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2008 calculus BC multiple choice

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AP Calculus 2008 BC Multiple Choice
en 4. Consider the series ∑ . If the ratio test is applied to the series, which of the following n =1 n ! inequalities results, implying that the series converges? (A) lim
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AP Calculus 2008 BC Multiple Choice
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14. The polynomial function f has selected values of its second derivative f " given in the table above. Which of the following statements must be true? (A) f is increasing on the interval ( 0, 2 ) .
(B) f is decreasing on the interval ( 0, 2 ) . (C) f has a local maximum at x = 1. (D) The graph of f has a point of inflection at x = 1. (E) The graph of f changes concavity in the interval ( 0, 2 ) .

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critical numbers – these are the values of the independent variable at which the first derivative is zero: f ' x = 0 . They also occur where the first derivative is undefined, such as at a cusp or sharp corner (like absolute value). These numbers are useful when looking for changes in the trend of the graph: either increasing or decreasing. Critical numbers also indicate relative extrema, either relative maxima or relative minima. absolute extrema – absolute maxima and minima must be found by using CRITICAL NUMBERS and the ENDPOINTS OF THE INTERVAL. Whichever value is furthest to the desired extreme is given as the absolute extrema. inflection – inflection is the change of the concavity of the graph. A graph is either concave up or concave down (a horizontal line would have both or none) Points of inflection occur where the second derivative is zero. Evaluation of the second derivative at any value yields the concavity; f ' ' x 0 ⇒ concave up ; f ' ' x 0 ⇒ concave down mean value theorem (for derivatives) – this theorem declares that there must be at least one point on a given interval for which the instantaneous slope (derivative) equals the average rate of change over that interval. The function must be continuous on the closed interval, and differentiable on the open interval. Symbolically, this is f b − f a expressed f ' c = where a , b is the open interval. b− a Rolle's theorem – this is a simpler version, and an application of the mean value theorem. It adds the condition that, on the interval a , b , f a = f b . This means that the average slope is zero, and there must be a point c at which f ' c = 0 position, velocity, and acceleration – that's the order, a t = v ' t = s' ' t or s t =∫ v t dt =∫ [∫ a t dt ] dt

AP Calculus AB考试2024历年真题全视角

AP Calculus AB考试2024历年真题全视角AP Calculus AB考试是全球范围内许多高中生梦寐以求的考试之一。

这一门考试对学生的数学技能、推理和分析能力有着较高的要求。

为了帮助考生更好地准备这一考试，本文将从全视角介绍AP CalculusAB考试的历年真题。

第一部分：选择题选择题是AP考试的常见题型，也是Calculus AB考试的一部分。

这一部分主要考察学生对基本概念、定义、定理和计算方法的理解和掌握程度。

历年的选择题主要围绕微积分的各个知识点展开，涵盖了函数、导数、微分、积分等内容。

考生需要通过解题，展示对于这些数学概念的理解和应用。

第二部分：解答题解答题是AP Calculus AB考试中更为复杂和综合的一部分。

这部分考题要求学生不仅对概念和方法具有扎实的理解，还需要能够灵活应用，解决实际问题。

在解答题中，学生需要回答一系列问题，这些问题可能涉及到函数的性质、图像的分析、极限、导数、微分方程、积分等。

解答题要求学生具备一定的推理和分析能力，能够运用所学知识解决实际问题。

第三部分：应用题AP Calculus AB考试中的应用题是一部分考生较为困惑的内容，因为它需要将数学知识应用到实际问题中，涉及到模型的建立和问题的求解。

应用题常常涉及到各个学科领域，如物理、生物、经济、工程等。

考生需要将所学的微积分知识与具体领域的问题相结合，通过建立数学模型，解决实际问题。

第四部分：历年真题的分析通过对历年真题的分析，我们可以发现一些考试的特点和趋势。

比如，近年来，AP Calculus AB考试更加注重对学生的应用能力和数学建模能力的考察。

而在解答题中，经常出现一些复合型的题目，要求学生综合运用多个概念和方法解决问题。

同时，历年真题中还提供了一些有用的参考资料，例如公式表和常见数学定理的总结。

这些参考资料可以帮助考生在考试中更快地找到解题思路，提高解题效率。

总结：通过全视角地了解AP Calculus AB考试的历年真题，我们可以更好地理解考试的要求和出题规律。

AP CALCULUS AB

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 微积分 AB 2007 (Form B) 真题与答案

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 =。

AP Calculus AB考试2024历年题目全解

AP Calculus AB考试2024历年题目全解2024年的AP Calculus AB考试历年题目全解[注意：本文以解题步骤为主要内容，为了更好地呈现题目解析，会包含大量的公式和符号。

请确保在阅读时有一定的数学基础。

]1. 第一题题目描述：给定一个函数 f(x)，求其在区间 [a，b] 上的定积分。

解析：根据题目所给函数 f(x) 的表达式，我们可以使用不定积分的方法解决这个问题。

首先，先求 f(x) 的原函数 F(x)，然后计算 F(b) -F(a) 即可得到定积分的结果。

具体步骤如下：1. 求函数 f(x) 的原函数 F(x)。

2. 计算 F(b) - F(a) 的值，即为所求定积分的结果。

2. 第二题题目描述：计算函数 f(x) 的导数。

解析：根据题目所给函数 f(x) 的表达式，我们可以使用求导的方法来计算其导数。

根据导数的定义，导数 f'(x) 可以通过计算 f(x) 的极限来得到。

具体步骤如下：1. 对函数 f(x) 求导。

2. 化简导数的表达式，得到最终结果。

3. 第三题题目描述：给定一个函数 f(x)，求其在某点 x=c 处的极限。

解析：根据题目所给函数 f(x) 的表达式，我们可以使用极限的定义来计算其在某点 x=c 处的极限。

根据极限的定义，我们需要分别计算 x 无限接近 c 时 f(x) 的左极限和右极限。

具体步骤如下：1. 计算 f(x) 在 c 的左极限。

2. 计算 f(x) 在 c 的右极限。

3. 判断左极限和右极限是否存在且相等，若相等则极限存在，否则不存在。

4. 第四题题目描述：给定一个函数 f(x)，求其在某点 x=c 处的导数。

解析：根据题目所给函数 f(x) 的表达式，我们可以使用导数的定义来计算其在某点 x=c 处的导数。

导数 f'(x) 可以通过计算 f(x) 的极限来得到。

具体步骤如下：1. 对函数 f(x) 求导。

2. 将 x 的值替换为 c，计算导数的值，即为所求导数的结果。

AP 微积分 AB 2008 (Form B) 真题及解析

(c) Let T be the time at which the car’s speed reaches 80 kilometers per hour.
Then, r(T ) = 80 or T = 0.331453 hours.
At time T, the car has gone
∫ x(T ) = T r (t ) dt = 10.794097 kilometers 0
AP® Calculus AB 2008 Scoring Guidelines
Form B
The College Board: Connecting Students to College Success
The 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 5,400 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.

AP Calculus AB历年真题精选2024

AP Calculus AB历年真题精选2024 2024年的AP Calculus AB考试即将到来，对于准备参加考试的学生来说，熟悉历年真题是非常重要的。

在本文中，我将为大家精选几道2024年AP Calculus AB考试的历年真题，帮助大家更好地备考。

一、选择题1. 求函数f(x) = x^3 - 3x^2 + 2x的导数。

解析：根据导数的定义，对函数f(x)求导数，我们可以将每一项的指数乘以系数，并减少指数。

因此，f'(x) = 3x^2 - 6x + 2。

2. 设函数g(x) = (x - 2)^2 + 3，求g(x)的最小值。

解析：由于(x - 2)^2的值永远大于等于0，所以g(x)的最小值就是当(x - 2)^2的值为0时，即x = 2。

代入函数g(x)，可以得到最小值为3。

二、计算题1. 求函数y = x^2 + 4x + 3的定积分，区间为[-1, 2]。

解析：我们可以先对函数y = x^2 + 4x + 3求不定积分，得到原函数F(x) = (1/3)x^3 + 2x^2 + 3x + C。

然后，我们将上下限带入原函数，得到积分值为F(2) - F(-1) = 11。

2. 求曲线y = x^3在x = 2处的切线方程。

解析：首先，我们需要求得曲线y = x^3的导数，即y' = 3x^2。

然后，代入x = 2，得到斜率k = 12。

由于切线方程的一般形式为y = kx +b，我们可以代入已知的点(x, y) = (2, 8)，解得b = -4。

因此，切线方程为y = 12x - 4。

三、证明题证明：对于任意实数x和y，有(x + y)^2 ≥ 4xy。

解析：我们可以展开(x + y)^2，得到x^2 + 2xy + y^2。

根据平方项的非负性，我们可以得出x^2 ≥ 0和y^2 ≥ 0。

因此，x^2 + y^2 ≥ 0，进而有x^2 + y^2 + 2xy ≥ 2xy。

AP_calculus参考书

Score
AP Score 5 4 3 2 1 Qualification Extremely well qualified Well qualified Qualified Possibly qualified No recommendation
Topic Outline for Calculus BC
1
by taking sufficiently close values of the domain.) • Understanding continuity in terms of limits. • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem). * Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form. II. Derivatives Concept of the derivative • Derivative presented graphically, numerically, and analytically. • Derivative interpreted as an instantaneous rate of change. • Derivative defined as the limit of the difference quotient. • Relationship between differentiability and continuity. Derivative at a point • Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. • Tangent line to a curve at a point and local linear approximation. • Instantaneous rate of change as the limit of average rate of change. • Approximate rate of change from graphs and tables of values. Derivative as a function • Corresponding characteristics of graphs of ƒ and ƒ∙. • Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’. • The Mean Value Theorem and its geometric interpretation. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Second derivatives • Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ’’∙. • Relationship between the concavity of ƒ and the sign of ƒ’∙. • Points of inflection as places where concavity changes. Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity. + Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration. • Optimization, both absolute (global) and relative (local) extrema. • Modeling rates of change, including related rates problems. • Use of implicit differentiation to find the derivative of an inverse function. • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration. • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations. + Numerical solution of differential equations using Euler’s method. + L’Hospital’s Rule, including its use in determining limits and convergence of improper integrals and series. Computation of derivatives • Knowledge of derivatives of basic functions, including power, exponential, logarithmic,

2024 AP Calculus AB历年真题全面解析

2024 AP Calculus AB历年真题全面解析随着人们对高等教育的追求日益增加，美国大学理事会（The College Board）推出的Advanced Placement（AP）考试成为了许多学生申请大学的重要标准之一。

尤其是对于准备攻读理工科学位的学生来说，AP Calculus AB考试是一个关键的考验。

本文将对2024年的AP Calculus AB历年真题进行全面的解析，以帮助学生更好地理解考试内容和考点，并为备考提供有效的指导。

一、题型解析AP Calculus AB考试题型主要包括选择题和解答题，题目涵盖了微积分的基本概念、定理和运算技巧。

选择题部分要求学生选择一个正确答案，解答题则要求学生写出详细的解题过程和答案。

选择题部分通常包含多项选择题和填空题。

在多项选择题中，学生需要在给出的选项中选择一个正确答案；而在填空题中，学生需要根据题目要求填写一个数值或一个表达式。

这部分题目主要考察学生对基本概念和运算技巧的掌握程度。

解答题部分包括问题求解和证明题。

问题求解题要求学生根据题目描述，运用所学的知识和技巧回答问题；证明题则要求学生证明一个给定的命题。

这部分题目主要考察学生的应用能力和分析能力。

二、难点解析1. 变量分离和微分方程变量分离和微分方程是微积分课程的重要内容之一。

在解答题中，经常会出现与变量分离和微分方程相关的问题。

学生需要掌握如何将微分方程转化为可分离变量的形式，并进行变量分离求解。

2. 极限和连续性极限和连续性是微积分的基础概念，也是整个微积分体系的重要支撑。

学生需要理解和熟练运用极限的定义，包括无穷大极限和无穷小极限。

对于连续函数的性质，学生需要掌握函数连续的必要条件和充分条件，以及连续函数的运算法则。

3. 导数和微分导数和微分是微积分的核心概念，也是微积分应用的基础。

学生需要理解导数的定义和性质，掌握常见函数的导数规则，并能够灵活运用导数进行函数的求导和近似计算。

ap-calculus-ab-and-bc-2019-exams-instructions

AP Calculus AB/BC ExamsRegularly Scheduled Exam Date: Tuesday morning, May 14, 2019Late-Testing Exam Date:Friday morning, May 24, 2019Before Distributing Exams: Check that the title on all exam covers is Calculus AB or Calculus BC. Be sure to distribute the correct exam—AB or BC—to the students. If there are any exam booklets with a different title, contact the AP coordinator immediately.What Proctors Need to Bring to This ExamExam packetsAnswer sheetsAP Student Packs2018-19 AP Coordinator’s ManualThis book—2018-19 AP Exam InstructionsAP Exam Seating Chart templateSchool Code and Homeschool/Self-Study Codes Extra graphing calculatorsPencil sharpener Container for students’ electronic devices (if needed) Extra No. 2 pencils with erasersExtra pens with black or dark blue inkExtra paperStaplerWatchSigns for the door to the testing room–“Exam in Progress”–“Phones of any kind are prohibited during thetest administration, including breaks”A P C a l c u l u s AB /BC E x a m s 2018-19 AP Exam InstructionsSEATING POLICY FOR AP CALCULUS AB AND CALCULUS BC EXAMSGraphing calculators are required to answer some of the questions on the AP Calculus Exams. Before starting the exam administration, make sure each student has a graphing calculator from the approved list on page 53 of the 2018-19 AP Coordinator’s Manual . If a student does not have a graphing calculator from the approved list, you may provide one from your supply. If the student does not want to use the calculator you provide or does not want to use a calculator at all, he or she must hand copy, date, and sign the release statement on page 52 of the AP Coordinator’s Manual .During the administration of Section I, Part B, and Section II, Part A, students may have no more than two graphing calculators on their desks. Calculators may not be shared. Calculator memories do not need to be cleared before or after the exam. Students with Hewlett-Packard 48–50 Series and Casio FX-9860 graphing calculators may use cards designed for use with these calculators. Proctors should make sure infrared ports (Hewlett-Packard) are not facing each other. Since graphing calculators can be used to store data, including text, proctors should monitor that students are using their calculators appropriately. Attempts by students to use the calculator to remove exam questions and/or answers from the room may result in the cancellation of AP Exam scores.The AP Calculus AB Exam and the AP Calculus BC Exam should be administeredsimultaneously. They may be administered in separate rooms, or in the same room if it is more convenient.SECTION I: Multiple Choice›Do not begin the exam instructions below until you have completed the appropriate General Instructions for your group.These exams include survey questions. The time allowed for the survey questions is in addition to the actual test-taking time.Make sure you begin the exams at the designated time. Remember, you must complete a seating chart for this exam. See pages 295–296 for a seating chart template and instructions. See the 2018-19 AP Coordinator’s Manual for exam seating requirements (pages 56–59).If you are giving the regularly scheduled exam, say:It is Tuesday morning, May 14, and you will be taking either the AP Calculus AB Exam or the AP Calculus BC Exam.If you are giving the alternate exam for late testing, say:It is Friday morning, May 24, and you will be taking either the AP Calculus AB Exam or the AP Calculus BC Exam.2018-19 AP Exam InstructionsIf you are giving the AP Calculus AB Exam, say:Look at your exam packet and confirm that the exam title is “AP Calculus AB.” Raise your hand if your exam packet contains any title other than “AP Calculus AB,” and I will help you.If you are giving the AP Calculus BC Exam, say:Look at your exam packet and confirm that the exam title is “AP Calculus BC.” Raise your hand if your exam packet contains any title other than “AP Calculus BC,” and I will help you.If you are giving both the AP Calculus AB Exam and AP Calculus BC Exam in the same room, say:Look at your exam packet and confirm that the exam title is “AP Calculus AB” or “AP Calculus BC,” depending upon which exam you are taking today. Raise your hand if your exam packet contains any other title and I will help you.Once you confirm that all students have the correct exam, say:In a moment, you will open the exam packet. By opening this packet, you agree to all of the AP Program’s policies and procedures outlined in the 2018-19 Bulletin for AP Students and Parents.Y ou may now remove the shrinkwrap from the outside only of your exam packet. Do not open the Section I booklet; do not remove the shrinkwrap from the Section II materials. Put the white seals and the shrinkwrapped Section II booklet aside. . . .Carefully remove the AP Exam label found near the top left of your exam booklet cover. Place it on page 1 of your answer sheet on the light blue box near the top right corner that reads “AP Exam Label.”If students accidentally place the exam label in the space for the number label or vice versa, advise them to leave the labels in place. They should not try to remove the label; their exam can still be processed correctly.Listen carefully to all my instructions. I will give you time to complete each step. Please look up after completing each step. Raise your hand if you have any questions.Give students enough time to complete each step. Don’t move on until all students are ready. Read the statements on the front cover of the Section I booklet. . . .Sign your name and write today’s date. . . .Now print your full legal name where indicated. . . .Turn to the back cover of your exam booklet and read it completely. . . .Give students a few minutes to read the entire cover.Are there any questions? . . .Y ou will now take the multiple-choice portion of the exam. Y ou should have in front of you the multiple-choice booklet and your answer sheet. Y ou may never discuss the multiple-choice exam content at any time in any form with anyone, including your teacher and other students. If you disclose the multiple-choice exam content through any means, your AP Exam score will be canceled. Open your answer sheet to page 2. Y ou must complete the answer sheet using a No. 2 pencil only. Mark all of your responses beginning on page 2 of your answer sheet, one response per question. Completely fill in the circles. If you need to erase, do so carefully and completely. No credit will be given forA P C a l c u l u s AB /BC E x a m s 2018-19 AP Exam Instructionsanything written in the exam booklet. Scratch paper is not allowed, but you may use the margins or any blank space in the exam booklet for scratch work. Section I is divided into two parts. Each part is timed separately, and you may work on each part only during the time allotted for it. Calculators are not allowed in Part A. Please put your calculators under your chair. Are there any questions? . . .Y ou have 1 hour for Part A. Part A questions are numbered 1 through 30. Mark your responses for these questions on page 2 of your answer sheet. Open your Section I booklet and begin.Note Start Time . Note Stop Time .Check that students are marking their answers in pencil on page 2 of their answer sheets and that they are not looking beyond Part A. The line of A’s at the top of each page will assist you in monitoring students’ work.After 50 minutes, say:There are 10 minutes remaining.After 10 minutes, say:Stop working on Part A and turn to page 24 in your Section I booklet. . . .On that page, you should see an area marked “PLACE SEAL HERE.” Making sure all of your other exam materials, including your answer sheet, are out of the way, take one of your seals and press it on that area and then fold the seal over the open edge to the front cover. Be sure you don’t seal the Part B section of the booklet or let the seal touch anything except the marked areas. . . .After all students have sealed Part A, say:Graphing calculators are required for Part B. Y ou may get your calculators from under your chair and place them on your desk. Part B questions are numbered 76 through 90. Fold your answer sheet so only page 3 is showing and mark your responses for these questions on that page. Y ou have 45 minutes for Part B. Y ou may begin.Note Start Time . Note Stop Time .Check that students have sealed their booklets properly and are now working on Part B. The large B’s in an alternating shaded pattern at the top of each page will assist you in monitoring their work. Proctors should make sure that students are using their calculators appropriately. Proctors should also make sure Hewlett-Packard calculators’ infrared ports are not facing each other.After 35 minutes, say:There are 10 minutes remaining.After 10 minutes, say:Stop working and turn to page 38. Y ou have 3 minutes to answerQuestions 91–94. These are survey questions and will not affect your score.Note that each survey question has five answer options. Y ou may not go back to work on any of the exam questions. . . .Give students approximately 3 minutes to answer the survey questions.2018-19 AP Exam Instructions Then say:Close your booklet and put your answer sheet on your desk, faceup. Make sure you have your AP number label and an AP Exam label on page 1 of your answer sheet. Sit quietly while I collect your answer sheets.Collect an answer sheet from each student. Check that each answer sheet has anAP number label and an AP Exam label.After all answer sheets have been collected, say:Now you must seal your Section I booklet. Remove the remaining white seals from the backing and press one on each area of your exam booklet cover marked “PLACE SEAL HERE.” Fold each seal over the back cover. When you have finished, place the booklet on your desk, faceup. I will now collect your Section I booklet. . . .Collect a Section I booklet from each student. Check that each student has signed the front cover of the sealed Section I booklet.There is a 10-minute break between Sections I and II.When all Section I materials have been collected and accounted for and you are ready for the break, say:Please listen carefully to these instructions before we take a 10-minute break. All items you placed under your chair at the beginning of this exam, including your Student Pack, must stay there, and you are not permitted to open or access them in any way. Leave your shrinkwrapped Section II packet on topof your desk during the break. Y ou are not allowed to consult teachers, other students, notes, textbooks, or any other resources during the break. Y ou may not make phone calls, send text messages, use your calculators, check email, use a social networking site, or access any electronic or communication device. Y ou may not leave the designated break area. Remember, you may never discuss the multiple-choice exam content with anyone, and if you disclose the content through any means, your AP Exam score will be canceled. Are thereany questions? . . .Y ou may begin your break. T esting will resume at . SECTION II: Free ResponseAfter the break, say:May I have everyone’s attention? Place your Student Pack on your desk. . . .Y ou may now remove the shrinkwrap from the Section II packet, but do not open the Section II exam booklet until you are told to do so. . . .Read the bulleted statements on the front cover of the exam booklet. Look up when you have finished. . . .Now take an AP number label from your Student Pack and place it on the shaded box. If you don’t have any AP number labels, write your AP number in the box. Look up when you have finished. . . .Read the last statement. . . .Using your pen, print the first, middle, and last initials of your legal name in the boxes and print today’s date where indicated. This constitutes your signature and your agreement to the statements on the front cover. . . .A P C a l c u l u s AB /BC E x a m s 2018-19 AP Exam InstructionsNow turn to the back cover. Using your pen, complete Items 1 through 3 under “Important Identification Information.”. . .Read Item 4. . . .Are there any questions? . . .If this is your last AP Exam, you may keep your Student Pack. Place it under your chair for now. Otherwise if you are taking any other AP Exams this year, leave your Student Pack on your desk and I will collect it now. . . .Read the remaining information on the back cover of the exam booklet, paying careful attention to the bulleted statements in the instructions. Do not open the exam booklet or break the seals in the exam booklet until you are told to do so. Look up when you have finished. . . .Collect the Student Packs from students who are taking any other AP Exams this year.Then say:Are there any questions? . . .Section II also has two parts that are timed separately. Y ou are responsible for pacing yourself and may proceed freely from one question to the next within each part. Graphing calculators are required for Part A, so you may keep your calculators on your desk. Y ou must write your answers in the appropriate space in the exam booklet using a No. 2 pencil or a pen with black or dark blue ink. Do not break the seals for Part B at this time. Are there any questions? . . .Y ou have 30 minutes to answer the questions in Part A. If you need more paper to complete your responses, raise your hand. At the top of each extra sheet of paper you use, write only: your AP number, the exam title, andthe question number you are working on.Do not write your name. Open your exam booklet and begin.Note Start Time . Note Stop Time .Check that students are working on Part A only and writing their answers in their exam booklets using pencils or pens with black or dark blue ink. The pages for the Part A questions are marked with large 1’s or 2’s at the top of each page to assist you in monitoring their work.After 20 minutes, say:There are 10 minutes remaining in Part A.After 10 minutes, say:Stop working on Part A. Calculators are not allowed for Part B. Please put all of your calculators under your chair. . . .Turn to page 13. Y ou have 1 hour for Part B. During this time you may go back to Part A, but you may not use your calculator. Remember to show your work and write your answer to each part of each problem in the appropriate space in the exam booklet. Are there any questions? . . .Using your finger, break open the seals on Part B. Do not peel the seals away from the booklet. Y ou may go on to the next page and begin Part B.Note Start Time. Note Stop Time .2018-19 AP Exam Instructions After 50 minutes, say:There are 10 minutes remaining in Part B.After 10 minutes, say:Stop working and close your exam booklet. Place it on your desk, faceup. . . . If any students used extra paper for a question in the free-response section, have those students staple the extra sheet(s) to the first page corresponding to that question in their free-response exam booklets. Complete an Incident Report after the exam and return these free-response booklets with the extra sheets attached in the Incident Report return envelope (see page 68 of the 2018-19 AP Coordinator’s Manual for complete details).Then say:Remain in your seat, without talking, while the exam materials are collected. . . . Collect a Section II exam booklet from each student. Check for the following:Exam booklet front cover: The student placed an AP number label on the shaded box and printed their initials and today’s date.Exam booklet back cover: The student completed the “Important IdentificationInformation” area.When all exam materials have been collected and accounted for, return to students any electronic devices you may have collected before the start of the exam.If you are giving the regularly scheduled exam, say:Y ou may not discuss or share the free-response exam content with anyone unless it is released on the College Board website in about two days. Y our AP Exam score results will be available online in July.If you are giving the alternate exam for late testing, say:None of the content in this exam may ever be discussed or shared in any way at any time. Y our AP Exam score results will be available online in July.If any students completed the AP number card at the beginning of this exam, say: Please remember to take your AP number card with you. Y ou will need the information on this card to view your scores and order AP score reporting services online.Then say:Y ou are now dismissed.After-Exam TasksBe sure to give the completed seating chart to the AP coordinator. Schools must retain seating charts for at least six months (unless the state or district requires that they be retained for a longer period of time). Schools should not return any seating charts in their exam shipments unless they are required as part of an Incident Report.NOTE: If you administered exams to students with accommodations, review the 2018-19 AP Coordinator’s Manual and the 2018-19 AP SSD Guidelines for information about completing the Nonstandard Administration Report (NAR) form, and returning these exams.A P C a l c u l u s AB /BC E x a m s 2018-19 AP Exam InstructionsThe exam proctor should complete the following tasks if asked to do so by the AP coordinator. Otherwise, the AP coordinator must complete these tasks:Complete an Incident Report for any students who used extra paper for the free-responsesection. (Incident Report forms are provided in the coordinator packets sent with the exam shipments.) These forms must be completed with a No. 2 pencil. It is best to complete a single Incident Report for multiple students per exam subject, per administration (regular or late testing), as long as all required information is provided. Include all exam booklets with extra sheets of paper in an Incident Report return envelope (see page 68 of the 2018-19 AP Coordinator’s Manual for complete details).Return all exam materials to secure storage until they are shipped back to theAP Program. (See page 27 of the 2018-19 AP Coordinator’s Manual for more information about secure storage.) Before storing materials, check the “School Use Only” section on page 1 of the answer sheet and:Fill in the appropriate section number circle in order to access a separateAP Instructional Planning Report (for regularly scheduled exams only) or subject score roster at the class section or teacher level. See “Post-Exam Activities” in the 2018-19 AP Coordinator’s Manual .Check your list of students who are eligible for fee reductions and fill in theappropriate circle on their registration answer sheets.。

2024年AP Calculus AB真题全方位解读

2024年AP Calculus AB真题全方位解读敬爱的读者朋友们，今天我将为大家带来对2024年AP Calculus AB真题的全方位解读。

Calculus AB作为高中阶段的高等数学科目，是一门考察学生对微积分概念、方法和应用的综合能力的考试。

通过对本次考试题目的详细解析，我将帮助大家更好地理解这门学科的要点和考试要求。

本次考试共分为两个部分：选择题和解答题。

以下将从宏观和微观两个层面对每个题目进行解析，帮助读者朋友们全面掌握考试要点。

一、选择题解析选择题是考察学生对概念理解和快速计算的能力的重要方式。

以下是本次考试中的几道典型选择题：1. 在区间[0, π] 上考虑函数 f(x) = sinx + cosx，那么 f'(x) 的值在以下哪个区间上均为正数？A. (0, π/2)B. (π/2, π)C. (π/4, 3π/4)D. (3π/4, π)解析：我们首先求出 f'(x) 的导函数为 f'(x) = cosx - sinx。

然后，我们通过观察 sinx 和 cosx 在[0, π] 区间上的图像变化，可以得知当π/4 < x < 3π/4 时，cosx - sinx 大于零，故选项 C 正确。

2. 函数 f(x) = x^3 - 5x^2 + 2x 的图像在以下哪个点处有一个局部极小值？A. (0, 0)B. (1, -1)C. (1, 2)D. (2, -4)解析：我们首先求出 f'(x) 的导函数为 f'(x) = 3x^2 - 10x + 2。

然后，我们将导函数的根代入 f(x) 中，计算函数值。

经计算可知，当 x = 1 时，f(x) 取得局部极小值，故选项 C 正确。

通过以上两个例题的解析，我们可以看出选择题在考察学生对概念和方法的理解上具有一定的难度。

但只要我们掌握了基本的概念和计算方法，就能够迎接这些挑战。

二、解答题解析解答题是考察学生对概念和计算方法的深入理解以及解决实际问题的能力。

AP各科考试教材推荐

AP各科考试教材推荐本文整理了AP各科考试教材推荐，包括AP考试常见问题15问，帮助大家快速了解AP考试，请看详细内容介绍。

1、问：什么是AP考试?答：AP是Advanced Placement的缩写，即大学预修课程。

2、问：参加AP考试有什么作用?答：AP成绩不但可以抵扣成功申请美国大学的同学入学后相应课程的学分，而且AP成绩也成为美国各大学录取学生的重要依据。

美国，是大部分留学生的首选之地，留学美国，首先要了解美国大学的教学制度，美国大学毕业成绩采用的是学分制，每一学分都要交取一定的费用，只有达到了一定的学分，才可以顺利毕业。

如果在去往美国留学之前，可以再国内赚取美国大学的学分，这样便可节省时间与费用，目前国内学习可以获得美国大学学分的办法有两个，其中一种方法便是通过AP课程考试或得学分，那么AP考试的好处在哪里呢?据悉，AP(美国大学预修课程考试)课程考试覆盖全球想去往美国留学的高中生，国内计划考取AP的学生要到西南财经大学美国留学预备学院，参加两周的考试。

此考试是由美国大学理事会主持的，属于高中生具有大学水平的课程，其目的是让全球最优秀的高中学生提前考取美国大学课程。

当然，人们最想知道的还是通过AP考试的好处都有哪些，AP考试的成绩是美国各所院校录取学生的重要参考之一，其次，AP考试的成绩也可以抵扣学生入学后的基础课程的学分。

家长及学生正是因为知道了AP课程的好处，近年来参加报名考试的人日益增多。

3、问：AP考试如何报名?答：报名网站：AP考试于每年3月前报名，5月考试6月底之前考生就可以收到成绩单。

(PS：一般北京的考场费用在600~700元/门，香港的考场费用为1000~1500港币/门。

)4、问：每年什么时候考AP?考试地点有哪些?成绩怎么算?答：AP考试全球统一在每年五月份的前二周的工作日进行。

AP考试成绩为5分制，满分5分的意义是极为优秀，4分为优秀，3分相当于合格。

每门课的考试时间约2-3个小时。

AP Calculus AB Syllabus

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 bc知识点

ap calculus bc知识点
以下是AP Calculus AB和BC的共同和不同知识点：
共同知识点：
1.一元函数的极限、连续性和微积分基本原理。

2.导数和微分的计算和应用，包括最值，凹凸性和曲线图的分析。

3.定积分和反常积分的计算和应用，包括积分方法和积分应用。

4.基本微积分定理和微积分基本公式的应用。

5.微分方程和其解法的基本知识。

AP Calculus BC的独立知识点：
1.定积分的应用，包括体积、表面积、质量、重心和扭曲面积的计算。

2.参数方程、极坐标和向量方程的微积分，包括曲线长度和速度的计算。

3.微积分学中的序列和级数，包括收敛和发散、收敛测试和幂级数的
收敛半径。

4.黎曼和与积分的近似值，包括梯形法和辛普森法。

5.微积分的基本应用，如曲线最值、广义平均值和微积分学的基本定
理的应用。

注：以上内容取自美国考试服务中心（The College Board）的官方
网站。

2024 AP Calculus AB历年题目精粹

2024 AP Calculus AB历年题目精粹在2024年的AP Calculus AB考试中，历年的题目一直是备考的重要内容。

通过分析历年的题目，我们可以更好地了解该科目的考察点和难点，为备考提供有力的参考。

本文将对2024年AP Calculus AB历年题目进行精选和整理，以便学生们更好地备考。

第一部分：选择题选择题是AP Calculus AB考试中的常见题型，涵盖了课程的各个知识点。

以下是2024年AP Calculus AB选择题中的精选题目。

题目一：设函数f(x) = (x^3 - 9x^2 + 15x - 7)^{\frac{1}{3}}，则f'(x)等于：A. 3x^2 - 18x + 15B. 3x^2 - 14xC. 3x^2 - 6x + 5D. 3x^2 - 6x - 2E. 3x^2 - 11x - 2题目二：已知函数f(x) = e^x，g(x) = \ln(x + 1)，则f(g(e))的值为：A. 1B. eC. e^2D. e^xE. 0题目三：对于函数f(x) = \frac{2}{x + 1}，以下哪个区间上f(x)是严格递增的？A. (-\infty, -1)B. (-1, 0)C. (0, 1)D. (1, \infty)E. (-\infty, \infty)第二部分：解答题解答题是AP Calculus AB考试中的较为复杂的题型，要求学生理解问题并进行完整的解答。

以下是2024年AP Calculus AB解答题中的精选题目。

题目一：已知曲线y = x^2 + 2x，求曲线与x轴之间的面积。

题目二：已知函数f(x) = 2x^3 - 3x^2 - 12x + 5，求f'(x)的零点。

题目三：已知函数f(x) = \frac{1}{2}x^2 - 2x + 4，求f(x)的极大值和极小值点。

第三部分：综合题综合题是AP Calculus AB考试中的综合考察题型，要求学生综合运用所学的知识和技巧进行解答。

ap04_calculusbc_formb_38589【AP Calculus】(2021年-202

AP®Calculus BC2004 Sample Student ResponsesForm BThe materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other usemust be sought from the Advanced Placement Program®. Teachers may reproduce them, in whole or in part, in limited quantities, for face-to-face teachingpurposes but may not mass distribute the materials, electronically or otherwise. This permission does not apply to any third-party copyrights contained herein. These materials and any copies made of them may not be resold, and the copyright noticesmust be retained as they appear here.The 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 4,500 schools, colleges, universities, and other educational organizations. Each year, the C ollege Board serves over three 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 b e st-known programs are the SAT®, thePSAT/NMSQT®, and the Advanced Placement Program®(AP®). The College Board is committed to the principles ofexcellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns.For further information, visit Copyright © 2004 College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, AP Vertical Teams, APCD, Pacesetter, Pre-AP, SAT, Student Search Service, and the acorn logo are registered trademarks of theCollege Entrance Examination Board. PSAT/NMSQT is a registered trademark of theCollege Entrance Examination Board and National Merit Scholarship Corporation. Educational Testing Service and ETS are registered trademarks of Educational Testing Service.Other products and services may be trademarks of their respective owners.For the College Board’s online home for AP professionals, visit AP Central at .。

Calculus第八版教学设计 (2)

Calculus第八版教学设计IntroductionCalculus is a basic and important course in mathematics that is essential for students who m to take up a career in engineering, physics, economics, or computer science. This course mnly covers the topics like limits, differentiation, integration, and application of derivatives and integrals. Calculus also provides the foundation for many advanced courses in mathematics. The m of this document is to design an effective teaching plan for Calculus course using the latest 8th edition of the subject.Learning objectivesThe primary objective of the Calculus course is to help students understand the fundamental concepts of differentiation and integration and their application in solving real-world problems. At the end of this course, the students should be able to:•Understand the concept of limits and infinity•Analyze the continuity of functions using limits•Apply the rules of differentiation, including product rule, chn rule, and quotient rule•Solve optimization problems using differentiation•Understand the concept of integration and antiderivatives•Compute definite integrals using integration by substitution and integration by parts•Understand the application of integration in calculation of areas and volumes•Be able to solve real-world problems related to calculus Course contentThe following is the outline of topics that will be covered during the Calculus course:1.Limits and continuity2.Derivatives and its applications3.Integration and its applications4.Differential equations5.Infinite series6.Additional topics (depending on the avlability of time)Teaching methodologyThe teaching methodology used for the Calculus course will include the following:LecturesLectures will be the primary mode of instruction. The lectures will be designed to provide students with a comprehensive understanding of the topics covered in the course. Lectures will be delivered using interactive multimedia tools and other innovative teaching techniques. Each lecture will be around 60 to 90 minutes in duration.Problem-solving sessionsProblem-solving sessions will be conducted in the form of tutorials. These sessions will provide students with an opportunity to apply the concepts learned in the lectures. During these sessions, students will be given a set of problems to solve, and they will be guided by the instructor to arrive at the correct solution. These sessions will be around 60 to 90 minutes in duration.Group discussionsGroup discussions will be conducted periodically. These sessionswill provide students with an opportunity to discuss their understanding of the topics covered in the course. Students will be divided into groups, and each group will be assigned a set of problems to solve. These sessions will be around 40 to 60 minutes in duration.AssignmentsAssignments will be given periodically to help students to practice and reinforce the concepts learned in the course. These assignments will consist of both theoretical and numerical problems. Students will be given around one week to complete the assignments.ExaminationsExaminations will be conducted at the end of each module. The examinations will include both theoretical and numerical questions. The duration of each examination will be around 120 minutes.AssessmentThe students’ performance in the course will be assessed based on the following components:•Assignments (20%)•Examinations (50%)•Group discussions (10%)•Problem-solving sessions (10%)•Attendance and participation in class (10%)ConclusionIn conclusion, the Calculus course is an essential course for students seeking a career in several fields like engineering, physics, economics, or computer science. The course content and the teaching methodology will help students understand the fundamental concepts of calculus, solve real-world problems, and provide a foundation for advanced courses in mathematics. The assessment mechanism will ensure that students learn and practice the concepts covered in the course.。

AP Calculus AB Syllabus

AP CALCULUS AB SYLLABUS STACY LINN 2010-2011RULE OF FOUROur textbook (along with the world of math) has adopted the rule of four. Just like Pre Calculus we will be exploring Calculus graphically, numerically, algebraically, and verbally. We will find the connections between graphs (with out equations) and data and how it relates to a graph. We still analyze traditional equations and how they relate to the graphs and numerical representations. Last, you will be required to write sentences to justify your thoughts for a problem. Often, there may be little mathematical manipulations, but a sentence to explain what is happening with a graph.TEXTBOOKSmith & Minton, Calculus: Early Transcendental Functions, McGraw-Hill, 3rd edition, 2007OTHER RESOURCESHowell & Montgomery, Be Prepared for the AP Calculus Exam, Skylit, 2005Finney, Demana, waits, & Kennedy, Calculus Graphical, Numerical, Algebraic, Prentice Hall, Third Edition, 2010UNIT 1: LIMITS AND CONTINUITY (17 DAYS)∙Verbal Definition of a limit∙Rate of change and Tangent Lines∙The concept of Limito Graphicallyo Tableo One and two sided limits∙Computation of Limitso At a pointPowell County High Schoolo Properties of Limitso End Point Behavior Model∙Continuity and Its Consequenceso Types of Discontinuity∙Limits Involving Infinityo AsymptotesUNIT 2: DIFFERENTIATION (21 DAYS)∙Tangent Lines and Velocityo Slope of Secant Lineo Slope of Tangent Lineo Average Velocityo Instantaneous Velocity∙The Derivative- The definition of f’o Numerical using the calculatoro Differentiability and continuityo Relating graphs of f and f’∙Computation of Derivatives: The Power Rule∙The Product and Quotient Rules∙The Chain Rule∙Derivatives of Trigonometric Functions∙Derivatives Exponential and Logarithmic Functions∙Implicit Differentiation and Inverse Trigonometric FunctionsCHAPTER 3: APPLICATIONS OF DIFFERENTIATION (18 DAYS)∙The Mean Value Theorem and Rolles Theorem∙Maximum and Minimum ValuesPowell County High Schoolo Critical Number∙Extreme Value Theorem∙Using the derivative to find:o Critical Points and Extreme Valueso When the function is increasing and decreasing (1st derivative test)o Points of Inflectiono When the functions is concave up and concave down (2nd derivative test)∙Optimization∙Related Rates∙Rates of Change in Economics and the SciencesCHAPTER 4: INTEGRATION (18 DAYS)∙Antiderivatives∙Sum and Sigma Notation∙Areao Approximating Area with Rectangleso Riemann Sum∙The Definite Integralo Overall Change in Positiono Average Value of a Functiono Integral Mean Value Theorem∙The Fundamental Theorem of Calculus (1 and 2) on a closed interval∙Integration by Substitution∙Numerical Integrationo Midpoint Ruleo Trapezoid Rule∙Natural Logarithm as an IntegralPowell County High SchoolCHAPTER 5 APPLICATIONS OF THE DEFINITE INTEGRAL (11DAYS)∙Area between curves∙Volume of Revolution∙Volume by SlicingCHAPTER 6 DIFFERENTIAL EQUATIONS AND MATHEMATICALMODELING (12 DAYS)∙Slope Fields∙Antiderivative and the indefinite integral∙Separable differential equations∙Exponential Growth and DecayREVIEW FOR THE AP EXAM (36 DAYS)To prepare students for the AP Exam we will spend the last 6 weeks reviewing for the exam. Due to our school often missing for snow this also allows for missed days to cover content. During that time period students will answer questions from the 1997, 1998, and 2003 exams during a timed environment. Students will work at night on assignments from the Be Prepared for the AP Calculus Exam, from Skylit publishing. Students will take the 1998, 2003, 2007 & 2008 free response questions. Students will be expected to write sentences and verbalize their justifications for these questions. I will use the document camera to share student work so students can discuss how the free response would be graded and how they can improve on their justifications.STUDENT EVALUATIONTESTSEach Unit will be followed by a test. These tests will assess the students basic skills learned in the chapter. Each test will also include opportunities for students to write and justify their solutions. For example if a student must find a numerical derivative, that would be asked to explain what that numerical value means in terms of the problem.RETAKING A TESTDoes not exist…however there are test corrections. Keep ReadingPowell County High SchoolTEST CORRECTIONSTest corrections are analysis of errors made on tests. Students will write analyses of mistakes and then correctly re-work the problems. They are evaluated by letter grades. I am attaching a full explanation of test corrections and how they are graded.LABSLabs are essential part of lessons. Labs are used to further understanding and to teach content. Labs always receive a letter grade.EXPLORATIONSExplorations are a short investigation that introduce topics and aid in the understanding of that days lesson. They are occasionally collected.HOMEWORKHomework assignments are those textbook exercises and handouts that are given on your assignment sheet. Assignment sheets are distributed at the beginning ofeach chapter.Homework will not be reviewed in class. After an assignment has been given, there will be an opportunity to ask for clarification on particular problems so that a help sheet can be completed. Help sheets will be available by the next school day in thetray on the classroom door.There will be periodic homework tests given that check to see that everyone iscompleting the homework. These will be evaluated by points and averaged into the homework score.MIDTERMStudents will take a midterm before Christmas Break that will be similar to an AP Exam based on the content covered. This will be graded according to the AP standards exam consisting of 30 multiple choice questions valued at two points each and 6 free response questions valued at ten points. All the questions on this exam have been taken from previously released AP exams or from the AP Calculus Course Description.SNOWPlease be aware that we do not have time for SNOW days therefore I will provide you with required assignments that should be completed should a snow day occur. These are not optional.Powell County High SchoolGRADING Grading Scale∙ A = 90 – 100∙ B = 80 –89∙ C = 70 – 79∙ D = 60 – 69∙ F = 59 and belowTests, Labs, and Test Corrections 60%Explorations, Homework and Review Problems 40%Powell County High SchoolAP Calculus Students:You have chosen an exciting path for your senior year here at Powell County High School! AP Calculus AB will be challenging and often time-consuming, but I promise this course will be beneficial! PCHS is 1 of 27 schools participating in a grant program through AdvanceKy. This program is providing our teachers and students in AP courses with tons of support that we did not have in the past. The goal of this grant is to increase the number of students taking AP courses and to increase the number of passing scores on AP exams (3, 4, or 5 on a scale of 1 to 5)I participated this summer in a week long training focused on teaching you AP Calculus. I received wonderful tips, exercises, lessons, slides, etc. to help you master the material, learn test taking strategies and to lead you toward our over-all goal- YOU PASSING THE EXAM!AdvanceKy does offer a $100 gift to you if you receive a qualifying score on the exam. You are also eligible for extra KEES money based on your qualifying scores (3-$200, 4-$250, 5-$300). Students that take AP courses increase college admission eligibility, they are more likely to KEEP their KEES awards in college, and consistently out perform non-AP students in college.I am very excited that we will be offering three 6-hour exam prep sessions before you take the exam. The tentative dates for these sessions are January 22nd at Montgomery County High School, February 26th at Clark County High School, and April 30th at Powell County High School. AdvanceKy will provide transportation to each session, yourlunch/snacks, and lots of prizes! This will be an awesome academic environment for you as students. The review sessions will be ran conference style so that you can choose four different sections that you want to attend. Through out the 3 sessions you have the potential of reviewing 12 AP Calculus topics, which will include sample multiple choice questions, free response questions and help with how to answer these questions. These sessions will be taught by teachers from different high schools and national consultants that are hired by AdvanceKy.These review sessions are not required. However, I highly encourage you to plan to attend. It is a great opportunity and support system that other students in the state/nation do not have.GOOD LUCK and I hope you have a GREAT SENIOR year!Powell County High SchoolMrs. Stacy LinnPowell County High School。