AP Calculus 2007 Released Multiple-Choice CALCULATOR REQUIRED (76

Course OverviewMy main objective in teaching AP Calculus AB is to enable students to appreciate the beauty of calculus and receive a strong foundation that will give them the tools to succeed in future mathematics courses. Students know that they will work hard, and our expectation is that this hard work will enable them to succeed in the course. We work together to help students discover the joys of calculus.Primary TextbookJon Rogawski, Calculus for AP, 2nd edition, New York: W.H.Freeman and company,2012. David S. Kahn, Cracking the AP Calculus BC exam, 2015 edition, New York, 2014.Course PlannerFunctions and Graphs (Chapter 0)Students complete this review of precalculus materials over the summer.Key Concept: Limits (Chapter 1)2 weeks1. Limits, Rates of Change, and Tangent Lines2. Limits : A Numerical and Graphical Approsch3. Basic Limit Laws4. Limits and Continuity5. Evaluating Limits Algebraically6. Trigonometric Limits7. Limits at Infinity8. Intermediate Value TheoremDifferentiation (Chapter 2)4 weeks1. Definition of the Derivative2. The Derivative as a Function3. The Product and Quotient Rules4. Rates of Change5.Higher Derivatives6. The Trigonometric Functions6. The Chain Rule7. Implicit Differentiation8. The Related RatesApplication of the Derivative (Chapter 3) 4 weeks1. Linear Approximation and Application2. Extreme values3. The mean Value Theorem and Monotonicity4. The Shape of a Graph5. Graph Sketching and Asymptotes6. Applied Optimization7. Newton’s Method8. AntiderivativesMidterm ExamKey Concept: The Integral (Chapter 4)4 weeks1.Approximating and Computing Area2. The Definite Integral3. The Foundamental Theorem of Calculus4. Net Change as the Integral of a Rate5. Substitution MethodApplication of the Integral (Chapter 5)4 weeks1. Area Between Two Curves2. Setting up Integrals3. Volumes of Revolution4. The Method of Cylindrical Shells5. Work and EnergyTeaching StrategiesOn the first day of school, I begin with Chapter 2 (Key Concept: The Derivative). This sets the tone for the year. As discussed in the Student Activities section below, I start with a Calculator-Based Laboratory (CBL) ball toss experiment. Students explore the concept of average rate of change and discover the concept of instantaneous rate of change. Thereafter, student exploration and discovery continue to be an important aspect of the remaining topics.Throughout the course, students work together on a regular basis, both formally and informally. At times, I set up groups to work on a particular activity, but students do not need to be told to work together. Our classroom has tables instead of desks to make it more conducive to group work. When students are working on a problem, they will often work alone initially but then turn to their partners to collaborate.In discovering new concepts, the class works as a whole. It is not necessary for students to raise their hands. If students have a thought to share, they are welcome to make a contribution. If they are so inclined, students will go up to the board to illustrate a point. At times, I am able to step back and just listen to the interaction among my students as they explore a topic.Technology can be used to help make calculus concepts come alive and it enables students to “see” what is being discussed. Students are issued TI-89 calculators. Our classroom also contains 10 computers and a SMART Board. We use TI-Navigator, TI Inter Active!, and CBLs. Topics are presented using the “rule of four”: graphically, numerically, algebraically, and orally. Through this multifaceted approach, students gain an in-depth understanding of the material.Student EvaluationStarting in October, I assign six AP free-response questions for students to work on for two weeks. Students may work on these questions with one other person and come to me for extra help. At the end of the two weeks, I randomly select one of these questions for a quiz. Students are graded as they would be graded on an AP Exam. Students are expected to explain the solutions to problems using written sentences. (Free-response questions and scoring guidelines are available on AP Central®.)When the second semester begins in February, we work on multiple-choice sections of AP Calculus Released Exams. Students have two weeks to complete a multiple-choice section. They may work on these questions with one other student and come to me for extra help. At the end of the two weeks, students hand in their work for grading. They are also quizzed on selected questions from the work they submit. The multiple-choice work alternates with the free-response work until the AP Exam administration.Student Activities1. On the first day of school, we begin the study of derivatives with a CBL experiment. Students toss a ball in the air and examine the height-versus-time graph generated. They fit the data with a quadratic equation for the position function to determine how high the ball went and how long it was in the air. Students compute the average velocity over a time interval. They are then asked to determine the velocity of the ball at exactly 0.06 seconds after the ball was tossed and explain how their answer was obtained. Finally, they zoom in on the graph of the position function near t = 0.06 until the graph looks like a line. Students find the slope of the line and compare it to their estimation of the instantaneous velocity at t = 0.06.2. When students are first learning about derivatives, I sketch the graph of an unknown function on the board. Each student comes up to the board and plots a point that would lie on the graph of the derivative of the function. We watch as the graph of the derivative unfolds. For example, I often graph a sine function on the board. At the maximum value of this function, the derivative would be zero. Students are anxious to plot these points. At the points of infection, the derivative would be at a maximum or minimum. These are also easy points to plot. Gradually, students fill in the remaining values, and we can see that the derivative of a sine function is a cosine function.3. During our study of related rates, students suck on Tootsie Roll Pops to deter-mine the rate of change of their r adius; they then calculate the rate of change of the Pops’ volume. Students measure the initial radius of a Pop with dental foss. They then suck on the Pop for 30 seconds, record its radius, suck for another 30 seconds, etc. They model the rate of change of the radius with some function of time. Students then use this rate of change to estimate the rate of change of the volume of the Pop when its radius is three-fourths of its original radius. This lab, “How Many Licks?,” can be found in Ellen Kamischke’s b ook, A Watched Cup Never Cools. [SC5]4. The study of optimization can be made more meaningful to students by asking them to design an optimum can. Students obtain a can of soda, soup, tuna, etc. They measure the height and diameter of the can and determine its volume. They then find the radius and height of the most cost-effective can that will hold the same volume and write an explanation, using well-written sentences, of the mathematics involved in making their determination. Students then construct the most cost-effective can, bring their original can and constructed can to school, and make a presentation to the class.5. After learning how to approximate a definite integral, students use these techniques to calculate the distance covered during a 20-minute drive with a friend or parent. Before beginning the drive, students record the car’s odometer read ing. Using the speedometer, they record the car’s speed at 1-minute intervals, noting any traffic conditions. At the end of the drive, they check the odometer reading again. Students then graph speed versus time and use integration techniques to approximate the distance traveled over the 20-minute interval.They compare this distance with the actual mileage determined by the odometer. Students are often amazed at the closeness of their approximation to the odometer reading. Students are to write a report on this project that includes an explanation of data collection, graphing of the data, interpretation of the data, and the closeness of their approximation to the odometer reading.6. As an introduction to slope fields, I use an activity from the AP CalculusTeacher’s Guide. Using the graphing-calculator screen with the grid turned on, I project a 3 x 3 grid onto the board and assign each student several coordinate points in the region (([1, 1], [1, 2]), etc.).For a given differential equation, each student computes the slope at his or her coordinate position and then goes to the board to draw a short line segment with the calculated slope and the coordinate point as the midpoint of the segment. For example, if dy/dx = y, the student with coordinates (1, 1) would go to the board and at the point (1, 1) draw a short line segment with a slope of 1. The student with coordinates (1, 2) would go to the board and at the point (1, 2) Draw a small line segment with a slope of 2. (It is important that the second student draw a line segment whose slope is steeper than the slope of the fi rst student’s line se gment.) Continuing in this fashion, the class would complete the slope field. The students then use their calculator to graph a solution found analytically to the differential equation. The students are asked to use the graph obtained with their calculator to interpret the results and to support their conclusions obtained with the slope field. At this point, all sorts of discussions can ensue.7. One of the most difficult concepts for me as a student of calculus was finding the volume of solids. If students cannot visualize the solids, they have a more difficult time understanding how to compute the volume. To enable students to “see” the solids of revolution, I have purchased several “open-up” party decorations from a party supply store. Instead of just di scussing how aline revolving around an axis forms a cone, students see the cone generated. (I found an open-up ice cream cone in one of the party stores.)I bring cans of Play-Doh into school and ask students to construct solids whose bases are bounded by two curves and whose cross sections are squares or equilateral triangles, etc. For example, students are given the graph of a circle and asked to construct a solid in which each cross section perpendicular to the base is an equilateral triangle. Students build the solids using Play-Doh and then use plastic knives or dental floss to cut through the solid and obtain the required cross sections.Students also use the Winplot program on the computer and see the solids come alive.We finish off this topic with two activities from Work Smarter Not Harder, a book of labs accompanied by a disk of calculator programs. Students can download a program onto their calculators that will enable them to enter a function, graph the function, and rotate the function about a line. The calculator will then display a cross section of the solid generated. Another program enables students to enter a function and display a cross section that is a square, isosceles right triangle, etc. By the time the students have completed these activities, they are quite comfortable with the topic.。

AP Calculus BC1. Important limits()001111011sin sin lim1, lim 1lim 1lim 10, ()lim lim , x x xt x t m m m m m m n n x x x ax ax bx be t e x m n P x a x a x a x a a m n b x b →→→∞→---→∞→∞==⎛⎫+=⇔+= ⎪⎝⎭<++++===+++⎧⎪⎪⎨x x sec )'(tan = x csc (cot)'-= x x x tan sec )'(sec = x x x cot csc )'(csc -=(6) 211)'(arcsin xx -=211)'(arccos xx --=211)'(arctan x x +=211)'cot (x x ar +-= 11)'sec (2-=x x x arc 11)'csc (2--=x x x arc2. Rules（1）If f (x )，g (x ) are differential ，a. )()())()((x g x f x g x f '±'='±；b. )()()()())()((x g x f x g x f x g x f '+'='，especially ，)())((x f C x Cf '='（C is a constant ）；c. )0)(( ,)()()()()())()((2≠'-'='x g x g x g x f x g x f x g x f ，especially ，21()()()()g x g x g x ''=-。

(2) Chain Rule )]([x g f y =⇒dxdudu dy dx dy ⋅= (3) Implicit Differentiation ()(()'r t f =3. Applications of Derivative Mean Value Theorem) is differentiable Tangent and normalFirst Derivative Test'f changes from - to +，local min 'f changes from + to -，local maxSecond Derivative Test ''()0f a >，local min ''()0f a <，local max(4) Absolute max/min: compare function value of critical point and endpoints.Steps: 1. find 'f ;2.solve '0f =;3. compare value at critical point and endpoints.1. Indefinite Integral:'()()Definition f x dx f x C =+⎰ Method ()()df x f x C =+⎰2. Methods for Indefinite Integral 不定积分方法 ① Formulas ：dx x C =+⎰（integrand 为1）, kdx kx C =+⎰, 11n nx x dx C n +=++⎰ln x xa a dx C a=+⎰ x xe dx e C =+⎰1sin 2 4x x C ++②()f u du3. Find the antiderivative of ()f u .③ Partial fraction （拆分）ln ||ln ||()()cx d A Bdx dx dx A x a B x b C x a x b x a x b +=+=-+-+----⎰⎰⎰(通分求A,B)④ Integration by partsudv uv vdu =-⎰⎰，（or ''uv dx uv vu dx =-⎰⎰）， Tabular Integration ， （story about ln x , sin x , and xe ）3. Improper Integral 反常积分（两种形式，无穷积分，瑕积分） ① Integral on infinite interval()lim (),()lim (),()()()lim ()lim ().ba ab bbaa c cbcaca b f x dx f x dx f x dx f x dx f x dx f x dx f x dx f x dx f x dx ∞→∞-∞→-∞∞∞-∞-∞→-∞→∞===+=+⎰⎰⎰⎰⎰⎰⎰⎰⎰② Integrand with infinite discontinuities()lim (),bca ac bbbf x dx f x dx -→=⎰⎰4.5. )'()x β⋅6. ①(,)c a b ∈. ②low upper function er function a⎣⎦L Right functio ef func on n ti )cy dy ⎤⎥⎣⎦③ V olume ：with known cross section()baA x dx ⎰, ()A x is the area of the cross section.Revolution ：2Inner 2Out rad er radiu ius s baR r dx π⎡⎤-⎣⎦⎰，Shell Method ：2baxydx π⎰（旋转轴为y-axis ） or 2baxydy π⎰（旋转轴为x-axis ）④ Length21x x ⎰or 21y y ⎰or 21t t ⎰.1. Separation Variable()()()()dy M x N y dy M x dx dx N y =⇒=⎰⎰ 2. Logistic Equation (1)(),lim ()1kt t dP P K kP P t P t K dt K e-→∞=-⇒==+ ,where is the environmental capacity. 3. Slope Field4. Euler’s Method ： 11n n n n x x x dyy y x dx ++=+∆⎧⎪⎨=+⋅∆⎪⎩n a ++，1n a∞=∑Test for ConvergenceAssume that the following limit exists: 1≥..③ Comparison TestAssume that there exists 0M >such that 0n n a b ≤≤for n M ≥. If1nn b∞=∑ converges, then If1nn a∞=∑ also converges. If1nn a∞=∑ diverges, then1nn b∞=∑ alsodiverges.Limit Comparison Test Let {}n a and {}n b be positive sequences. Assume limnn na Lb →∞=a) If 0L >,then1nn a∞=∑ converges if and only if1nn b∞=∑ converges.b) If L =∞, and1nn a∞=∑ converges, then1nn b∞=∑ converges.c) If 0L =, and1nn b∞=∑ converges, then1nn a∞=∑ converges.3. Four Series① The geometric seriesnar∞converges if ||1r <and diverges otherwise.0, lim n n a a →∞>>>>4. ()(012a a x c a x +-+-()n∞∑is either anonnegative number ( )or infinity( ).If i s finite, converges absolutely whenx c R -< and diverges when x c R ->. If R =∞, ()F x converges absolutely for all x .（ii ）Term-by-term Differentiation and IntegrationPossible convergence at the endpointsAssume that ()0()nn n F x a x c ∞==-∑ has radius of convergence 0R > . Then is differentiable on(, )c R c R -+[or for all x if R =∞]. Furthermore, we can integrate and differentiate term by term.For (, )x c R c R ∈-+,()()110'() () (A any constant)1n n n n n n a F x na x c F x dx x c A n ∞∞-+===-=-++∑∑⎰；These series have the same radius of convergence R .()(!n f a n ++()(0)!n f n ++and x ).Interval of convergence all real numbers初等函数基本知识点小结一、指数函数（一）指数与指数幂的运算1．∈N◆2ma n◆3（112（一）对数1．对数的概念：一般地，如果N a x =)1,0(≠>a a ，那么数x 叫做以．a 为底．．N 的对数，记作：N x a log =（a — 底数，N — 真数，N a log — 对数式）说明：注意底数的限制0>a ，且1≠a 两个重要对数：○1 常用对数：以10为底的对数N lg ； ○2 自然对数：以无理数 71828.2=e 为底的对数N ln ． 指数式与对数式的互化幂值 真数如果a ○1 ○2 ○3 abln ln（1）1，+∞）． ○2 21、幂函数定义：一般地，形如αx y =)(R a ∈的函数称为幂函数，其中α为常数．2、幂函数性质归纳．（1）所有的幂函数在（0，+∞）都有定义并且图象都过点（1，1）；（2）0>α时，幂函数的图象通过原点，并且在区间),0[+∞上是增函数．特别地，当1>α时，幂函数的图象下凸；当10<<α时，幂函数的图象上凸；（3）0<α时，幂函数的图象在区间),0(+∞上是减函数．在第一象限内，当x 从右边趋向原点时，图象在y 轴右方无限地逼近y 轴正半轴，当x 趋于∞+时，图象在x 轴上方无限地逼近x 轴正半轴．三角函数基本知识点小结sin1. 2. 3.sin(α±β(tan a a sin2αa 2tan =Trigonometric Function and Inverse Trigonometric Function。

AP CALCULUS ABSYLLABUSFORSCHOOL YEAR: 2007 –2008INSTRUCTOR: ANN MILSTEADCENTRAL HIGH SCHOOLPOLLOK, TEXASCourse Overview and Brief DescriptionAP 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 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.Goals from the AP Calculus Course Description•Students should be able to work with functions numerically, graphically, analytically, and verbally…•The derivative should be understood as the instantaneous rate of change of a function and as the local linear approximation of the function…•The definite integral should be understood as the limit of a Riemann sum and as the net accumulation of a rate of change…•The relationship between derivatives and the definite integral should be understood in terms of both parts of the Fundamental Theorem ofCalculus…•Students learn to communicate about mathematics verbally and in writing…•Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral…•Students learn to use technology to analyze problems, experiment, and verify and interpret results…•Students are expected to learn to judge the reasonableness of their solutions…•Students develop an appreciation of the wonderful world of calculus and for their personal accomplishment in learning calculus…Teaching StrategiesConnections 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. Solutions to problems are found graphically, numerically, analytically, and verbally in order to demonstrate knowledge of the calculus curriculum being studied. In addition, proper vocabulary and symbolism are used in the classroom and expected of the students.Students jump right into Calculus, Chapter 1, Section 1, the first day of school. Precalculus is reviewed as needed. Students are taught proper form in putting their work on paper, justifying their solutions, and how to state their solutions in written form.Students are encouraged to ask questions immediately during lecture. No hands raised in this math class. Consequently, problems are cleared up quickly and no classmates are left behind and in a quandary due to a lack of understanding.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”. Study groups are formed early in the school year and employ the use of cooperative learning techniques for daily assignments with access to the instructor as needed. The instructor strives for a positive learning environment in the classroom. Students practice on questions from old AP exams on a weekly basis. A set is due each week. In addition, students build a notebook (which includes handouts, lab sheets, notes, charts, projects, and homework) to take to college with them to use as a study aid in future math courses.Examples of some (but not all) homework are illustrated by the instructor. Students are expected to extend their knowledge to problems that are different from the homework examples.Graphing Calculators and TechnologyGraphing calculators are used on a daily basis to reinforce calculus concepts and interpret results. Students are provided with a TI-83+ and a TI-89 Titanium by the school which they may take with them and use at home for the school year. Demonstrations are done on occasion with the TI-200 calculator. Our students are very comfortable with the TI-83+, which they have been using since Algebra I. The TI-89 is not used until the spring semester. Students are expected to find solutions with the calculator and without the calculator.Students will be able to do the following with their graphing calculators:1.Plot the graph of a function with an arbitrary viewing window2.Find the zeros of functions (solve equations numerically)3.Numerically calculate the derivative of a function4.Numerically calculate the value of a definite integral1Early in the course students use their calculators to approximate and arrive at a reasonable conclusion numerically of what the slope of a tangent line is to some quadratic function at a particular point on that function. This activity then leads to further investigation by the student doing the same thing graphically and analytically.In addition, a computer projector is used to demonstrate calculus concepts and a TI-CBL unit is used for labs, demonstrations, and to collect data to further enhance studies. Some of the software used in this class is Geometer’s Sketchpad and Calculus in Motion. Also, students have access to the Internet in the classroom for research. And, the class has access to a computer lab (on request) in order to work on the APCD Calculus AB2 software for which we have a site license.Primary TextbookLarson, Ron, et al. Calculus with Analytic Geometry, Eighth and Advanced Placement Edition, Boston: Houghton Mifflin Company, 2006.Each student is issued a copy of the primary text.Some Examples of Textbook AssignmentsSection ProblemsChapter 1: Limits and Their Properties1.1 1, 2, 5, 7, 8, 9 and set up notebook with handouts1.2 1-25 odd, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 59, 63, 65, 671.3 5-61 odd, 67, 69, 71, 73, 75, 77, 78, 83, 84, 85, 86, 87, 101,103, 113, 115 1.4 1-19 odd, 25, 29-51 odd, 57, 59, 61, 63, 69, 71, 75, 77, 83, 85, 87, 91, 105 1.5 1-47 odd, 53, 55, 57, 58, 59, 61, 62, 633.531, 3, 5, 7, 15-33 odd, 41, 45, 51, 85, 87Course Planner, Pacing Guide, and Topic OutlineThe pacing guide has 142 teaching days including 10 days of formal assessment. Precalculus is reviewed as needed throughout the course. The sections listed fulfill the requirements of content demanded by the AP Course Description Guide. A two-week review period precedes the AP Calculus AB exam.1 The College Board. AP Calculus AB Course Description2 The College Board. APCD Calculus AB3 Deliberately out of orderSection Topic Number ofDaysChapter 1: Limits and Their Properties1.1 A Preview of Calculus 21.2 Finding Limits Graphically and Numerically 21.3 Evaluating Limits Analytically 21.4 Continuity and One-Sided Limits 21.5 Infinite Limits 23.54Limits at Infinity 2Review and Assessment 3Chapter 2: Differentiation2.1 The Derivative and Tangent Line Problem 52.2 Basic Differentiation Rules and Rates of Change 432.3 Product and Quotient Rules and Higher-OrderDerivatives2.4 The Chain Rule 32.5 Implicit Differentiation 32.6 Related Rates 3Review and Assessment 3Chapter 3: Applications of Differentiation3.1 Extrema on an Interval 33.2 Rolle’s Theorem and the Mean Value Theorem 33.3 Increasing and Decreasing Functions and2The First Derivative Test3.4 Concavity and the Second Derivative Test 23.6 A Summary of Curve Sketching 2Review and Assessment 33.7 Optimization Problems 53.9 Differentials 3Review and Assessment 3Chapter 4: Integration4.1 Antiderivatives and Indefinite Integration 34.2 Area 34.3 Riemann Sums and Definite Integrals 34.4 The Fundamental Theorem of Calculus 34.5 Integration by Substitution 34.6 Numerical Integration 1Review and Assessment 3Chapter 5: Logarithmic, Exponential,and Other Transcendental Functions5.1 The Natural Logarithmic Function: Differentiation 35.2 The Natural Logarithmic Function: Integration 24 Deliberately out of order5.3 Inverse Functions 3 5.4 Exponential Functions: Differentiation and Integration 3 5.5 Bases Other than e and Applications 2Review and Assessment 3 5.6 Inverse Trigonometric Functions: Differentiation 2 5.7 Inverse Trigonometric Functions: Integration 1Review and Assessment 3Chapter 6: Differential Equations6.1 Slope Fields 3 6.2 Differential Equations: Growth and Decay 5 6.3 Separation of Variables and the Logistic Equation 3Review and Assessment 3Chapter 7: Applications of Integration7.1 Area of a Region Between Two Curves 37.2 The Integral as Net Change Over a Specific Period ofTime(4.5 Exercise 115, Ch. 4 Review Exercises 93 and 94)Volume: The Disk Method (Includes disks, washersand volumes of solids with known cross sections) 6 5Review and Assessment 3Chapter 8: Integration Techniques, L’Hopital’sRule and Improper Integrals8.1 Basic Integration Rules 2Review and Assessment 3AP Calculus ExamMay 2008After the AP ExamResearch topics on calculus applicationsSelected topics from AP Calculus BCStudent ActivitiesStudents review parent functions, domain, and range early in the school year. In addition, Precalculus is reviewed throughout the course as needed. Students approach their study of calculus with a multi-representational view (i.e. graphically, numerically, analytically, and verbally).On a daily basis students are working in and adding to their notebooks. Cooperative learning groups are frequently used in the classroom on assignments. In addition, students often work at the board or at the overhead projector desk to demonstrate calculus solutions to their classmates. Students participate and work together on lab assignments.During the teacher’s lecture 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. Students use technology on a daily basis; however, practice with the TI-89 calculator is not done until the spring semester. Students also have a weekly practice on questions from old AP exams.Students participate in a number of lab activities. For example: “What Goes Down, Must Come Up” from the book A Watched Cup Never Cools is a lab activity in which students use their calculators along with the TI-CBL unit and motion detector to investigate average velocity as a numeric derivative of position, average acceleration as a numeric derivative of velocity, derivative as a slope of a tangent to a curve, and differentiability of a curve. Another lab activity in which students participate is “ As Easy as Pie” in which graphing calculators and cake pans are used to find the volume of a cylinder. 5Students are directed to carefully read all sections in their textbooks that are assigned by their instructor. In addition, they will have worksheets, lab activities, experiments, reviews, and projects.Students may attend a morning tutorial session beginning at 7 a.m. if further individual help is needed.Student EvaluationThe pacing guide listed above provides 10 days of assessments. In addition, students will be assessed with a number of other types of evaluation. For example: homework assignments, daily quizzes, pop test, three week tests, nine week tests, midterm exam, final exam, AP free-response questions, AP multiple-choice questions, and cooperative learning projects. Students will take a full-length practice exam prior to sitting for the AP Calculus AB Exam.A high level of expectation is maintained at all times. Frequent daily assessments keep students ever mindful of keeping up with their work and staying current in their studies.On all work, solutions alone will not be given credit. Answers must be accompanied by the appropriate work. Scrambled versions of tests are administered in order to maintain honor and integrity in the classroom.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.5 Kamischke, Ellen. A Watched Cup Never CoolsSome Examples of Important ResourcesThe College Board. AP Calculus AB Course DescriptionThe College Board. Released Exams for AP Calculus ABKamischke, Ellen. A Watched Cup Never Cools: Lab Activities for Calculus and Precalculus. Emeryville, CA. Key Curriculum PressRoberts, A. Wayne. Resources for Calculus Collection, 5 volumes. Published by MAAVol. 1: Learning by DiscoveryVol. 2: Calculus Problems for a New CenturyVol. 3. Applications of CalculusVol. 4: Problems for Student InvestigationVol. 5: Readings for CalculusAnderson, Frank. Review for the AP Calculus AB Examination: The TwoWeek Difference. Atlanta, GA. Andco Educational ServiceAudrey Weeks. Calculus in Motion. SoftwareThe College Board. Advanced Placement APCD Calculus AB. New York.SoftwareWebsite Resources(Quicker to search by Google to get to these websites)AP CentralHandley Math PageDr. MathMath Forum。

AP® Calculus AB2007 Scoring GuidelinesForm BThe College Board: Connecting Students to College SuccessThe College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,000 schools, colleges, universities, and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns.© 2007 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation.Permission to use copyrighted College Board materials may be requested online at:/inquiry/cbpermit.html.Visit the College Board on the Web: .AP Central is the official online home for the AP Program: .Question 1Let R be the region bounded by the graph of 22x x y e −= and thehorizontal line and let S be the region bounded by the graph of2,y =22x x y e −= and the horizontal lines and 1y =2,y =as shown above. (a) Find the area of R . (b) Find the area of S .(c) Write, but do not evaluate, an integral expression that gives thevolume of the solid generated when R is rotated about the horizontal line 1.y =222x x e −= when 0.446057,1.553943x =Let and 0.446057P = 1.553943Q =(a) Area of ()2220.5Qx x P14dx −=−=⌠⌡⎪⎩R e3 : ⎪⎨ 1 : integrand1 : limits 1 : answer ⎧(b) when 2221x x e −=0,x =Area of S e Area of R()22201x x dx −=−−⌠⌡− Area of2.06016= 1.546R =OR()()()222220110.219064 1.1078860.219064 1.546Px x x x Qe dx Q P e d −−−+−⋅+−=++=⌠⌠⌡⌡1x⎪⎩3 : ⎪⎨ 1 : integrand 1 : limits 1 : answer ⎧(c) Volume ()()2222121Qx x P e d π−⎛⎞−−⎜⎟⎝⎠⌠⎮⌡x =−3 : {2 : integrand1 : constant and limitsQuestion 2A particle moves along the x -axis so that its velocity v at timeis given by The graph of v is shown abovefor 0t ≥()()2sin .v t t=0t ≤≤ The position of the particle at time t is ()x t and its position at time is 0t =()05x =.(a) Find the acceleration of the particle at time t 3.=(b) Find the total distance traveled by the particle from time 0t =to t 3.==(c) Find the position of the particle at time t 3.(d) For 0t ≤≤ find the time t at which the particleis farthest to the right. Explain your answer.Question 3The wind chill is the temperature, in degrees Fahrenheit ()F ,° a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity v , in miles per hour ()mph . If the air temperature is 32 then the wind chill is given by and is valid for 56F,°()0.1655.622.1W v v =−0.v ≤≤ (a) Find ()20.W ′ Using correct units, explain the meaning of ()20W ′ in terms of the wind chill.(b) Find the average rate of change of W over the interval 560.v ≤≤ Find the value of v at which theinstantaneous rate of change of W is equal to the average rate of change of W over the interval 560.v ≤≤ (c) Over the time interval hours, the air temperature is a constant 32 At time the windvelocity is mph. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at 0t ≤≤4 F.°0,t =20v =3t = hours? Indicate units of measure.Question 4Let f be a function defined on the closed interval 55x −≤≤ with ()13f=. The graph of ,f ′ the derivative of f , consists of two semicircles and two line segments, as shown above.(a) For −< find all values x at which f has arelative maximum. Justify your answer.5x 5,<5,<(b) For −< find all values x at which the graph of fhas a point of inflection. Justify your answer.5x (c) Find all intervals on which the graph of f is concave upand also has positive slope. Explain your reasoning.(d) Find the absolute minimum value of ()f x over the closed interval 5x 5.−≤≤ Explain your reasoning.Question 5Consider the differential equation 11.2dy x y dx =+−(a) On the axes provided, sketch a slope field for the given differential equationat the nine points indicated.(Note: Use the axes provided in the exam booklet.)(b) Find 2d ydxin terms of x and y . Describe the region in the xy -plane inwhich all solution curves to the differential equation are concave up.(c) Let ()y f x = be a particular solution to the differential equation with theinitial condition ()01f =. Does f have a relative minimum, a relative maximum, or neither at Justify your answer. 0?x =(d) Find the values of the constants m and b , for which y mx b =+ is asolution to the differential equation.y −1Question 6Let f be a twice-differentiable function such that ()2f 5= and ()52f .= Let g be the function given by ()()().g x f f x =(a) Explain why there must be a value c for 25c << such that () 1.f c =−′(b) Show that Use this result to explain why there must be a value k for 2 such that()()2g g =′′5.5k <<()0.g k =′′ (c) Show that if ()0f x =′′ for all x , then the graph of g does not have a point of inflection. (d) Let ()().h x f x x =− Explain why there must be a value r for 25r << such that ()0.h r =。