AP AB Calculus Syllabus

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1 AP AB Calculus Syllabus Pre-Requisites Three years of High School Mathematics (including Precalculus) with emphasis on functions and their behavior and modeling real world applications. Course Overview Calculus provides students a framework for putting together many of the pieces of their mathematical education into a coherent whole and provides students with a powerful tool for understanding the world around them. This AP Calculus AB is an enriched mathematics course and curriculum that is designed to help students in their understanding of the calculus curriculum and to provide and prepare them for the mathematics needed to be successful in post secondary studies. Students should not be merely skilled in the various techniques of calculus; they should appreciate it as an integrated system for observing, studying, and describing motion and change. Students are introduced to the wonderful and exciting world of higher mathematics through a comprehensive study of all of the objectives outlined in the AP Calculus Course Description. In addition, students are encouraged to take the AP Calculus AB exam. The following Curricular and Resource requirements below will be sited: Curricular Requirements C 1 The teacher has read the most recent AP Calculus Course Description, available as a free download on the AP Calculus AB Course Home Page. C 2 The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description. C The 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. C 4 The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences. C 5 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. Resource Requirements R 1 The school ensures that each student has a college-level calculus textbook (supplemented when necessary to meet the curricular requirements) for individual use inside and outside of the classroom. R 2 The school ensures that each student has a graphing calculator for individual use inside and outside of the classroom, with all the required capabilities listed in the AP Calculus Course Description. (A list of approved graphing calculators is available on AP Central on the AP Calculus AB Course Home Page.) 1

2 Course Planner, Guide, and Topic Outline [C1 and C2]. Note: Our school is on a following combined schedule: every subject is taught three classes a week one class lasting 4 minutes (one period) on Monday and two classes lasting 88 minutes each (two periods) and meeting every other day. This way the equivalent of the five days (periods) a week is achieved. The primary text used: Stewart, James. Calculus 7e. Brooks/Cole, Cengage Learning, [R1]. Below is the sequence of our AP Calculus AB course. Section numbers refer to the primary text. All timelines are approximate and include review and testing times. These will be adjusted according to student comprehension. Unit Section-description Planned (in periods) 1 Functions and Limits 14 Graphing Calculators and Computers (Appendix G). Calculus AB item I Four ways to represent a function. Calculus AB item I Mathematical Models: A Catalog of Essential Functions. Prerequisite for later AP Calculus Material 1. New Functions from Old Functions. Prerequisite for later AP Calculus Material The Tangent and Velocity Problems. Calculus AB item I.2(a) The Limit of a Function. Calculus AB items I.2(c), I.(a), and I.(b) Calculating Limits Using the Limits Laws. Calculus AB item I.2(b) Continuity. Calculus AB items I.4(a)-(c) 2 Unit Review. Include Principles of Problem solving (p.97)and AP AB Review questions AP1-1 Unit 1 Test day 2 2 Derivatives Derivates and Rates of Change. Calculus AB items II.1, II. Writing Project Early Methods for Finding Tangents assigned for homework 2.2 The Derivative as a Function. Calculus AB items II.1, II.(a), II.4(a), II.5(f) 2 2. Differentiation Formulas. Calculus AB items II.1(b), II.6(a), II.6(b). Applied Project Building a better Roller Coaster 2.4 Derivatives of Trigonometric Functions. Calculus AB item II.6(a) The Chain Rule. Calculus AB item II.6(c) Applied Project Where Should a Pilot Start Descent? Implicit Differentiation. Calculus AB item II.6(c) Lab project Families of Implicit Curves Rates of Change in the Natural and Social Sciences. Calculus AB items II.2(c), II.(d), II.5(f) 2.8 Related rates. Calculus AB items II.(d), II.5(d)

3 2.9 Linear Approximations and Differentials. Calculus AB item II.2(b) 2 Unit Review. Include Problems Plus (p.194) and AP AB Review questions AP2-1 Unit 2 Test day 2 Applications of Differentiation 21.1 Maximum and Minimum Values. Calculus AB items I.4(c), II.5(c) Applied Project The Calculus of Rainbows.2 The Mean value Theorem. Calculus AB item II.(c) 1. How Derivatives Affect the Shape of a Graph. Calculus AB items II.(b), II.4(b),(c), II.5(a),(c).4 Limits and Infinity: Horizontal Asymptotes. Calculus AB items I.(a),(b) 1.5 Summary of Curve Sketching. (optional material, skip if time issue will be addressed during the Review and Preparation for the AP exam).6 Graphing with Calculus and Calculators. Calculus AB item I Optimization Problems. Calculus AB item II.5(c) Applied Project The Shape of a Can.9 Antiderivatives. Calculus AB items II.4(a), III.4(a),(c), III.5(a) Unit Review. Include Problems Plus (p.41) and AP AB Review questions AP-1 Unit Test day 2 MIDTERM Exam 4 Integrals Areas and Distances. Calculus AB items III.1(a), III The Definite Integral. Calculus AB items III.1(a),(b),(c), III.6 Discovery Project Area Functions 4. The Fundamental Theorem of Calculus. Calculus AB items III.1(b), III.2, III..(a),(b) Indefinite integrals and the Net Change Theorem. Calculus AB items III.1(b), III.4(a), III.5(a) 4.5 The Substitution Rule. Calculus AB item III.4(b) 1 Unit Review. Include Problems Plus (p.41) and AP AB Review questions AP4-1 Unit 4 Test day 2 5 Applications of Integration Areas between curves. Calculus AB item III.2 Applied Project The Gini Index 5.2 Volume. Calculus AB item III. 5.5 Average Value of a Function. Calculus AB item III.2 Applied Project Calculus and Baseball Unit Review. Include Problems Plus (p.80) and AP AB Review questions AP5-1 Unit 5 Test day Inverse Functions (Exponential, Logarithmic, and Inverse Trigonometric functions) 6.1 Inverse Functions. Calculus AB item II.5(e) 1 12

4 6.2 Exponential Functions and their Derivatives. Calculus AB items I.(c), II.6(a) 1 6. Logarithmic Functions. Calculus AB item I.(c) Derivative of Logarithmic Functions. Calculus AB item II.6(a) Exponential Growth and Decay. Calculus AB items II.(d), III.5(b) Inverse Trigonometric Functions. Calculus AB item II.6(a) 1 Unit Review. Include Problems Plus (p.485) and AP AB Review questions AP6-1 Unit 6 Test day 2 7 Techniques of Integration Trigonometric Integrals. Calculus AB item III.4(b) Appropriate Integration. Calculus AB item III.6 1 Unit Review. Include Problems Plus (p.557) and AP AB Review questions AP7-1 2 Unit 7 Test day 2 8 Differential equations (Chapter 9) Modeling with Differential Equations. Calculus AB items II.(d), III.5(b),(c) Direction Fields and Euler s Method. Calculus AB items II.5(g), Skip Euler s Method Calculus BC II.5(b) 9. Separable Equations. Calculus AB item III.5(b) 2 Unit Review. Include Problems Plus (p.657) and AP AB Review questions AP9-1 2 Unit 9 Test day 2 9 Review and Preparation for the AP Exam AP Exam 11 Post AP Exam topics and various projects 8 12 Preparation for the Final Exam 5 1 Final Exam 2 Explaining the Content In Unit One, multiple representations of functions are stressed: verbal, numerical, visual and algebraic. A discussion of mathematical models leads to a review of standard functions from these points of view. Students develop a library of basic functions, with their various characteristics (domain, continuity, symmetry, end behavior, etc.). Throughout the course, these functions are referred back to again and again so that students begin to see those basic functions arise from information given in exercises. The material of limits is motivated by a prior discussion of the tangent and velocity problems. Limits are treated from descriptive, graphical, numerical, and algebraic point of view. A limit exploration is done using graphing calculators. Using tables, students use differing values of h that they select to determine the effect on the limit of a function. After completing Unit One students will be able to calculate average and instantaneous speeds; define and evaluate limits analytically (algebraically); use the concept of limits to define continuity; identify continuous functions graphically and analytically; apply the Intermediate Value and Squeeze Theorems; find equations of the tangent line and normal line to a curve at a given point. [C2], [C], [C4], and [C5] 4

5 In Unit Two, the limit definition of the derivative is compared and contrasted with the graphing calculator approach. This gives both an appreciation for the power of the calculator, but also a demonstration of its occasional deficiencies (i.e. finding the derivative of the absolute value function at x = 0). The material on derivatives is covered in two sections in order to give students more time to get used to the idea of a derivative as a function. Higher derivatives are introduced in section 2.2. After completing Unit Two students will be able to interpret the derivative in varied applied contexts, including velocity, speed, and acceleration; explain the relationship between the continuity and differentiability; visually match the graph of a function with its derivative function and explain all relevant characteristics; graph the derivative of a function given numerically with data;; calculate derivatives of basic functions and properly apply the Power Rule, Trig Rules/Identities, Product Rule, and Quotient Rule; apply the Chain Rule to find the derivative of a composition of two functions; find derivatives of functions that are defined implicitly. Students will explore and investigate functions differentiable at x=a as well as those not differentiable at x=a algebraically and graphically. [C2], [C], [C4], and [C5] Before starting Unit Three, students have already investigated some of the applications of derivatives, and because they know the differentiation rules; they are in a better position to pursue the applications of differentiation in greater depth. In Unit Three, they will learn how derivatives affect the shape of a graph of a function, and in particular, how they help us locate maximum and minimum values of functions. Corresponding characteristics of the function and its first and second derivatives, concavity and points of inflection (p ) will be emphasized. Many practical problems will be introduced, requiring minimizing a cost or maximizing an area, or somehow finding the best possible outcome of a situation. In particular, students will be able to investigate the optimal shape of a can and to explain the location of the rainbow in the sky. After completing Unit Three students will be able to find relative extrema using the First Derivative Test and the Second Derivative Test; prove and apply the Mean Value Theorem; better understand the Second Derivative: analyze and sketch graphs of polynomial, radical, rational and trigonometric functions by determining intervals of increase/decrease and concavity, finding extrema and inflection points and using previously studied techniques. Graphing with Technology emphasizes the interaction between the calculus and calculators and the analysis of families of curves. [C2], [C], [C4], and [C5]. Previously, in Unit Two students have used the tangent and velocity problems to introduce the derivative. In much the same way, by using the area problem and the distance problem, Unit Four introduces definite integral, with sigma notation introduced as needed. Emphasis is placed on explaining the meanings of integrals in various contexts and estimating their values from graphs and tables. After completing Unit Four students will be able to approximate the area under the graph of a function by using rectangle approximation methods; interpret the area under a curve as a net accumulation of a rate of change; express the area under the curve as a definite integral and as a limit of Riemann sums; apply the basic properties of definite integrals in order to evaluate them; find the antiderivative (indefinite integral) of a function using knowledge of derivatives; perform the process of indefinite integration using basic integration rules; understand and apply the Fundamental Theorem of Calculus to evaluate definite integrals; perform substitution rule. Students will use the graphing calculator to evaluate the indefinite integrals in order to illustrate and check that their answer is reasonable (by graphing both the function and its antiderivative). [C2], [C], [C4], and [C5]. 5

6 In Unit Five, students will explore some of the applications of the definite integral by using it to compare areas between curves and volumes of solids. The common theme will be following the general method which is similar to the one they used to find areas under curves: break up a quantity Q into a large number of small parts, approximate Q by Riemann sum, take the limit and express Q as an integral; finally evaluate the integral using the Fundamental Theorem of Calculus or the Midpoint rule. Significant time will be spent in describing the physical aspects of definite integrals. Interpreting the meaning of the integral in terms of appropriate units is elevated in importance rather than just focusing on the numerical answer. After completing Unit Five, students will be able to apply their knowledge of definite integrals to find the area between two curves and volume of major solids; use integration techniques to calculate volume of a cross-section of a known geometric figure; find the average value of a function on a closed interval by applying the Mean Value Theorem. Students will use Graphing Calculator when needed in order to illustrate and check that their answer is reasonable. [C2], [C], [C4], and [C5]. In Unit Six, students will explore in depth two most important functions - the exponential and its inverse function, the logarithmic function. They will investigate their properties, compute their derivatives, and will use them to describe exponential growth and decay in biology, physics, chemistry, and other sciences. After completing Unit Six, students will be able to use process of logarithmic differentiation to find derivatives of non-logarithmic functions; integrate functions whose antiderivative involves a logarithmic function; integrate trigonometric functions; find the derivative of the inverse function; define, recognize, and utilize properties of the natural exponential function; find the derivative and integrate natural exponential functions; define the inverse trigonometric functions both analytically and graphically and be able to evaluate them and to solve equations involving them. Students will use the graphing calculator to evaluate the functions in order to illustrate and check that their answer is reasonable [C2], [C], [C4], and [C5]. In Unit Seven, students will develop techniques for using basic integration formulas to obtain indefinite integrals of more complicated functions. Students will discover that integration is not as straightforward as differentiation there are no rules that absolutely guarantee obtain an indefinite integral of a function. After completing Unit Seven, students will learn methods that are special to particular classes of functions, such as trigonometric functions; approximate the definite integrals of functions represented algebraically, graphically or numerically by using the Trapezoid Rule. Students will use the graphing calculator to evaluate the integrals in order to illustrate and check that their answer is reasonable [C2], [C], [C4], and [C5]. In Unit Eight students will understand and appreciate the importance of applications of calculus in solving differential equations, modeling scientific processes. Students will discover that although it is often impossible to find an explicit formula for the solution of a differential equation, many times the graphical and numerical approaches provide the needed information. After completing Unit Eight students will be able to determine the general solution to a differential equation analytically and graphically via a slope field; draw and interpret a slope field as a solution of a differential equation; solve the first order separable differential equations and use them in modeling; use differential equations to model and solve growth and decay problems. Students can connect slope fields with the library of functions developed in Unit One to theorize about the eventual solutions of differential equations. Students will use Technology (TI slopefields program) and the graphing calculator in order to illustrate and check that their answers are reasonable [C2], [C], [C4], and [C5]. 6

7 Review and Preparation for the AP Exam Students will be the major drivers in review. We will spend one or two classes on each of the previous units, with some sample AP questions given to students each day. For homework, students will receive questions (including free-response questions) from the previous AP exams that will be reviewed next time in class. I will encourage students to have a dialog together, as they discuss why an answer is correct, including defending their position if their answers is different from the stated answer. Practice exams will be given, scored (using scoring guidelines similar to those developed by the College Board) and analyzed. Some will be done in groups while others will be completed individually. After the AP Exam After the AP exam we will cover some skipped topics, and students will work on a number of projects mentioned above. They will also help to develop new projects and activities for the next year class as well as do research on famous mathematician and present to the class; prepare f or the Final Exam. Examples of Additional Student Activities [C], [C4] Trig Derivatives Activity: Students work in small groups to derive a formula for the derivative of secant, tangent, cosecant, and cotangent using their knowledge of derivatives (power rule, product and quotient rules, derivatives of sine and cosine) and trig identities. Each group finds one of the derivatives and the groups present their results to the class on the whiteboard. Slope Field Activity: Students, working in small groups, are assigned a differential equation and dy selected sample points. They calculate the value of at each point and plot the slope field on dx grid chart paper. After all groups have completed their graph, the class makes generalizations about the slope fields. The class will then make predictions about the solution curves for the differential equations based on the slope fields. Teaching Strategies [C], [C4], and [C5] Students are directed to carefully read all sections in their textbooks that are assigned by their instructor. In addition, they will have in-class and lab activities, experiments, reviews, and projects. Teacher Presentations I try to present all topics in many different ways. Among these are graphical, numerical, analytical and verbal approaches to almost all problems. I have access to PowerPoint lectures that accompany our text. While I do not use them in class when I am presenting the material, I do make them available as a resource for students who are either absent or who feel they need a bit more instruction on a particular topic. I use Internet demonstrations when I learn of them from the Electronic Discussion Group. 7

8 I use virtual TI SmartView software when I need to demonstrate a numerical approach to problems. Occasionally, trips will be made to the computer lab for explorations through webbased resources, Geometer Sketchpad, to help students visualize the different topics I am presenting. During the teacher s presentation and modeling of example problems, students are encouraged to jump in, ask questions, and participate in a class discussion of the day s lesson. Students are comfortable and free to learn in this math class. Graphing Calculators Instruction will be given using primarily the TI-84 Plus. There will be frequent use of the TI 84 within the classroom and on homework and on assessments. Students are already familiar with how to use them, having been required to use one in Algebra 2 and Precalculus. The graphing calculator allows the student to support their work graphically, make conjectures regarding the behavior of functions and limits among other topics, thus allowing students to view problems in a variety of ways. The calculator helps students develop a visual understanding of the material. Students learn the four required Calculator skills needed for the AP exam: graphing a function with an appropriate window, finding roots and points of intersection, finding numerical derivatives and approximating definite integrals. Calculators are encouraged to be used at the beginning of an exercise to get an idea of what is going on, or at the end of an exercise to verify analytical results. They are also used for explorations and investigation of new or unfamiliar topics. Students are expected to find solutions with the calculator and without the calculator. Investigative Approach Instructor s lecture is used only to introduce topics and give explanations where needed. The bulk of class time is given to investigations and practice (group work assignments from Teacher Recourse Guide by Douglas Shaw), making sure to pause and give opportunity to verbalize in correct mathematical terms what is being done. Memorization of formulas and methods is encouraged only to the extent that a student can explain in written or spoken words the significance of the particular formula or method. Students are continually told that arriving at an answer is not an end in itself; they must be able to explain how and why they got there. A lot of informal questioning is done to force students to verbalize what they claim they understand. Cooperative Learning Students are seated at tables of -4 students to facilitate group work. Most in-class practice is done with other students; collaboration is encouraged and expected. Reporting of group results is a common practice, with each member expected to understand what other members have contributed. Frequently, students will show their work on the white board or project their work on the screen (by handwriting on the Toshiba tablet PC screen or using document camera), and then answer questions from classmates regarding particular elements of their solutions. In addition, students participate and work together on lab assignments. Students are made comfortable early in the year with going to the white board, asking questions of their teacher, and working with their classmates. Students learn the first week of school to give their classmates put-ups and not put-downs. Consequently, problems are cleared up quickly and no classmates are left behind and in a quandary due to a lack of understanding. The instructor strives for a positive learning environment in the classroom. Study Groups 8

9 Study groups are formed early in the school year. Informal study groups are often formed by students for working outside of school hours. These groups employ the use of cooperative learning techniques for daily and weekly assignments with access to the instructor as needed. The Rule of Four This rule is continually set before the students, almost on a daily basis. Whenever possible, using at least two representations (algebraic, numerical, graphical, and verbal) of an example is used to give not only differing perspectives, but a deeper knowledge of the topic under study. In particular, students are challenged to not be overly reliant on analytical methods that become routine exercises that can be done without understanding exactly what was accomplished. Students become accustomed to hearing why? and asked to connect different representations from the Rule of Four. Answers to quiz and test questions can be required in any of the Rule of Four methods. Connections in Mathematics Connections in mathematics are stressed frequently. For instance: not all students realize at the beginning of the study of limits that the definition relates back to the study of slope in Algebra I. For comprehension of calculus concepts, students must make the mathematical connections to previous learning in order to have a true understanding of new calculus concepts and applications. Vocabulary and Symbolism Proper vocabulary and symbolism are used in the classroom and expected of the students. Students are taught proper form in putting their work on paper, justifying their solutions, and how to state their solutions in written form. Weekly AP Assignments Students practice on questions from released AP exams and AP-type questions on a weekly basis. At the beginning of each week students receive a packet of the week, designed to target the last two weeks topics. A set is due by the end of each week. The packet of the week includes one or two free response questions to solve. Students are required to explain and justify their answers. Expert Problems Expert problems (problems from a daily or weekly assignment or AP Questions) will be assigned to individual students and in the event a classmate has a question or concern regarding a homework problem, he/she, being the expert will come before the class and present the problem and discuss any areas of concern. The use of a document camera makes this a very effective teaching strategy and allows the expert the opportunity to communicate their reasoning verbally as well as the opportunity to receive feedback on their written work from their classmates. Many discussions may arise as to what is sufficient justification and what is not. Calculus Portfolio Students build a portfolio (which includes handouts, investigations, lab sheets, notes, charts, projects, weekly assignments, and homework) to take to college with them to use as a study aid in future math courses. 9

10 Daily Warm up and Procedures At the beginning of each class, students are given a short warm-up problem that may deal with the current material or a review problem or up to multiple choice questions to complete. Different methods of solution are shared and discussed. After going over the warm-up problem, students will review and discuss the homework in their group. They go over any difficult exercises from the homework assignment. This allows those students who got it to teach those who are still getting it. After assignments are corrected, students must first seek solutions from one another before asking the instructor. Examples of some (but not all) relevant homework problems are illustrated by the instructor. Students are expected to extend their knowledge to problems that are different from the homework examples. Often I will have students present their solutions on the board. I encourage as many different approaches to a problem as I can foster. I expect that the students will use numerical methods or technology to ensure that their answers are reasonable. Students grade each other s homework and submit their group homework log at the end of the week. Technology; Software, Calculators [R2], [C5] Graphing calculators are used on a daily basis to reinforce calculus concepts and interpret results. Students are provided with TI-84 Plus by the school which they may take with them and use at home for the school year. Our students are very comfortable with the TI-84 Plus, which they have been using since Algebra 2. Some quizzes and tests are done with calculator, some are done without. For questions that would require a calculator to complete, students are instructed to set up an expression that would be entered into a calculator to fully answer the question. Emphasis is placed on knowing how to arrive at an appropriate and reasonable conclusion, not just punching the right buttons to get the right answer. Students are shown how to calculate the numerical integral on their calculator and are expected to verify results when they have access to a calculator. Student Evaluation [C], [C4] Questions from previous AP Exams and from the textbook will strongly influence assessment during the year. All concepts will be assessed in each of the four approaches; graphically, numerically, analytically, and verbally. The pacing guide listed above provides 8 days for the Chapter Tests. Chapter tests are divided into a calculator part and a non-calculator part. Both parts contain multiple-choice questions as well as free response questions. At least one (usually more) of the free response questions is pulled from past AP Exams. In addition, students will be assessed with a number of other types of evaluation. For example: homework assignments, weekly AP assignments, quizzes, midterm exam, final exam, and cooperative learning projects. The midterm exam will follow the format of the tests and be similar to the AP exam format. In April students will take a full-length practice exam(s) given on a Saturday prior to sitting for the AP Calculus AB Exam. 10

11 Formal assessments make up 80% of a student s grade: 50 % from tests and 0% from weekly AP assignments and quizzes. Homework assignments and keeping a portfolio with all notes, assignments, quizzes, tests, and any supplemental material contribute the final 20% of the grade. Homework is awarded a score based on a combination of effort, completeness and correctness. Assignments are intended to show students if they have an adequate grasp of new material. They are only the first step to developing long-term retention of material and are thus weighted accordingly. At least one question in every assignment (usually more) is a word problem or exercise in which students must apply what they have learned. Attention is given to questions that require an explanation for an answer. This comes in a variety of ways: showing a detailed, step-by-step connection between a question and its answer; connecting the given information in a question to conditions for a theorem that is used; giving a written justification for the use of a particular method. Use of complete sentences is encouraged rather than simply using mathematical shorthand to present solutions. On all work, solutions alone will not be given credit. Answers must be accompanied by the appropriate work. Test questions may include any material that the instructor has taught from the first day of school. Likewise, students are also held accountable for math concepts taught in previous grade levels. No extra credit work will be extended to students. A high level of expectation is maintained at all times. Frequent assessments keep students ever mindful of keeping up with their work and staying current in their studies. Primary Textbook [R1] Stewart, James. Calculus 7e. Brooks/Cole, Cengage Learning, Some Examples of Textbook Assignments For each section, a set of essential Core Exercises is provided. Each exercise in that assignment is classified as Descriptive, Algebraic, Numeric, and/or Graphic Section Problems Chapter 1: Functions and Limits 1.1, 5, 7, 9, 11, 1, 15, 2, 25, 27, 29, 1,, 5, 7, 9, 41, 42, 4, 45, 47, 49, 51, 5, 55, 61, 6, 64, 69, 72, 7, 75, 77 and set up notebook with handouts Exercise D* A* N* G* Exercise D A N G Exercise D A N G X X 5 X 5 X 5 X 55 X X 7 X 7 X 61 X 9 X 9 X X 6 X 11 X X 41 X X 64 X 1 X X 42 X X 65 X 15 X 4 X X 69 X 11

12 2 X X 45 X X 72 X 24 X X 46 X X 7 X X 25 X 47 X X 75 X X 27 X 49 X X 77 X X 29 X 51 X 1 X *D Discrete, A* - Algebraic, N* - Numeric, G* - Graphical Supplemental material: Shaw, Douglas. Teacher s Resource Guide for the Advanced Placement Program* to accompany Single Variable Calculus with Vector Functions by James Stuart. 7 th edition. Brooks/Cole, Cengage Learning, St. Andre, Richard. Study Guide for Stewart s Single Variable Calculus. 7 th edition. Brooks/Cole, Cengage Learning, Cade, Sharon, Phea Caldwell, and Jeff Lucia. Fast track to a 5. Preparing for the AP Calculus AB and Calculus BC Examinations to accompany Calculus and Single Variable Calculus 6 th edition by James Stuart and Calculus and Single Variable Calculus with Vector Functions 7 th edition by James Stuart. Brooks/Cole, Cengage Learning, Finney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy. Calculus: Graphical, Numerical, Algebraic. Third Edition, Boston: Pearson Prentice Hall, 2 Forrester, Paul A., Calculus: Concepts and Applications, Berkley: Key Curriculum Press, 2004 Supplements teacher generated worksheets and released versions of AP exams. Geometer s Sketchpad Software, Key Curriculum Press Website Resources Stuart Calculus Home page AP Calculus AB Course Home Page Free AP exams 12

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