GOVERNOR LIVINGSTON HIGH SCHOOL MATHEMATICS DEPARTMENT CALCULUS, AP CALCULUS AB/BC #MAY1212/MAY1130/MAY1131

Transcription

1 BERKELEY HEIGHTS PUBLIC SCHOOLS BERKELEY HEIGHTS, NEW JERSEY GOVERNOR LIVINGSTON HIGH SCHOOL MATHEMATICS DEPARTMENT CALCULUS, AP CALCULUS AB/BC #MAY1212/MAY1130/MAY1131 Curriculum Guide November 2011 Mrs. Judith Rattner, Superintendent Mrs. Patricia Qualshie, Assistant Superintendent Mrs. Susan Rembetsy, District Supervisor Developed by: William Ayres Steven Ciarrocca Theresa Pisano This curriculum may be modified through varying techniques, strategies, and materials, as per an individual student s Individualized Educational Plan (IEP). Approved by the Berkeley Heights Board of Education at the regular meeting held on November 17, 2011.

2 TABLE OF CONTENTS Page Vision Statement... 1 Mission Statement... 2 Course Proficiencies... 3 Course Objectives... 3 Student Proficiencies... 5 Methods of Evaluation... 8 Course Outline/Student Objectives... 9 Resources/Activities Guide Suggested Audio-Visual/Computer Aids Suggested Materials Resources for Students Resources for Teacher... 15

3 VISION STATEMENT The study of Calculus at all levels challenges students, to analyze changes in physical quantities, in ways they have not considered prior to this course. Pre-calculus mathematics is more static, whereas calculus is more dynamic. Applying principals of Calculus is necessary to analyze certain concepts, such as finding the velocity of an accelerating object. Emphasis on conceptual understanding of the theorems will be stressed at all levels. Techniques of proving, evaluating, and applying theorems will also be stressed, within the AP Calculus courses. Students will be provided with the fundamentals of Calculus, needed to successfully make the transition from high school mathematics to college mathematics courses. AP Calculus is designed for college-bound students who have demonstrated high mathematical ability. AP Calculus prepares students for taking the AP Calculus (AB) and/or (BC) test and students are encouraged, but not required, to do so. Berkeley Heights Public Schools 1

4 MISSION STATEMENT Throughout the study of Calculus two concepts are stressed: Calculus is the mathematics of change Calculus is the mathematic tool used to find the equation of tangent lines, areas, volumes, and other concepts that model real- life situations Topics to be included in these courses are: Functions and their properties Limits Derivatives and their applications, including finding minimums and maximizations Integration and its applications, including finding area between curves and volumes of solids of revolution The Natural Logarithmic function Comprehensive exercises are provided that are applications of theorems. Critical-thinking exercises and theoretical problems are emphasized. An extension of topics will also be covered, at the AP level. Technology will be integrated, throughout the instruction of the course, and will be used to expand the problem-solving capabilities of the students. Calculus AB is an extension of the content of Calculus, with the primary differences being the additional subject material and greater rigor expected in the AB course. Students develop the ability to think abstractly, as well as in a logical and analytical way. Throughout the presentation of concepts and methods, emphasis is placed on understanding and connecting unifying themes. Calculus BC is an extension of the content of AP Calculus AB, and additional subject material will be covered throughout the completion of this course. The students will be held responsible for individual work while also being expected to function productively, in a cooperative learning environment. Five (5.0) credits will be earned for successful completion of this full year course. National and state standards are integrated throughout the curriculum. Berkeley Heights Public Schools 2

5 COURSE PROFICIENCIES COURSE OBJECTIVES 1. To understand the concept of limits and evaluate limits numerically, graphically, and analytically. (F12/IF4-7; 8.1/12A1) 2. To determine continuity of a function, at a point and on an interval; to apply the Intermediate Value Theorem to functions that are continuous over a given interval. (8.1/12A1) 3. To understand the relationship between the derivative and the slope of a curve, at a given point; to use the limit definition to find the derivative of functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric. (F12/LE1-3) 4. To use the constant rule, power rule, sum and difference rules, product and quotient rules, chain rule, and implicit differentiation technique to find the derivative of functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric. (F12/TFI,7) 5. To evaluate derivatives of parametric, polar, and vector functions and analyze planar curves given in these forms. (F12/IF8; N12/CN1-8,VM1-5) 6. To use the derivative to find extrema, intervals over which the function is increasing or decreasing, and intervals over which the function is concave up or down. (F12/IF4-9; A12/APR3-5) 7. To solve related rate, optimization, and rectilinear motion problems, using the derivative; to apply differentials to make approximations. (F12/LE1,IF7; N12/NQ3) 8. To solve separable differential equations and logistic differential equations and use them in modeling; to find numerical solutions of differential equations using Euler s Method. (N12/NQ3) 9. To estimate the area of a plane region, using rectangles, trapezoids, and Simpson s Rule; to calculate the exact area of a plane region, using limits. (G12/GPE7) 10. To use integration techniques to find anti-derivatives of functions. (A12/CED1-2) 11. To understand the relationship between the derivative and the definite integral and evaluate definite integrals, using the Fundamental Theorem of Calculus. (A12/REI10) Berkeley Heights Public Schools 3

6 COURSE PROFICIENCIES (continued) 12. To use L Hôpital s Rule to evaluate limits and determine convergence of improper integrals and evaluate improper integrals, as limits of definite integrals; use integration by parts and partial fraction techniques, to find anti-derivatives. (A12/REI5,CED4) 13. To solve rectilinear motion, area, and volume problems, using integration techniques. (G12/MG1-2,SRT8) 14. To define and study geometric, harmonic, and alternating series as partial sums and areas of rectangles; to define and test for convergence, using the integral, ratio, and comparison tests. (F12/BF2,LE2) 15. To develop and apply Taylor and Maclaurin series polynomials for various functions (e x, sin x, cos x, 1 / (1-x)) and centered at x = a; to manipulate Taylor series with various shortcuts (substitution, differentiation, and anti-differentiation) and find Lagrange error bounds for Taylor Polynomials. (F12/BF2) 16. To examine functions defined by Power Series and find their radii and intervals of convergence. (F12/BF2) Berkeley Heights Public Schools 4

7 STUDENT PROFICIENCIES The student will be able to: 1. Develop an informal notion of a limit. (F12/BF1,IF4) 2. Evaluate limits numerically, graphically, and analytically. (F12/IF4-5,LE1-4; 8.1/12A1) 3. Determine continuity of a function at a point and on an interval. (F12/IF1) 4. Apply the Intermediate Value Theorem. (F12/LE2) 5. Understand the concept of the slope of a curve. (F12/LE2-3) 6. Calculate derivatives using the limit definition. (A12/CED3-4) 7. Determine derivative of a function, using the constant, power, sum and difference, product and quotient, and chain rules. (F12/IF8) 8. Use implicit differentiation to find the derivative of a function. (F12/IF8,BF1) 9. Utilize the graphing calculator, to determine continuity and differentiability of a function. (F12/BF1) 10. Apply Limit Definition and Rule of Differentiation to various families of functions, including power, exponential, logarithmic, trigonometric and inverse trigonometric. (F12/TF1-9) 11. Evaluate derivatives of parametric, polar, and vector functions. (F12/TF5-7) 12. Use derivatives to find extrema of a function, on a given interval. (A12/REI10) 13. Apply derivatives to identify increasing and decreasing intervals. (F12/IF7,8a) 14. Determine concavity of a function, by applying derivatives. (F12/IF5,7c,e,8a) 15. Apply the derivative to related rate problems. (F12/IF5,7c,e,8a) 16. Apply the derivative to optimization problems. (F12/IF7) 17. Use the derivative to find zeros of a function. (F12/IF7) Berkeley Heights Public Schools 5

8 STUDENT PROFICIENCIES (continued) 18. Apply the derivative to rectilinear motion. (F12/BF1,LE2) 19. Make approximations by applying differentials. (F12/IF6,8) 20. Recognize the Geometric Interpretation of Differential Equations Via Slope Fields and the Relationship Between Slope Fields and Solution Curves for Differential Equations. 21. Analyze planar curves given in parametric, polar, and vector forms, including velocity and acceleration vectors. (N12/CM6-7,VM2,3,5) 22. Solve separable differential equations. 23. Find numerical solutions of differential equations using Euler s Method. 24. Solve logistic differential equations and use them in modeling. (F12/LE1) 25. Apply L Hôpital s Rule to evaluate a limit. (F12/IF8b) 26. Find anti-derivatives, using basic integration techniques. 27. Write and evaluate a sum, using sigma notation. (F12/IF3) 28. Use rectangles, trapezoids, and Simpson s Rule to approximate the area of a plane region. (G12/GPE7) 29. Use limits to find the area of a plane region. 30. Evaluate definite integrals, using limits and properties of integrals. 31. Understand the relationship between the derivative and the definite integral; evaluate a definite integral, using the Fundamental Theorem of Calculus. 32. Understand and use the Mean Value Theorem for integrals. 33. Find the average value of a function over a closed interval. (G12/GPE7) 34. Use U-substitution to evaluate integrals. (F12/IF7-8) 35. Use L Hôpital s Rule to determine the convergence of improper integrals and series. (F12/IF8b) Berkeley Heights Public Schools 6

9 STUDENT PROFICIENCIES (continued) 36. Evaluate improper integrals, as limits of definite integrals. 37. Use integration by parts and partial fraction techniques, to find anti-derivatives. (F12/IF7-8) 38. Solve rectilinear motion problems using definite integrals. (F12/IF2) 39. Find the area between a curve and an axis. (F12/IF2,7) 40. Find the area between two curves. (G12/GPE7) 41. Use the Disk Method, to find the volume of a solid of revolution. (G12/GMD3-4) 42. Use the Shell Method, to find the volume of a solid of revolution. (G12/GMD3-4) 43. Define a series, as a sequence of partial sums. (F12/BF1) 44. Study geometric, harmonic, and alternating series and the convergent or divergent nature of them. (F12/BF2) 45. Explore the terms of series, as areas of rectangles and their relationship to improper integrals, including the integral test and use in testing the convergence of the p-series. 46. Use the Ratio Test and the Comparison Test to determine convergence and divergence. 47. Compare graphically the Taylor approximating polynomial to that of its corresponding functions. 48. Develop the Maclaurin Series polynomials for the functions e x, sin x cos x, and 1/(1 x). 49. Apply the Maclaurin Series and the general Taylor Series centered at x = a. 50. Manipulate Taylor Series formally and with shortcuts, including substitution, differentiation, anti-differentiation, and the formation of new series from known series. 51. Examine functions defined by power series. 52. Find the radius and interval of convergence, for power series. 53. Find Lagrange error bounds, for Taylor polynomials. Berkeley Heights Public Schools 7

10 METHODS OF EVALUATION 1. Homework and class work. 2. Class participation. 3. Tests and quizzes. 4. Projects. 5. Notebooks. 6. Cooperative learning assignments. 7. Mid-term and final examinations. Berkeley Heights Public Schools 8

11 SCOPE AND SEQUENCE COURSE OUTLINE/STUDENT OBJECTIVE The student will be able to: NJ Core National Curriculum Standards Standards/ Grade A12/CED1 F12/IF1-3,6 7,8a F12/BF1a,b, 2,5 F12/LEI1,2,4 F12/TF3 F12/LE2-3 F12/IF4-5 F12/LE1-4 A12/CED 1,2 A12/REI1 F12/BF1 F12/IF4-5 Strands & Indicators Course Outline/Student Objectives 8.1/12 A1 I. Functions (2 Weeks) (Regular Only) A. Understand and Describe the General Behavior of Functions B. Sketch the Graph of a Function C. Analyze and Explain the General Properties of Functions 1. Determine domain and range 2. Find intercepts 3. Determine axis and origin symmetry D. Identify and Apply Fundamental Types of Transformations of Functions E. Recognize a Function as the Composite of Two Other Functions F. Review Concepts of Linear Modeling G. Understand the Role of Functions, as a Unifying Concept in Mathematics H. Use Functions to Model Real World Phenomena 8.1/12 A1 II. Limits (1-2 Weeks) A. Develop the Informal Notion of a Limit B. Evaluate Limits, Using a Variety of Techniques 1. Numerically 2. Graphically (by hand and using a calculator) 3. Analytically C. Understand and Evaluate One Sided Limits D. Understand and Determine Continuity at a Point on an Interval E. Understand and Apply the Intermediate Value Theorem 8.1/12 A1 III. Limits And The Derivative (2-3 Weeks) A. Develop and Explain the Concept of the Slope of a Curve and Use That Concept to Discuss the Information Contained in Graphs Berkeley Heights Public Schools 9

12 F12/LE1-4 F12/TF1-9 A12/APR3-5 A12/REI10 F12/BF1 F12/LE1 F12/IF4,6,7c, e,7,8a,b F12/TF1-9 N12/NQ3 N12/CM6-7 N12/VM2,3,5 III. Limits And The Derivative (continued) B. Use the Limit Definition to Find the Derivative of Functions 1. Power 2. Exponential 3. Logarithmic 4. Trigonometric 5. Inverse Trigonometric C. Understand the Relationship Between Differentiability and Continuity D. Understand and Apply the Basic Rules of Differentiation 1. Constant rule 2. Power rule 3. Sum and difference rules 4. Product and quotient rules 5. Chain rule E. Use Implicit Differentiation, to Find the Derivative of a Function F. Apply Limit Definition and Rules of Differentiation to Various Families of Functions 1. Power 2. Exponential 3. Logarithmic 4. Trigonometric 5. Inverse Trigonometric G. Evaluate Derivatives of Parametric, Polar, and Vector Functions H. Use Graphing Calculator to Determine Continuity and Differentiability of a Function 8.1/12 A1 IV. Applications Of Differentiation (3-4 Weeks) A. Understand the Definition of Extrema and Find the Extrema on an Interval B. Understand How Derivatives Relate to Increasing and Decreasing Intervals C. Understand How Derivatives Relate to Concavity D. Use a Graphing Calculator to Observe Extrema, Increasing/Decreasing Intervals, and Concavity of Functions E. Apply the Derivative to Related Rate Problems F. Apply the Derivative to Optimization Problems G. Apply the Derivative to Find Zeros of a Function Berkeley Heights Public Schools 10

13 F12/IF2,7-8 G12/GPE7 IV. Applications Of Differentiation (continued) H. Apply the Derivative to Rectilinear Motion I. Make Approximations, by Applying Differentials J. Recognize the Geometric Interpretation of Differential Equations Via Slope Fields, and the Relationship Between Slope Fields and Solution Curves for Differential Equations K. Analyze Planar Curves Given in Parametric, Polar, and Vector Form, Including Velocity and Acceleration Vectors L. Solve Separable Differential Equations M. Find Numerical Solutions of Differential Equations, Using Euler s Method N. Solve Logistic Differential Equations and Use Them in Modeling O. Understand L Hôpital s Rule and Apply it to Evaluate a Limit 8.1/12 A1 V. Integrals (3-4 Weeks) A. Write the General Solution for a Differential Equation B. Use Basic Integration Rules to Find Antiderivatives C. Use Sigma Notation to Write and Evaluate a Sum D. Approximate the Area of a Plane Region, Using Rectangles, Trapezoids, and Simpson s Rule E. Find the Area of a Plane Region, Using Limits F. Evaluate Definite Integrals, by Using Limits as Well as Properties of Definite Integrals G. Understand the Relationship Between the Derivative and the Definite Integral and Evaluate a Definite Integral, Using the Fundamental Theorem of Calculus H. Understand and Use the Mean Value Theorem for Integrals I. Find the Average Value of a Function Over a Closed Interval J. Use U-Substitution to Evaluate Integrals K. Use L Hôpital s Rule to Determine the Convergence of Improper Integrals and Series. L. Evaluate Improper Integrals as Limits of Definite Integrals M. Use the Integration by Parts and Partial Fraction Techniques to Find Anti-derivatives. Berkeley Heights Public Schools 11

14 F12/BF1-2 G12/GMD /12 A1 VI. Applications Of Integration A. Apply the Definite Integral to Rectilinear Motion B. Find the Area Between a Function and an Axis C. Find the Area between Two Curves D. Find the Volume of a Solid of Revolution F12/BF /12 A1 VII. Polynomial Approximations And Series (3 Weeks) A. Define a Series as a Sequence of Partial Sums and Define Convergence, in Terms of the Limit of the Sequence of Partial Sums B. Study Geometric, Harmonic, and Alternating Series and the Convergent or Divergent Nature of Them C. Explore the Terms of Series as Areas of Rectangles and Their Relationship to Improper Integrals, Including the Integral Test and Its Use in Testing the Convergence of the p-series. D. Use the Ratio Test and the Comparison Test to Determine Convergence or Divergence E. Compare Graphically the Taylor Approximating Polynomial to That of Its Corresponding Functions F. Develop the Maclaurin Series Polynomials for the Functions e x, sin x, cos x, and 1/(1 x) G. Apply the Maclaurin Series and the General Taylor Series Centered at x = a H. Manipulate Taylor Series Formally and with Shortcuts, Including Substitution, Differentiation, Anti-Differentiation, and the Formation of New Series from Known Series I. Examine Functions Defined by Power Series J. Find the Radius and Interval of Convergence for Power Series K. Find the Lagrange Error Bounds for Taylor Polynomials. Note: The New Jersey Core Curriculum Content Standards can be accessed at Note: Bolded items are for AP Calculus only, BC required, AB as time allows. Berkeley Heights Public Schools 12

15 RESOURCES/ACTIVITES GUIDE Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable, 8 th ed. Boston, Massachusetts: Houghton Mifflin Company, Houghton Mifflin Company Resource Materials: HM mathspace CD-ROM HM ClassPrep with HM Testing CD-Rom Complete Solutions Guide Instructor s Resource Manual Study and Solutions Guide for students Supplementary Textbooks: Anton, Howard. Calculus: Brief Edition, 7 th ed. New York, New York: Wiley Publications, Smith, Robert T. and Roland B. Minton. Calculus, 2 nd ed. New York, New York: McGraw-Hill, Stewart, James. Calculus, 5 th ed. Belmont, California: Thompson-Brooks/Cole, Berkeley Heights Public Schools 13

16 SUGGESTED AUDIO-VISUAL/COMPUTER AIDS 1. Texas Instruments graphing calculator 2. Graphing calculator view screen for overhead projector 3. Overhead projector and transparencies 4. LCD projector and laptop computer 5. Websites: math.college.hmco.com Houghton Mifflin textbook companion website. apcentral.collegeboard.com Official College Board Advanced Placement Program website. archives.math.utk.edu/visual.calculus/ Calculus tutorials with visual demonstrations. mathforum.org/dr.math Contains archive of student questions and allows students to ask questions. calculus.org Contains a listing of student and teacher online resources, for the study of calculus. calc101.com Provides step by step help for common calculus topics. Berkeley Heights Public Schools 14

17 SUGGESTED MATERIALS Resources for Students Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable. 8 th ed. Boston, Massachusetts: Houghton Mifflin Company, Houghton Mifflin Company Resource Materials: HM mathspace CD-ROM Study and Solutions Guide for Students Resources for Teacher Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable. 8 th ed. Boston, Massachusetts: Houghton Mifflin Company, Houghton Mifflin Company Resource Materials: HM mathspace CD-ROM HM ClassPrep with HM Testing CD-Rom Complete Solutions Guide Instructor s Resource Manual Anton, Howard. Calculus: Brief Edition 7 th ed. New York, New York: Wiley Publications, Smith, Robert T., and Roland B. Minton. Calculus. 2 nd ed. New York, New York: McGraw-Hill, Stewart, James. Calculus. 5 th ed. Belmont, California: Thompson-Brooks/Cole, Berkeley Heights Public Schools 15