VIEWS: 88 PAGES: 18 POSTED ON: 8/8/2012 Public Domain
AP Calculus AB Syllabus This course is designed to teach the student to solve problems using the concepts of calculus graphically, numerically, analytically, and verbally. The main objective of this course is for the student to learn to use calculus to model and solve real life problems. This course will give students a strong foundation to succeed in any future endeavors in mathematics. RESOURCE REQUIREMENTS (Resource Requirements #1 and #2) Text: Calculus of a Single Variable (Sixth Edition) By Larson, Hostetler, Edwards 1998 Supplementary Materials: College Board AP Central AP Calculus Institute University of Georgia Summer 2006 Activities and Resource Manual by Dane Marshall Instructor: Dane Marshall TI Interactive! Texas Instruments, 2001 Derive 5 Texas Instruments, 2000 . Prerequisites: Students in this class are required to have completed Honors Precalculus or must be taking Honors Precalculus. Text: Precalculus (Fourth Edition) By Larson & Hostetler 1997 Graphing calculators: Each student is required to have a TI-83 or TI-84. I have a TI- 89 for use in the classroom. The calculator is used extensively in this class to explore, discover, and reinforce concepts that are learned. Students are allowed to use calculators on all assignments and on most assessments. Grading Policies: Our school year is divided into four 9-week grading periods. Each nine week grade is 42.5% of the semester grade and the semester exam is 15%. Each nine-week grade will consist of assessments from tests, quizzes, and projects. Homework is not graded but required. Page 1 of 18 Assessments: Tests are designed in the same format of the AP Exam – multiple- choice and free-response. Questions from previous AP Exams influence the free- response and text test generator questions are used for the multiple-choice. Teacher designed problems are also used. Time Line: The time spent on each chapter is an estimate and varies each year depending on the caliber of students in this class. Each semester has 90 days. The approximate times include time for teaching and assessment. The twelve missing days are due to four scheduled exam days and to at least eight days of review for the AP Calculus AB Exam. A day consists of 49 minutes of class instruction. CURRICULAR REQUIREMENTS (Curricular Requirement #2, #3, #4, #5) The Course Outline is a list of the topics from the text that are covered and the sequence in which those topics are covered. Supplementary materials are used to enhance the objectives learned in the text or to fulfill AP Course Audit requirements. The Standards required by the State of Tennessee are also included. The Student Syllabus by Nine Weeks is a detailed outline of problems and notes that have been assigned during the 2006-2007 school year. This syllabus will vary slightly from year to year as the course continues to become more rigorous. The Class Activities Section is a sampling of activities used in this course. Course Outline (Curricular Requirement #2) Course description: Calculus is an advanced mathematics course that uses meaningful problems and appropriate technology to develop concepts and applications related to continuity and discontinuity of functions and differentiation, and integration. Prerequisite Text: Calculus of a Single Variable (Sixth Edition) By Larson, Hostetler, Edwards 1998 Chapter P Preparation for Calculus (Prerequisite and/or Summer Assignment and 1 day in class) P.1 Graphs and models The graph of an equation Intercepts of a graph Symmetry of a graph Points of intersection Page 2 of 18 Mathematical models P.2 Linear Models and Rates of Change The slope of a line Equations of a line Ratios and Rates of change Graphing linear models Parallel and perpendicular lines P.3 Functions and Their Graph Functions and function notation Domain and range of a function The graph of a function Transformations of functions Classifications and combinations of functions P.4 Fitting Models to Data Linear Regressions Quadratic regressions Trigonometric regressions Tennessee Standard 1.0: Functions Students will expand the concept of functions to include the analysis and interpretation of both continuous and discontinuous functions in problem situations and the development of the concept of limit. Learning Expectations: Students will: 1.1 demonstrate an understanding of the concepts and applications related to a variety of continuous functions; 1.2 calculate and estimate limits; 1.3 represent a variety of functions graphically; 1.4 use graphical representations to demonstrate an understanding of asymptotes; 1.5 use a variety of methods to analyze and interpret functions; 1.6 apply functions in problem situations Student Performance Indicators: analyze the graphs of polynomial, rational, radical, and transcendental functions using appropriate technology; predict and explain the observed local and global behavior of a function; calculate limits using algebra; estimate limits from graphs or tables of data. demonstrate an understanding of asymptotes in terms of graphical behavior; describe asymptotic behavior in terms of infinite limits and limits at infinity; compare relative magnitudes of functions and their rates of change. Page 3 of 18 demonstrate an understanding continuity in terms of limits; demonstrate a geometric understanding of graphs of continuous functions. Text: Calculus of a Single Variable (Sixth Edition) By Larson, Hostetler, Edwards 1998 Chapter 1 Limits and Properties (approximately 20 days) 1.1 Preview of Calculus 1.2 Finding Limits Graphically and Analytically An introduction to limits Limits that fail to exist A formal definition of a limit 1.3 Evaluating Limits Analytically Properties of limits Strategies for finding limits Cancellation and rationalization techniques The squeeze theorem 1.4 Continuity and One-sided Limits Continuity at a point and on an open interval One-sided limits and continuity on a closed interval Properties of continuity The Intermediate Value Theorem 1.5 Infinite Limits Infinite limits Vertical asymptotes Tennessee Standard 2.0: Derivatives Students will extend the concept of slope of a line to develop the concept of derivative. Learning Expectations: The student will: 2.1 define, represent and interpret the concept of derivative; 2.2 use the derivative of a function to characterize the function and vice versa; 2.1 connect the relationships among a function and its first and second derivative; 2.2 apply basic rules for differentiation; 2.3 apply derivatives in problem situations. Student Performance Indicators: Page 4 of 18 represent the concept of the derivative geometrically, numerically, and analytically; interpret the derivative as an instantaneous rate of change; define the derivative as the limit of the difference quotient; articulate the relationship between differentiability and continuity. articulate corresponding characteristics of graphs of f and f ´ ; communicate the relationship between the increasing and decreasing behavior f and the sign of f ´ ; demonstrate an understanding of the Mean Value Theorem and its geometric consequence; translate verbal descriptions into equations involving derivatives and vice versa. articulate corresponding characteristics of the graphs of f , f ´ , and f ´´ ; communicate the relationship between the concavity of f and the sign of f ´´ ; identify points of inflection; analyze curves using the notions of monotonicity and concavity; optimization, both absolute (global) and relative (local) extrema; model rates of change, including related rates problems; use implicit differentiation to find the derivative of an inverse function; interpret the derivative as a rate of change in varied applied contexts; apply basic rules for the derivative of basic functions and their sum, product, and quotient; use the chain rule and implicit differentiation Text: Calculus of a Single Variable (Sixth Edition) By Larson, Hostetler, Edwards 1998 Chapter 2 Differentiation (approximately 30 days) 2.1 The derivative and the Tangent Line Problem The tangent line problem The derivative of a function Differentiability and continuity 2.2 Basic Differentiation Rules and Rates of Change The Constant Rule The Power Rule The Constant Multiple Rule The Sum and Difference Rules Derivatives of sine and cosine functions Rates of Change 2.3 The Product and Quotient Rules and Higher-order Derivatives The Product Rule The Quotient Rule Derivatives of trigonometric functions Higher-order derivatives 2.4 The Chain Rule The Chain Rule The General Power Rule Simplifying derivatives Page 5 of 18 Trigonometric functions and the Chain Rule 2.5 Implicit Differentiation Implicit and explicit functions Implicit differentiation 2.6 Related Rates Finding related rates Problems with related rates Chapter 3 Applications of Differentiation (approximately 27 days) 3.1 Extrema on an Interval Extrema of a function Relative extrema and critical numbers Finding the extrema on a closed interval 3.2 Rolle’s Theorem and the Mean Value Theorem 3.3 Increasing and Decreasing Functions and the First Derivative Test 3.4 Concavity and the Second Derivative Test 3.5 Limits at Infinity 3.6 A Summary of Curve Sketching 3.8 Newton’s Method 3.7 Optimization Problems 3.10 Business and Economic Applications 3.9 Differentials Slope fields (supplementary material) Rectilinear motion (supplementary material) Tennessee Standard 3.0: Integrals Students will develop the concepts of integrals and their applications. Learning Expectations: The student will: 3.1 define and apply basic properties of definite integrals; 3.2 evaluate or approximate define integrals; Page 6 of 18 3.3 apply techniques of antidifferentiation. Student Performance Indicators: communicate the relationship between a Riemann sum and a definite integral; apply basic properties of definite integrals; evaluate definite integrals using the Fundamental Theorem; apply techniques of antidifferentiation; find specific antiderivatives using initial conditions, including applications to motion along a line; use separable differential equations in modeling; use Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values Text: Calculus of a Single Variable (Sixth Edition) By Larson, Hostetler, Edwards 1998 Chapter 4 Integration (approximately 30 days) 4.1 Antiderivatives and the Indefinite Integration Antiderivatives Notation for antiderivatives Basic integration rules Initial conditions and particular solution 4.2 Area Sigma notation Area The area of a plane region Upper and lower sums 4.3 Riemann Sums and Definite Integrals Riemann sums Definite integrals Properties of definite integrals 4.4 The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus The Mean Value Theorem of Integrals Average value of a function The Second Fundamental Theorem of Calculus The distance traveled by a particle (Supplementary material) 4.5 Integration by Substitution Pattern recognition Change of variables The general Power Rule for integration Change of variables for definite integral Page 7 of 18 Integration of even and odd functions 4.6 Numerical Integration The Trapezoidal Rule Simpson’s Rule Error analysis Chapter 5 Logarithmic, Exponential, and other Transcendental Functions (approximately 30 days) 5.1 The Natural Logarithmic Function and Differentiation The natural logarithmic function The number e The derivative of the natural logarithmic function 5.2 The natural logarithmic function and integration Log Rule for integration Integrals of trigonometric functions 5.3 Inverse Functions Inverse functions Existence of an inverse function Derivative of an inverse function 5.4 Exponential Functions: Differentiation and Integration The natural exponential function Derivatives of exponential functions Integrals of exponential functions 5.5 Bases of than e and Applications Bases other than e Differentiation and integration Applications of exponential functions 5.6 Differentiation Equations: Growth and Decay Differential equations Growth and decay models 5.7 Differential Equations: Separation of Variable General and particular solutions Separation of variables Homogeneous differential equations Applications 5.8 Inverse Trigonometric Functions and Differentiation Inverse trigonometric functions Derivatives of inverse trigonometric functions Review of basic differentiation rules Page 8 of 18 5.9 Inverse Trigonometric Functions and Integration Integrals involving inverse trigonometric functions Completing the square Review of basic integration rules 5.10 Hyperbolic Functions Hyperbolic Functions Differentiation and integration of hyperbolic functions Differentiation and integration of inverse hyperbolic functions Chapter 6 Applications of Integration (approximately 18 days) 6.1 Area of a Region between Two Curves Area of a region between two curves Area of a region between intersecting curves 6.2 Volume: The Disc Method The Disc Method The Washer Method Solids with known cross-sections 6.3 Volume: The Shell Method The Shell Method Comparison of Disc and Shell Method 6.4 Arc Lengths and Surfaces of Revolution Arc length Area of a surface of revolution Page 9 of 18 Student Syllabus by Nine Weeks (Curricular Requirement #2) First Nine Weeks AP Calculus AB Optional Summer Assignment Section P1 (Page 3 - 7) Optional Summer Assignment Page 9 (1 - 4,7,13,15,17,23,25,27,29) Optional Summer Assignment Section P1 (Page 3 - 7) Optional Summer Assignment Page 9 (31 - 51 odd, 55 ) Optional Summer Assignment Section P2 ( Page 11 - 16) Optional Summer Assignment Page17 ( 1 - 9 odd, 15,17,21-25 odd, 32,36,37) Optional Summer Assignment Section P3 Optional Summer Assignment Page 28 (3 - 7, 11 - 21, 35,41,47,49) Optional Summer Assignment Section P4 (Page9 #55) Optional Summer Assignment Page34 ( 1 - 5,7,9,13,15) Optional Summer Assignment Review Page 37 ( 1- 7 odd,21,25,29-33,42) NO TEST over Chapter P 8/7/06 1. Section 1.1 8/7/06 2. Page 46 ( 1,3,5,7,11) 8/8/06 3. Section 1.2 8/8/06 4. Page 53 ( 1 - 19 odd) 8/9/06 5. Section 1.2 8/9/06 6. Page 53 ( 21 - 37 odd) 8/10/06 7. Section 1.3 8/10/06 8. Page 64 ( 1 - 27 odd) 8/11/06 9. Section 1.3 8/11/06 10. Page 64 ( 29 - 43 odd) 8/14/2006 - 8/16/06 11. Section 1.3 8/14/2006 - 8/16/06 12. Page 64 ( 45 - 67 odd 87 89 90 91 92) and Limits of Trigonometric Functions WS 13. Review for Test (Page 87 #3 - 20 all) 8/17/06 TEST 1.1 - 1.3 8/18/06 14. Section 1.4 8/18/06 15. Page 75 ( 1 - 25 odd) 8/21/06 16. Section 1.4 cont'd 8/21/06 17. Page 75 ( 27 -51 odd) 8/22/06 18. Section 1.4 cont'd 8/22/06 19. Page 75 ( 57 - 81 odd omit 71,77) 8/23/06 - 8/24/06 Free-response Problem: Continuity 8/25/06 TEST 1.1 - 1.4 8/28/06 20. Section 1.5 8/28/06 21. Page 84 ( 1 - 43 odd) 8/29/06 22. Section 1.5 8/29/06 23. Page 84 (16,24,38,47 - 52 all) 8/29/06 24. Page 84 (48 - 52 all) 8/30/06 24. Section 2.1 8/30/06 25. Page 99 (1,3,5 - 15 odd) Page 10 of 18 8/31/06 26. Section 2.1 8/31/06 27. Page 99 (17,19,23,25, 51 - 60 odd) 9/1/06 28. Review Page87 ( 23 - 31 35 - 43) Page99 even 9/8/06 TEST 1.1 - 2.1 9/7/06 29. Section 2.2 9/7/06 30. Page 110 ( 1 - 12) 9/11/06 31. Section 2.2 9/11/06 32. Page 110 ( 13 - 42 odd) 9/12/06 33. Section 2.2 9/12/06 34. Page 111 ( 43 - 51 59 65 71 72 73 74 79 81) 9/13/06 35. Section 2.3 9/13/06 36. Page 121 ( 1 - 21 odd) 9/14/06 37. Section 2.3 9/14/06 38. Page 121 (23 - 41 odd) 9/20/06 39. SECTION 2.3 9/20/06 40. Page 122 (47 - 61, 65 73 - 81ODD) 9/21/06 41. Section 2.4 9/21/06 42. Page 130 ( 1 - 25 odd) 9/25/06 43. Section 2.4 9/25/06 44. Page 130 (27, 29, 43 - 51, ODD) 9/26/06 45. Section 2.4 9/26/06 46. Page 130 ( 53 - 77 odd) 9/27/06 47. Review for test 2.2 2.3 2.4 Page 150 ( 7 - 73) 9/28/06 48. Section 2.4 9/28/06 49. Page 130 ( 53 - 77 even) 9/29/06 TEST 1.1 - 2.4 10/2/06 50. Section 2.5 10/2/06 51. Page 139 ( 9 - 25 odd) 52. Section 2.5 53. Page 139 ( 9 - 25 even) 10/3/06 52. Section 2.5 10/3/06 53. Page 139 ( 9 - 25 even) 10/3/06 54. Section 2.5 10/3/06 55. Page 139 (27, 31, 35, 41, 45) 10/4/06 56. Section 2.5 10/4/06 57. Page 150 ( 7 - 57 pick and choose) 10/5/06 TEST 2.5 10/6/06 58. Section 2.6 10/6/06 59. Page 146 ( 1 - 7 all, 9) Second Nine Weeks AP Calculus AB 10/16/06 1 Worksheet on related rates 10/17/2006 - 10/24/06 2 Worksheet #2 on related rates 10/17/2006 - 10/24/06 3 Section 2.6 (OMIT) Page 11 of 18 10/17/2006 - 10/24/06 4 Page 146 (11,12,13,14,16) (OMIT) 10/17/2006 - 10/24/06 5 Section 2.6 (OMIT) 10/17/2006 - 10/24/06 6 Page146 (17 - 21 all) (OMIT) 10/17/2006 - 10/24/06 7 Section 2.6 (OMIT) 10/17/2006 - 10/24/06 8 Page 146 ( 22,24,25,29,30,31) (OMIT) 10/27/06 TEST 2.6 10/25/06 9 Section 3.1 10/25/06 10 Page 160 ( 1 - 21) 10/26/06 11 Section 3.1 10/25/06 12 Page 160 ( 23 - 40) omit 13 Section 3.1 omit 14 Page160 (4) 10/31/06 15 Section 3.2 10/31/06 16 Page 167 (1 - 21 odd 27 29 ) 11/1/06 17 Section 3.3 11/3/06 18 Page 176 ( 1 - 19 odd) 11/6/006 19 Section 3.3 and even problems Page176 11/1/06 19 Page 176 (21 -47 odd) omit 20 Review for test 3.1 3.2 3.3 11/7/06 21 Section 3.4 11/7/06 22 Page 184 (7 - 19 odd) 11/8/06 23 Section 3.4 11/8/06 24 Page 184 ( 21 - 35 odd) 11/9/06 25 Section 3.4 11/9/06 26 Page 184 (21 - 36 even) 11/10/06 Test 3.1 3.2 3.3 3.4 11/13/06 27 Section 3.5 11/13/06 28 Page 193 ( 1 - 23 odd) 11/14/06 29 Section 3.5 11/14/06 30 Page 193 ( 35 - 51 odd) 11/15/06 31 Section 3.6 11/15/06 32 Page 202 ( 7, 9, 29, 33, 37) omit 33 Section 3.6 omit 34 Page 202 (11 13 31 35 39) omit 35 Review for Test 3.5 3.6 omit 36 Page 202 ( 17 21 29 31) OPTIONAL 11/17/06 TEST 3.1 3.2 3.3 3.4 3.5 3.6 11/20/06 37 Section 3.7 11/20/06 38 Page 210 ( 1- 10) 11/20/06 39 Section 3.7 11/20/06 40 WS 3.7 11/21/2006 - 11/28/06 41 Optimization WS (44 - 58) 42 Review for Test 3.7 (Quiz 3.7 B) 11/30/06 TEST 3.7 11/29/06 43 Section 3.8 11/29/06 44 Page 219 ( 1- 12 all) 11/29/06 45 Section 3.8 Page 12 of 18 11/29/06 46 Page 219 (13 - 16, 21 - 24, 27, 29) 12/1/06 47 Section 3.9 & Rectilinear Measurement 12/1/06 48 Page 226 ( 7 - 19 odd) 12/1/06 49 Section 3.9 12/1/06 50 Page 226 (21 23 25 27 39 41) omit 51 Worksheet 3.8 3.9 12/4/06 52 Section 3.10 12/4/06 53 Page 232 ( 3 - 15 odd, 31 33) omit 54 Review for test Take-home due 12/08/06 TEST 3.8 3.9 3.10 12/6/2006 - 12/14/06 PROJECT PRESENTATIONS (Free-response AP Problems) Semester Exam Third Nine Weeks AP Calculus AB 1/3/2007 1 Section 4.1 1/3/2007 2 Page 249 ( 1 - 36 odd) 1/4/2007 3 Section 4.1 1/4/2007 4 Page 249 ( 1 - 36 even) 1/5/2007 5 Section 4.1 & Introduction to Slope Fields 1/5/2007 6 WS on antiderivatives 1/8/2007 7 Section 4.1 1/8/2007 8 Page 249 (37 - 65 odd) 1/9/2007 9 Review for TEST 4.1 1/9/2007 10 Review sheet 1/11/2007 TEST 4.1 1/10/2007 11 Section 4.2 1/10/2007 12 Page 261 ( 1 - 9 & 15 - 21 odd) 1/12/2007 – 01/16/07 13 Section 4.2 1/16/2007 14 Page 261 ( 23 - 33 all) 01/17/07 15 Section 4.2 01/17/07 16 Page 261 (35-38) 01/18/07 17 Section 4.2 01/18/07 /01/19/27 18 Page 262 ( 41 - 44 all) 01/22/07 19 Review for Test 01/22/07 20 Page 262 ( 45 - 48) 01/23/07 TEST 4.2 01/24/07 21 Section 4.3 01/24/07 22 Page 271 ( 1 - 19 odd) 01/25/07 23 Section 4.3 01/25/07 24 Page 272 (21,23,25,29,43) 01/26/07 25 Review for Test 4.3 01/26/07 26 Page 211 ( 2 -30 EVEN 43) 01/29/07 / 01/30/07 TEST 4.3 / TI-Interactive Activity: Area Under the Curve 01/31/047 27 Section 4.4 01/31/07 28 Page 283 ( 1 - 21 odd) Page 13 of 18 02/01/07 29 Section 4.4 02/01/07 30 Page283 ( 23 - 41, 47, 49 67 69 73 79) 02/02/07 31 Review for Test 02/05/07 TEST 4.4 32 Section 4.5 33 Page 291 ( 1 - 20 all) 34 Section 4.5 35 Page 297 ( 21 -51 odd) 36 Section 4.5 37 Page 297 ( 65 - 75 odd) 38 Review for test 4.5 TEST 4.5 39 Section 4.6 40 Page 304 ( 1 - 19 odd Trapezoidal Rule) 41 Section 4.6 Simpson’s Rule 42 Page 304 ( 1 – 19 odd) 43 Review for test Test 4.6 44 Section 5.1 45 Page 318 (21 27 29 37 - 65 odd) 46 Section 5.1 47 Page 318 ( 67 69 73 75 83 85) 48 Section 5-2 49 Page 327 ( 1 - 9, 13, 23, 27,29,33, 37,47) 50 Review for TEST 5.1 5.2 51 Handout TEST 5.1 5.2 52 Section 5.3 53 Page 335 ( 3 7 9-12 13 19 23 51 53 57 61 73 79 89) 54 Section 5.4 55 Page 344 ( 1-7 29 - 47 odd 49 57) 56 Review for test 5.3 5.4 Test 5.3 5.4 Fourth Nine Weeks AP Calculus AB 1 Section 5.5 2 Page 354 (1 - 25 29 - 43 odd) 3 Section 5.5 4 Page 354 ( 47 51 - 75 odd) 5 Section 5.5 6 Page 354 ( 30 - 48 even, 45, 53 55 57 59) 7 Section 5.6 8 Page 363 ( 1- 25 odd) 9 Section 5.6 10 Page 363 ( 27 - odd) Page 14 of 18 11 Review for Test TEST 5.5 5.6 12 Section 5.7 13 Page 374 ( 1 - 23 odd) 14 Section 5.7 15 Page 374 ( 25 - 59 odd) 16 Section 5.7 TEST 5.7 17 Page 375 (61 - 85 odd) 18 Section 5.8 19 Page 383 ( 1 - 40 odd) 20 Section 5.8 21 Page 383 (41 - 61 odd) Test 5.8 22 Section 5.9 23 Page 390 ( 1 - 31 odd) 24 Section 5.9 25 Page 390 (33 - 41 odd) 26 Page 390 ( 34 - 40 even) TEST 5.9 27 Section 5.10 28 Page 400 ( 1 - 29 odd) 29 Section 5.10 30 Page 400 (31 - 51 odd) 31 Section 5.10 32 Page 400 ( 53 - 77 odd) TEST 5.10 33 Section 6.1 34 Page 413 (1- 20) odd 35 Section 6.1 36 Page 413 (21 - 31,65)odd 37 Section 6.2 38 Page 423 ( 1- 9 all) 39 Section 6.2 Test 6.1 6.2 40 Page 423 ( 11- 19 odd) 41 Section 6.1 & 6.2 42 Page 413 (21 - 31,65) EVEN Page 423 12 - 20 even 43 Section 6.3 44 Page 432 (1 - 11 odd) 45 Section 6.3 46 Page 432 (13 - 23 odd) 47 Section 6.3 48 Page 432 (13 - 23 even) Test 6.3 PROJECT PRESENTATIONS (Free-response AP Problems) Final Exam Page 15 of 18 Class Activities Section (Curricular Requirement #3, #4, #5) Calculators (Curricular Requirement #5) In my class, calculators are used daily to explore, discover, and reinforce concepts. Examples of use include finding a root; sketching a function in a specified window; approximating the derivative at a point; approximating the value of a definite integral Project Presentations (Curricular Requirement #3, #4, #5) Each student is given a notebook of previous AP CALCULUS AB Exam Free-response problems. Each student is required to choose 3-4 problems. At the end of each semester, each student is required to solve the problems and explain the solutions to the class. Area and Pre-Riemann Sums (Curricular Requirement #3, #4, #5) As a introduction to area under a curve, we graph a simple function and use rectangles to estimate the area. We use 3 rectangles and find the area using both right- and left-endpoints; then 6 rectangles; then 12 rectangles. Students come to the conclusion that the exact area is between the two and the more rectangles, the better estimate of area. We discuss and derive a summation formula needed to solve each of these problems. This is checked on the calculator. This leads to finding Page 16 of 18 the summation formula for every function as the number of rectangles increase indefinitely. Students are then given a part of a piece of graph paper that has been randomly cut in half. Using skills learned in the above activity, they have to find the area under the curve and find the person that has the matching half. Area under a Curve (Curricular Requirement #3, #4, #5) TI Interactive! Activity 10 Texas Instruments, 2001 Students first investigate the area under a curve by summing midpoint rectangles. The same method could be used with left-endpoint and right-endpoint. By the end of the activity, the students are able to find the numerical integral for any continuous function and the indefinite integral of the function. Page 17 of 18 Page 18 of 18