Notes about changes to Approved Syllabus # 43080v2

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1 Notes about changes to Approved Syllabus # 43080v2 1. An update to the syllabus was necessary because of a county wide adoption of new textbooks for AP Calculus. 2. No changes were made to the Course Outline other than to eliminate references to specific textbook sections. 3. In the course syllabus, all section number references, lesson titles, and assignments have been changed to correlate to the new text and the order in which some topics are taught have been rearranged. 4. Since I also teach an AP Calculus AB course, the first semester of this syllabus for BC is identical to the syllabus submitted for the AB course, #300586v1, currently under review.

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3 Course Syllabus and Outline AP Calculus BC COURSE DESCRIPTION AND BACKGROUND INFORMATION AP Calculus BC is a college level course covering material traditionally taught in the first two semesters of college calculus. The course is taught over two semesters consisting of 90 minute daily classes. Students need a strong foundation to be ready for the rigorous work required throughout the year. Completing the summer review packet before the beginning of the course will ensure a proper background. This packet consists of review material studied during Algebra II and Analysis. Students should expect to work approximately 10 hours on the assignment. The packet will be collected on the first day of the semester and will be given a grade that will be based on completeness of solutions and accuracy. In preparation for the AP test, students need to show all work with logical steps. You must show your work for problems in the review packet. Do not list only an answer. Students enrolled in AP Calculus BC will be using a graphing calculator throughout the course since a graphing calculator is required on the AP test. Students will be issued a TI 89 calculator to be used in class during the year and, with parental permission, students may take the calculator home to use as well. The success of each student in the AP Calculus program depends upon diligent effort and practice of newly learned skills. Although a suggested assignment is given for each lesson, completion of the assignment is optional and homework is not graded or checked for completion. The previous night s assignment will be reviewed in class each day and there will be ample opportunity to ask questions. Understanding calculus requires analyzing problems algebraically, numerically, graphically, as well as verbally and you will be expected to communicate your knowledge and understanding of concepts in several ways. As part of the school wide writing goal, not only will you be expected to justify your solutions to some test questions by writing explanations using complete sentences, but will also be expected to respond to open ended writing prompts that may be included with each major test. For example, you may be asked to compare and contrast different methods of determining maximum and minimum values of functions or to explain how the Fundamental Theorem of Calculus relates integration and differentiation. In addition, there will be several opportunities for you to work together in a group with other students to solve problems and present your solutions to the class.

4 TEXTBOOKS Rogawski, Jon. Calculus: Early Transcendentals. New York: W.H. Freeman and Company Foerster, Paul A. Calculus: Concepts and Applications (2 nd Edition). Emeryville, CA: Key Curriculum Press 2005 REQUIRED SUPPLIES Notebook (spiral or looseleaf), paper, pencils TI 89 Graphing Calculator (calculators are available to students who do not own a TI 89) Composition or Spiral Notebook for journal AP Calculus BC Course Outline 1. Limits and Continuity Evaluate the limit of a function graphically, numerically, and algebraically and investigate limits both with and without the use of a calculator. Calculate limits using Limit Laws Determine limit existence and explore limits involving infinity Investigate and determine continuity/discontinuity of a function and relate to limits Identify the three conditions that must be satisfied in order for a function to be considered continuous at a point Classify discontinuities as removable, jump, or infinite Understand and use the Intermediate Value, Extreme Value, and Sandwich Theorems Find equations of vertical and horizontal asymptotes using limits Compare and contrast relative rates of change 2. Introduction to Differential Calculus Find slopes and equations of tangent lines and normal lines to a curve at a point Investigate the relationship between the slope of the curve at a point and the slope of the tangent line at that point using a graphing calculator Explain the relationship between continuity and differentiability, and identify situations for which a function might be continuous, but not differentiable, and apply the Intermediate value Theorem Evaluate and apply tangents, velocities, and other rates of change Explain the relationship between average velocity and instantaneous velocity and calculate each appropriately Define and find the derivative of a function using the difference quotient Use Linear Approximation methods to estimate function values and rates of change

5 Compare functions and their derivatives Explore the derivative as a function both with and without using the calculator. 3. Differentiation Rules for Functions (constant, linear, polynomial, exponential, trigonometric, and logarithmic) Use Power Rule, Constant Multiple Rule, Sum Rule, Difference Rule, Product Rule, Quotient Rule, and Chain Rule Use Implicit and Logarithmic Differentiation 4. Applications of Differentiation Solve related rate problems Analyze graphs of functions comparing with first and second derivative graphs and apply the First and Second Derivative Tests Find relative and absolute extreme values, points of inflection, concavity, intervals where a function increases or decreases using the graphs of f, f, and f as well as numerical methods Use L Hopital s Rule, Newton s Method, Rolle s Theorem, and Mean Value Theorem Solve optimization and minimal path problems and investigate the use of calculus in economic applications Solve particle motion problems involving position, velocity, and acceleration and explore the relationships between these measurements and derivatives 5. Introduction to Integral Calculus Find antiderivatives graphically, numerically, and algebraically both with and without the use of a calculator Explore the relationships between area under a curve, distance and other accumulation functions, and the definite integral Explore the definition of an antiderivative using the limit of a sum Use Riemann Sums with left, right, and midpoint evaluation points Evaluate and use properties of definite integrals State, interpret, and use the Fundamental Theorem of Calculus Integrate functions using substitution, integration by parts, and by using partial fractions with non repeating linear factors Estimate definite integrals using approximation methods that involve graphs, charts of values, and algebraic representations Use the Trapezoidal Rule to approximate the area under the curve Solve particle motion problems involving position, velocity, and acceleration and explore the relationships between these measurements and derivatives Investigate indeterminate forms, improper integrals, and L Hopital s Rule

6 6. Applications of Integration Find the area under a curve or between two curves. Use the calculator to determine points of intersection and for graphing Find the volume of a solid of revolution using the disk and washer methods and finding volumes of solids with known cross sections Find the average value of a function, the arc length, and the area of a surface of revolution, and solve problems involving work for Hooke s Law, pumping, propulsion, and lifting. Find the distance traveled by a particle along a line 7. Differential Equations Solve first order separable differential equations, finding both general solutions and solutions to initial value problems Use differential equations to model growth and decay and the logistic model Explore solutions to differential equations through slope fields and Euler s Method and discuss the relationship between the slope field solution and the algebraic solution to differential equations. Construct slope fields manually as well as with the calculator 8. Parametric, Polar, and Vector Equations Investigate and determine derivatives and integrals as well as slopes, tangent lines, and normal lines of parametric, polar, and vector functions. Find the area of a region of a polar curve or between two polar curves. Find the arc length and area of a surface of revolution in parametric, polar, and vector form Solve initial value problems for particle motion involving acceleration, velocity, and position for parametric and vector valued functions. 9. Sequences and Series Determine whether a sequence is convergent or divergent using algebraic, numerical, and graphical investigations Express a repeating decimal as an infinite series and as a ratio of integers Identify a geometric series, find the sum of a finite geometric series or a convergent infinite geometric series, and use geometric series to solve application problems Investigate series as a sequence of partial sums and determine whether an infinite series is convergent or divergent using the following tests: a. telescopic series b. nth term test for divergence c. integral test d. limit comparison test e. direct comparison test

7 f. ratio test g. root test h. p series test (including the harmonic series) i. alternating series test j. geometric series test Determine whether a series converges absolutely or conditionally, or diverges Determine the error bounds for estimating the sum of a series when using the integral test or alternating series test Find the term for which a series can be approximated with a given accuracy 10. Polynomial Approximations Represent polynomial functions as power series Find the Taylor series expansion about x = c for a given function x 1 Identify the Maclaurin series for the functions e, sin x, cos x, and 1 - x Investigate derivatives and integrals of power series and form new series from known series Determine the radius and interval of convergence of power series Estimate the Lagrange error for Taylor polynomials Estimate function values using Taylor series 11. Review for AP Test

8 AP Calculus BC Bold and italicized assignments are due for a grade Day Topic Assignment Part 1: Limits, Continuity, and Differential Calculus 1 Introduction to Calculus; issue textbooks and calculators; collect summer packets. Section 2.1 Limits, Rates of Change, and Tangent Lines p. 66 # 3, 5, 6, 7, 8 16 (even),19, 22 23, Section 2.2 Limits: A Numerical and Graphical Approach p. 76 # 2, 6, (mult. of 3) 3 Section 2.2 (cont d) p. 77 # 37 39, (even) Section 2.3 Basic Limit Laws p. 82 # 2 28 (even) 4 Section 2.4 Limits and Continuity p. 90 # 2 4, 6 27 (mult. 3) 5 Section 2.4 (cont d) p. 92 # (even), (odd), 64, 68, Worksheet , 76, 82, 84 6 Section 2.5 Evaluating Limits Algebraically p. 97# 4, 8, 12, 16 20, 26, 28, 31, 32, 36, 46, 50 p. 257 # 56, 60, 66, 70 Group Work: Take it to the Limit 7 Review 8 Quiz Section 2.6 Trigonometric Limits and the Squeeze Theorem p 102 #1 4, 6, 10, 13, 16, 18, 22, 26, 34, 36, Section 2.7 Intermediate Value Theorem p. 106 # 1, 4, 6, 7, 12, 17, 22, 24

9 11 Review 12 Chapter 2 Test 13 Section 3.1 Definition of the Derivative p. 124 # 5, 6 10 (even), 11 14, 22, 28, 32, 34, 60, 66 39, 41, 46, 50 51, 54, 58, 14 Section 3.2 The Derivative as a Function p. 139 # 2, 6, 10, 12, (even), 25, 28, 34, 35, 37, 40, Section 3.2 (cont d) p. 140 # 47, 48, 49, 50, 52, 54, 55, 71, 76, Section 3.3 Product and Quotient Rules p. 148 # (even), 39, 44, Section 3.4 Rates of Change p. 157 # 9, 10, 12, 13, 20, 22, 23, (even), 35, 36, Section 3.4 (cont d) Rates of Change Word Problems 19 Review 20 Quiz: Section 3.5 Higher Derivatives p. 165 # (even), 30, 37, 39 41, Section 3.6 Derivatives of Trig Functions p.170 # 6, 8, 11, 20, 22, 26, 30, 32, 33, 35, 38, 43, 46, 49 Trig Derivative Proofs 23 Section 3.7 Chain Rule p. 178 # 12, 16, 18, 20, (mult. 3), Section 3.8 Implicit Differentiation p. 185 #10, 13, 24, 27, 30, 32, 36, 37, 43, 50, Section 3.9 Derivatives of Inverse Functions p. 191 # 4 20, (even), (odd)

10 26 Section 3.10 Derivatives of General Exponential p. 197 #2 20 (even), (odd), 43, 44 and Logarithmic Functions 27 Review 28 Chapter 3 Test includes both a non calculator and calculator portion be prepared to work derivatives without the aid of a calculator! 29 Section 3.11 Related Rates Group Work: Word Problems Part 1 p. 204 # 9 12, 14, Section 3.11 (cont d) Group Work: Word Problems Part 2 p. 205 # 15 17, 27 32, 35 38, 44 : 31 Section 3.11 (cont d) 32 Section 4.1 Linear Approximations p. 218 # 8 16 (even), 25, 27, 32, 33, 44 46, 51, 55, 57, 62, Section 4.2 Extreme Values and Rolle s Theorem p. 227 # 1 9 (odd), 15, 20, 27, 31, 37, 44, 49, 34 Review 60, 66, Quiz: Section 4.3 Mean Value Theorem and Montonicity p. 237 # 4, 6, 8, 13, 15 19( odd), 19, 21, 28, 30, 1 st Derivative Test 34, 39, 48, 52, Section 4.4 The Shape of a Graph p. 243 #1, 2, 5, (even), 20, 22, 23, 27 29, 2 nd Derivative Test 36, 39, 42, 45, 49, Section 4.5 Graph Sketching and Asymptotes p. 256 # 4, , 25, 30, 33, 42, 44, 77, 86, 89

11 39 Section 4.5 (cont d) Laptop Activity Group Work Section 4.5 (cont d) AP Problems 1987 #4, 1989 #5, 1991 #5, 1996 #1 41 Review 42 Quiz: Section 4.6 Applied Optimization Group Work: Word Problems Part 1 p. 265 # 4, 6, 8, 9, 15, 16, 20, 24, 31, Section 4.6 (cont d) Minimal Path Problems Group Work: Word Problems Part 2 Exploration 68 p. 267 # 36, 44, 59, Section 4.7 L Hopital s Rule p. 277 # 7, 12, 13, 21, 26, 33, 40, 42, 44, 48, Midterm Review 47 Midterm 48 Midterm Part 2: Integral Calculus 49 Section 4.9 Antiderivatives p. 292 # 5 8, 18, 24, 28, 30, 32, 35, 38, Section 4.9 (cont d) p. 293 # 47, 52, 57, 61, 64, 68, 74, 75, Section 5.1 Approximating and Computing Area p. 308 # 3, 8, 10, 13, 17, 19, 26, 31, 40, 42, 64 LRAM, RRAM, and MRAM Section 7.1 Numerical Integration p. 2, 6, 11, 30, 32,

12 Trapezoidal Rule 52 Section 5.2 The Definite Integral Group Work Sections 5.2 p. 321 # 3, 7, 13, 14, 18, 28, 36, 40, 43, 53, 56, 58, 61, 71, 53 Section 5.3 The Fundamental Theorem of Calculus p. 329 #15, 19, 22, 31, 32, 39, 44, 49 Part 1 54 Section 5.4 The Fundamental Theorem of Calculus p. 335 # 3, 6, 10, 13, (even), 31, 37, 39, Part 2 42, Section 5.5 Net or Total Change as the Integral of a Rate p. 341 # 2 14 (even), 15, 17, 20, Section 5.6 Substitution Method p. 349 # 35, 38, 40, 51, 54, 57, 66, 78, 82, Section 5.7 Further Transcendental Functions p. 355 # 6, 7, 9, 16, 20, 22, 25, 31, 37, 38, 46, 56, 62, Section 7.2 Integration by Parts p. 7, 9, 14, 15, 45, 48, (odd) 59 Review 60 Chapter 5 Test includes both a non calculator and calculator portion be prepared to work integrals without the aid of a calculator! 61 Section 6.1 Area Between Two Curves p. 379 # 9, 10, 16, 24, 28, 30, 33, 40, 44, Section 6.2 Volume as the Integral Of Cross Sectional p. 390 # 9 12, 38, 40, 42, 49, 54, 56, Area, MVT for Integrals, and Average Value 63 Section 6.3 Volumes of Revolution p. 398 # 9, 12, 14, 16, 20, 22, Disk and Washer Methods about x and y axis 64 Section 6.3 (cont d) Volumes of Revolution p. 399 # 35, (even), 48

13 Disk and Washer Methods about y = k or x = k 65 Section 6.3 (cont d) Practice Section 6.4 The Method of Cylindrical Shells p. 405 # 4, 8, 12, 14, 16, 20, 23, 25, 29, AP Problems 1998 # 1, 2000 #1, 2001 #1, 2002 #1, 2002B #1 68 AP Problems (cont d) 69 Review 70 Chapter 6 Test 71 Section 9.1 Solving Separable Differential Equations p. 521 # 5, 9, 16, 18, 19, 21, 23, (even) 72 Section 5.8 Exponential Growth and Decay Exploration # 42 p. 366 # 1, 4, 6, 8, 11, 14, 18, 20, 34, 37, Section 9.3 Graphical Methods (Slope Fields) Slope Fields Packet 74 Section 9.3 (cont d) p. 537 # 2 8 (even), Differential Equations/Mathematical Modeling Word Problems 76 Differential Equations/Mathematical Modeling Word Problems 77 AP Problems 2000 #6, 2002B #5, 2003 #5, 2004B #5 78 Newton s Law of Cooling Chill Out Lab with CBL Group Work 2&3, Section Differential Equations and Slope Fields Review

14 80 Chapter 9 Test 81 Cumulative Review 82 AP Problems 2003 Multiple Choice Part 1 83 AP Problems 2003 Multiple Choice Part 2 84 AP Problems 2003 Free Response Part 1 85 AP Problems 2003 Free Response Part 2 86 FINAL EXAM Free Response 87 FINAL EXAM Multiple Choice ***End of First Semester*** **Second Semester** Day Topic Assignment Part 1: Additional Techniques and Applications of Integration 1 Section 7.3 Trigonometric Integrals p. 439 #3, 5, 9 21 (odd), 24, 31, 35, 47 2 Section 7.4 Trigonometric Substitution p. 446 #13 29 (odd), (odd) 3 Section 7.6 The Method of Partial Fractions p. 464 # Section 7.7 Improper Integrals p. 474 #5, 9, 11, 13, 15, 21, 33, 35, 51, 60, 65 5 Section 7.7 (cont d) Group Work 5.10

15 6 Review Chapter 7 AP Problems 7 Quiz Additional Techniques of Integration 8 Lesson 6.5 Work and Energy Word Problems Part 1 Hooke s Law and Propulsion Problems 9 Lesson 6.5 (cont d) Word Problems Part 2 Pumping 10 Lesson 6.5 (cont d) Word Problems Part 3 Lifting 11 Lesson 6.5 (cont d) p. 413 # 2 7, 13, 17, 19, 26, 29, Section 9.3 Graphical and Numerical Methods Group Work: Euler with Care Exploration 45 p. 537 # AP Problems BC 1998 #4, BC 1999 #6, BC 2001 #5 14 Section 9.4 The Logistic Equation p. 543 #5, 6, 8, 9 Investigation 15 Quiz: Work and Energy, Euler s Method, Logistic Model Part 2: Calculus with Parametric Equations, Polar Coordinates, and Vectors 16 Section 11.1 Parametric Equations TI 89 calculator activity Derivatives and Tangent Lines p. 629 # 8 16 (even), (odd), 37, , 81, 84 Parametric Worksheet #1

16 17 Section 8.1 Arc Length and Surface Area p. 485 #7, 9, 33, 36, 42 Section 11.2 Arc Length sand Speed p. 3, 6, 10, 17, 18, 20, 22 Parametric Worksheet #2 18 Section 11.2 (cont d) Group Work Particle Motion 19 Section 13.2 Calculus of Vector Valued Functions p. 753 #8, 9, 16, 17, 18, 42, 46, 47 Explorations 69 and Section 13.3 Arc Length and Speed with Vectors Vector Worksheet #1 21 Sections 13.3 (cont d) Particle Motion with Vectors Word Problems Forrester text p Vector Worksheet #2 22 Section 13.3 (cont d) Activity: Tracking with Vectors 23 AP Problems BC 1993 #2, 1995 #1, 1996 #6, 1997 #1, 1998 #6 24 AP Problems (cont d) BC 1999 #1, 2000 #4, 2001 #1, 2002 #3 25 Section 11.3 Polar Coordinates p. 646 #5, 6, 13, 14, 19, 20, 25, Derivatives, and tangent Lines Worksheet 26 Section 11.4 Area and Arc Length in Polar Coordinates p. 653 #1 5, 7,9, 24, 26 Group Work 27 Section 11.4 (cont d) Area Between Two Polar Curves p. 653 # Section 11.4 (cont d) Area Between Two Polar Curves Group Work 29 Review 30 Group Project Presentations

17 31 Group Project Presentations 32 Test: Calculus with Parametric Equations, Polar Coordinates, and Vectors Part 3: Sequences and Series 33 Section 10.1 Sequences p. 561 #3, 5, 7, 11, 14, 18, 19, 23, 43, 47, 51, Section 10.1 (cont d) Worksheet 10.1 (in groups) 35 Section 10.2 Summing an Infinite Series p. 571 #1, 5, 9, 13, 15, 17, (odd) 36 Section 10.2 (continued) Group work: Exploration Section 10.2 (continued) Geometric Series as Mathematical Models 38 Section 10.2 (cont d) 39 Quiz Sections 10.1 and Section 10.3 Convergence of Series with Positive Terms p. 581 #3 13 (odd) Integral and p series Tests Worksheet 10.3 A 41 Section 10.3 (cont d) ` Worksheet 10.3 Comparison Tests 42 Section 10.4 Absolute and Conditional Convergence p. 588 #3 25 (odd) Alternating Series Test 43 Section 10.5 The Radio and Root Tests p. 592 # 5 17 (odd), (odd), Section 10.5 (cont d) Group Work: What s My Ratio?

18 45 Lessons 10.3 and10.4 (cont d) Estimating Sums p. 582 #78 81 p. 588 # 17, Review 47 Quiz Sections Section 8.4 Taylor Polynomials p. 509 # 1, 3, 6, 8, 13, 23, 25, 27, 49 Section 8.4 (cont d) Error Bound p. 509 #15, 17, 29, 32, Section 10.6 Power Series p. 603 #1, 6, 8, 9, 13, 16, 19, 22, 23, 25 Radius and Interval of Convergence 51 Section 10.6 (cont d) p. 604 # (odd), 44, 47 Expanding a Function as a Power Series Group Work Differentiating and Integrating a Power Series 52 Section 10.7 Taylor and Maclaurin Series Explorations 80 and Section 10.7 (cont d) p. 615 # 3 15 (odd), 21, 22, 25, (odd) 54 AP Problems BC 1997 #2, 1998 #3, 1999 #6, 2000 #3 55 AP Problems BC 2001 #6, 2002 #6, 2002B #6 56 Review 57 Test: Sections 8.4, Prepare for AP Test

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