fixed point of x, f x continue this pattern until they pick a point very close to the fixed point x, f x. Example of Interpretation of Results:. They will Using the example above, students will notice that our secant line is very close to slope of the curve at the fixed point x, f x. They will explore this further by zooming in on their original function close to the fixed point and noticing that it appears almost linear over a small fixed interval. Example of Supporting Conclusions: The first function students experiment with is usually a quadratic. I continue this experimentation with polynomial, exponential and logarithmic functions. AP Calculus BC Page 17 of 17

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