AP 微积分BC选择题Section2练习
2015年AP微积分BCfree-response-questions
f x be the particular solution to the differential equation with the initial condition f 2 (c) Let y Does f have a relative minimum, a relative maximum, or neither at x 2 ? Justify your answer.
(d) Find the values of the constants m and b for which y
3.
mx b is a solution to the differential equation.
© 2015 The College Board. Visit the College Board on the Web: .
4, so that f x
neither at x
1 . Determine whether f has a relative minimum, a relative maximum, or x 4x 2. Justify your answer.
2
(c) Find the value of k for which f has a critical point at x (d) Let k Find
2
x 2 kx
2
.
(a) Let k
3, so that f x
1 . Write an equation for the line tangent to the graph of f at the point x 3x
2
whose x-coordinate is 4. (b) Let k
AP 微积分BC 选择题样卷一
AP Calculus Practice ExamBC Version - Section I - Part ACalculators ARE NOT Permitted On This Portion Of The Exam28 Questions - 55 Minutes1) GivenFind dy/dx.a)b)c)d)e)2) Give the volume of the solid generated by revolving the region bounded by the graph of y = ln(x), the x-axis, the lines x = 1 and x = e, about the y-axis.a)b)c)d)e)3) The graph of the derivative of f is shown below.Find the area bounded between the graph of f and the x-axis over the interval [-2,1], given that f(0) = 1.a)b)c)d)e)4) Determine dy/dt, given thatanda)b)c)d)e)5) The functionis invertible. Give the slope of the normal line to the graph of f -1 at x = 3.a)b)c)d)e)6) Determinea)b)c)d)e)7) Give the polar representation for the circle of radius 2 centered at ( 0 , 2 ).a)b)c)d)e)8) Determinea)b)c)d)e)9) Determinea)b)c)d)e)10) Give the radius of convergence for the seriesa)b)c)d)e)11) Determinea)b)c)d)e)12) The position of a particle moving along the x-axis at time t is given byAt which of the following values of t will the particle change direction I) t = 1/8II) t = 1/6III) t = 1IV) t = 2a) I, II and IIIb) I and IIc) I, III and IVd) II, III and IVe) III and IV13) Determinea)b)c)d)e)14) Determine the y-intercept of the tangent line to the curveat x = 4.a)b)c)d)e)15) The function f is graphed below.Give the number of values of c that satisfy the conclusion of the Mean Value Theorem for derivatives on the interval [2,5].a)b)c)d)e)16) Give the average value of the functionon the interval [1,3].a)b)c)d)e)17) A rectangle has both a changing height and a changing width, but the height and width change so that the area of the rectangle is always 20 square feet. Give the rate of change of the width (in ft/sec) when the height is 5 feet, if the height is decreasing at that moment at the rate of 1/2 ft/sec.a)b)c)d)e)18) The graph of the derivative of f is shown below.Give the number of values of x in the interval [-3,3] where the graph of f has inflection.a)b)c)d)e)19) A rectangle has its base on the x-axis and its vertices on the positive portion of the parabolaWhat is the maximum possible area of this rectanglea)b)c)d)e)20) Computea)b)c)d)e)21) Determinea)b)c)d)e)22) Determinea)b)c)d)e)23) Give the exact value ofa)b)c)d)e)24) Determinea)b)c)d)e)25) Give the derivative ofa)b)c)d)e)26) Give the first 3 nonzero terms in the Taylor series expansion about x = 0 for the functiona)b)c)d)e)27) Determinea)b)c)d)e)28) Which of the following series converge(s)a) B onlyb) A, B and Cc) B and Cd) A and Be) A and C1) d)2) e)3) b)4) b)5) e)6) a)7) c)8) c)9) b)10) d)11) c)12) c)13) c)14) e)15) a)16) d)17) a)18) b)19) a)20) d)21) b)22) b)23) a)24) c)25) d)26) b)27) b)28) c)。
ap微积分试题
ap微积分试题摘要：一、引言二、AP 微积分简介1.AP 微积分的概念2.AP 微积分的重要性三、AP 微积分试题类型1.选择题2.填空题3.解答题四、如何准备AP 微积分考试1.学习基础知识2.大量练习3.制定学习计划五、AP 微积分考试策略1.分配考试时间2.注意审题3.答题技巧六、总结正文：【引言】AP 微积分是美国大学理事会主办的一项考试，它涵盖了微积分的基本概念、原理和方法。
对于想要进入美国大学的学生来说，通过AP 微积分考试可以获得大学学分，降低大学学习压力。
因此，如何顺利通过AP 微积分考试成为了许多学生关心的问题。
本文将介绍AP 微积分试题的相关内容，并给出一些备考建议。
【AP 微积分简介】AP 微积分主要包括两个部分：微积分AB 和微积分BC。
其中，微积分AB 涵盖了微积分的基本概念和一元函数的微分学与积分学；微积分BC 在AB 的基础上，增加了多元函数的微分学与积分学以及向量分析等内容。
无论是AB 还是BC，考试都分为选择题和解答题两部分，共计10 个问题。
【AP 微积分试题类型】AP 微积分的试题类型主要包括选择题、填空题和解答题。
选择题要求考生从四个选项中选择一个正确答案；填空题要求考生填写一个或多个数值或表达式；解答题则要求考生详细阐述解题过程，展示自己的计算和分析能力。
【如何准备AP 微积分考试】要想顺利通过AP 微积分考试，首先需要掌握微积分的基本概念、原理和方法。
这包括理解极限、导数、积分等基本概念，熟练掌握求导、积分、微分方程等基本方法。
此外，大量的练习也是提高考试成绩的关键。
通过做题，考生可以熟悉题型，提高解题速度和准确率。
最后，制定合理的学习计划，确保自己在考试前充分准备。
【AP 微积分考试策略】在考试过程中，合理的分配时间是提高考试成绩的关键。
一般来说，考生应该把大部分时间用于解答题，因为这部分题目分值较高。
在解答题中，要注意审题，确保理解题目的要求。
在解答过程中，要清晰地展示自己的思路，合理运用公式和定理。
AP 微积分BC 选择题样卷一,DOC
AP Calculus Practice ExamBC Version - Section I - Part ACalculators ARE NOT Permitted On This Portion Of The Exam28 Questions - 55 Minutes1) GivenFind dy/dx.a)b)c)d)e)2) Give the volume of the solid generated by revolving the region bounded by the graph of y = ln(x), the x-axis, the lines x = 1 and x = e, about the y-axis.a)b)c)d)e)3) The graph of the derivative of f is shown below.Find the area bounded between the graph of f and the x-axis over the interval [-2,1], given that f(0) = 1.a)b)c)d)e)4) Determine dy/dt, given thatanda)b)c)d)e)5) The functionis invertible. Give the slope of the normal line to the graph of f -1 at x = 3.a)b)c)d)e)6) Determinea)b)c)d)e)7) Give the polar representation for the circle of radius 2 centered at ( 0 , 2 ).a)b)c)d)e)8) Determinea)b)c)d)e)9) Determinea)b)c)d)e)10) Give the radius of convergence for the seriesa)b)c)d)e)11) Determinea)b)c)d)e)12) The position of a particle moving along the x-axis at time t is given byAt which of the following values of t will the particle change direction?I) t = 1/8II) t = 1/6III) t = 1IV) t = 2a) I, II and IIIb) I and IIc) I, III and IVd) II, III and IVe) III and IV13) Determinea)b)c)d)e)14) Determine the y-intercept of the tangent line to the curveat x = 4.a)b)c)d)e)15) The function f is graphed below.Give the number of values of c that satisfy the conclusion of the Mean Value Theorem for derivatives on the interval [2,5].a)b)c)d)e)16) Give the average value of the functionon the interval [1,3].a)b)c)d)e)17) A rectangle has both a changing height and a changing width, but the height and width change so that the area of the rectangle is always 20 square feet. Give the rate of change of the width (in ft/sec) when the height is 5 feet, if the height is decreasing at that moment at the rate of 1/2 ft/sec.a)b)c)d)e)18) The graph of the derivative of f is shown below.Give the number of values of x in the interval [-3,3] where the graph of f has inflection.a)b)c)d)e)19) A rectangle has its base on the x-axis and its vertices on the positive portion of the parabolaWhat is the maximum possible area of this rectangle?a)b)c)d)e)20) Computea)b)c)d)e)21) Determinea)b)c)d)e)22) Determinea)b)c)d)e)23) Give the exact value ofa)b)c)d)e)24) Determinea)b)c)d)e)25) Give the derivative ofa)b)c)d)e)26) Give the first 3 nonzero terms in the Taylor series expansion about x = 0 for the functiona)b)c)d)e)27) Determinea)b)c)d)e)28) Which of the following series converge(s)?a) B onlyb) A, B and Cc) B and Cd) A and Be) A and C1) d)2) e)3) b)4) b)5) e)6) a)7) c)8) c)9) b)10) d)11) c)12) c)13) c)14) e)15) a)16) d)17) a)18) b)19) a)20) d)21) b)22) b)23) a)24) c)25) d)26) b)27) b)28) c)。
AP微积分BC模考卷2023
AP 微积分 BC 模考卷 2023一、选择题（每题1分，共5分）1. 若函数f(x)在点x=a处可导，则f'(a)表示的是（）A. f(x)在x=a处的斜率B. f(x)在x=a处的函数值C. f(x)在x=a处的切线方程D. f(x)在x=a处的曲率A. lim(x→∞) f(x) = LB. lim(x→0) f(x) = LC. lim(x→a) f(x) = ∞D. lim(x→∞) f'(x) = L3. 若函数f(x) = x^3 3x在x=1处的导数为0，则（）A. x=1是f(x)的极大值点B. x=1是f(x)的极小值点C. x=1是f(x)的拐点D. x=1是f(x)的驻点A. ∫(0,1) x dxB. ∫(1,∞) 1/x^2 dxC. ∫(∞,∞) e^(x^2) dxD. ∫(0,2π) sin(x) dx5. 若f(x) = e^(2x)，则f''(x)是（）A. 2e^(2x)B. 4e^(2x)C. e^(2x)D. 2e^x二、判断题（每题1分，共5分）6. 若函数在闭区间上连续，则该函数在该区间上一定可积。
（）7. 若f'(x) > 0，则f(x)是单调递增函数。
（）8. 泰勒公式可以用来近似任何可导函数。
（）9. 第一类间断点处的函数一定不可导。
（）10. 两个函数的导数相等，则这两个函数一定相同。
（）三、填空题（每题1分，共5分）11. 函数f(x) = x^2在x=0处的导数f'(0) = ______。
12. 若f(x) = 3x^3 4x^2 + 2x，则f'(x) = ______。
13. ∫(0,π) sin(x) dx = ______。
14. 函数f(x) = e^x的n阶导数f^(n)(x) = ______。
15. 曲线y = x^3在点(1,1)处的切线方程是______。
历年AP微积分真题
历年AP微积分真题Introduction：微积分是数学的一个重要分支，被广泛应用于科学、工程、经济学等领域。
AP微积分考试是美国高中学生常参加的一项重要考试，通过该考试可以获得大学学分。
本文将回顾历年AP微积分真题，帮助读者了解该考试的内容和难度。
Section 1: Differential CalculusDifferential calculus is concerned with the study of rates of change and slopes of curves. This section of the AP Calculus exam tests students' understanding and application of derivative concepts.1. Example question:Find the derivative of the function f(x) = 3x^2 + 2x - 5.Solution:Using the power rule, we differentiate term by term to obtain f'(x) = 6x + 2.2. Example question:A particle moves along a straight line with position function s(t) = 4t^3 - 6t^2 + 2t + 1. Find the velocity function.Solution:To find the velocity function, we differentiate the position function with respect to time. Thus, v(t) = s'(t) = 12t^2 - 12t + 2.Section 2: Integral CalculusIntegral calculus focuses on the accumulation of quantities and finding areas under curves. This section of the AP Calculus exam examines students' ability to calculate definite and indefinite integrals.3. Example question:Evaluate the definite integral ∫(4x^3 + 2x - 1)dx from x = 1 to x = 3.Solution:Using the power rule and the constant rule, we integrate term by term and evaluate the integral to obtain 110.4. Example question:Fin d the indefinite integral ∫(5e^x + 3/x)dx.Solution:Integrating term by term, we obtain the indefinite integral as 5e^x +3ln|x| + C, where C is the constant of integration.Section 3: Applications of CalculusCalculus is widely used in various real-world applications such as physics, economics, and biology. This section of the AP Calculus exam assesses students' ability to apply calculus concepts to solve practical problems.5. Example question:A tank contains 500 liters of water with a salt concentration of 0.2 grams per liter. Brine with a concentration of 1 gram per liter enters the tank at a rate of 5 liters per minute. The mixture is continuously stirred and drained at a rate of 3 liters per minute. Find the salt concentration in the tank after 10 minutes.Solution:Using the principles of differential equations, we set up a rate of change equation and solve it to find the salt concentration to be approximately 0.439 grams per liter after 10 minutes.Conclusion:The AP Calculus exam covers a wide range of topics in both differential and integral calculus. By reviewing past exam questions, students can gain a better understanding of the exam format and level of difficulty. Mastering calculus concepts and their applications is crucial for success in this exam and for a deeper understanding of the field of mathematics.。
AP-微积分BC-选择题样卷一
13) Determine
a)
b)
c)
d)
e)
14) Determine they-intercept of the tangent line to the curve
atx= 4.
a)
b)
c)
d)
e)
15) The functionfis graphed below.
Give the number of values ofcthat satisfy the conclusion of the Mean Value Theorem for derivatives on the interval [2,5].
At which of the following values oftwill the particle change direction?
I)t= 1/8
II)t= 1/6
III)t= 1
IV)t= 2
a) I, II and III
b) I and II
c) I, III and IV
d) II, III and IV
a)
b)
c)
d)
e)
26) Give the first 3 nonzero terms in the Taye function
a)
b)
c)
d)
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27) Determine
a)
b)
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d)
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28) Which of the following series converge(s)?
a)
b)
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d)
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10) Give the radius of convergence for the series
AP_微积分BC_选择题样卷一
AP Calculus Practice ExamBC Version - Section I - Part A Calculators ARE NOT Permitted On This Portion Of The Exam28 Questions - 55 Minutes1) GivenFind dy/dx.a)b)c)d)e)2) Give the volume of the solid generated by revolving the region bounded by the graph of y = ln(x), the x-axis, the lines x = 1 and x = e, about the y-axis.a)b)c)d)e)3) The graph of the derivative of f is shown below.Find the area bounded between the graph of f and the x-axis over the interval [-2,1], given that f(0) = 1.a)b)c)d)e)4) Determine dy/dt, given thatanda)b)c)d)e)5) The functionis invertible. Give the slope of the normal line to the graph of f -1 at x = 3.a)b)c)d)e)6) Determinea)b)c)d)7) Give the polar representation for the circle of radius 2 centered at ( 0 , 2 ).a)b)c)d)e)8) Determinea)b)c)d)e)9) Determinea)b)c)e)10) Give the radius of convergence for the seriesa)b)c)d)e)11) Determinea)b)c)d)e)12) The position of a particle moving along the x-axis at time t is given byAt which of the following values of t will the particle change direction?I) t = 1/8II) t = 1/6III) t = 1IV) t = 2a) I, II and IIIb) I and IIc) I, III and IVd) II, III and IVe) III and IV13) Determinea)b)c)d)e)14) Determine the y-intercept of the tangent line to the curveat x = 4.a)b)c)d)e)15) The function f is graphed below.Give the number of values of c that satisfy the conclusion of the Mean Value Theorem for derivatives on the interval [2,5].a)b)c)d)e)16) Give the average value of the functionon the interval [1,3].a)b)c)d)e)17) A rectangle has both a changing height and a changing width, but the height and width change so that the area of the rectangle is always 20 square feet. Give the rate of change of the width (in ft/sec) when the height is 5 feet, if the height is decreasing at that moment at the rate of 1/2 ft/sec.a)b)c)d)e)18) The graph of the derivative of f is shown below.Give the number of values of x in the interval [-3,3] where the graph of f has inflection.a)b)c)d)e)19) A rectangle has its base on the x-axis and its vertices on the positive portion of the parabolaWhat is the maximum possible area of this rectangle?a)b)c)d)e)20) Computea)b)c)d)e)21) Determinea)b)c)d)e)22) Determinea)b)c)d)e)23) Give the exact value ofa)b)c)d)e)24) Determinea)b)c)d)25) Give the derivative ofa)b)c)d)e)26) Give the first 3 nonzero terms in the Taylor series expansion about x = 0 for the functiona)b)c)d)e)27) Determinea)c)d)e)28) Which of the following series converge(s)?a) B onlyb) A, B and Cc) B and Cd) A and Be) A and C1) d)2) e)3) b)4) b)5) e)6) a)7) c)8) c)9) b)10) d)11) c)12) c)13) c)14) e)15) a)16) d)17) a)18) b)19) a)20) d)21) b)22) b)23) a)24) c)25) d)26) b)27) b)28) c)。
AP微积分BC 2023年真题 附答案和评分标准 AP Calculus BC 2023 Real
AP微积分BC 2023年真题附答案和评分标准 AP Calculus BC2023 Real一、选择题1. 问题描述这个问题是关于……2. 解答过程解答过程如下： - 第一步：…… - 第二步：…… - 第三步：……3. 答案和评分标准答案为：A评分标准如下： - 如果只给出了答案，得0分。
- 如果给出了正确的解答过程，得1分。
二、填空题1. 问题描述这个问题是关于……2. 解答过程解答过程如下： - 第一步：…… - 第二步：…… - 第三步：……3. 答案和评分标准答案为：50评分标准如下： - 如果只给出了答案，得0分。
- 如果给出了正确的解答过程，得1分。
三、解答题1. 问题描述这个问题是关于……2. 解答过程解答过程如下： - 第一步：…… - 第二步：…… - 第三步：……3. 答案和评分标准答案为：解答过程如下：解答步骤1解答步骤2解答步骤3评分标准如下： - 如果只给出了答案而没有解答步骤，得0分。
- 如果给出了解答步骤但部分错误，得1分。
- 如果给出了正确的解答步骤，得2分。
四、简答题1. 问题描述这个问题是关于……2. 解答过程解答过程如下： - 第一步：…… - 第二步：…… - 第三步：……3. 答案和评分标准答案为：……评分标准如下： - 如果只给出了答案而没有解答步骤，得0分。
- 如果给出了解答步骤但部分错误，得1分。
- 如果给出了正确的解答步骤，得2分。
五、解决问题1. 问题描述这个问题是关于……2. 解答过程解答过程如下： - 第一步：…… - 第二步：…… - 第三步：……3. 答案和评分标准答案为：……评分标准如下： - 如果只给出了答案而没有解答步骤，得0分。
- 如果给出了解答步骤但部分错误，得1分。
- 如果给出了正确的解答步骤，得2分。
六、总结通过完成这道AP微积分BC 2023年真题的解答，我们学习了……总体而言，这道题目涵盖了……Markdown文本格式的输出使得我们能够清晰地呈现问题描述、解答过程、答案和评分标准，这对于学生来说非常有帮助。
AP微积分BC选择题样卷一.docx
AP Calculus PraCtiCe EXam BCVerSion - SeCti On I - Part ACalculators ARE NoT Permitted On ThiS Portio n Of The EXam28 QUeStiOns - 55 MinUteS1) GiVe n弓2珀—χ y = 4Find dy∕dx.- 42”)_尹6,一Xa) b)C)8e c~3Jf) -j∕6y —兀d)眦(一2对_尹b)*τr (e1 2 3 -e)C)y^(≡4+l) d)6b y - xe)2 GiVe the volume of the solid gen erated by revo IVing the regi on boun ded by the graph of y = ln( x), the x-axis, the IineS X = 1 and X = e, about the y-axis.r (/-1)a)6y+x6y+χ*7Γ(∕+i)e)3) The graph Of the derivative Of f is ShOWn below.Find the area boun ded betwee n the graph ofinterval [-2,1], given that f (0) = 1.13~4~a)29~↑2b)亘JC)31^1Γd)11~4~e)4) Determ ine dy/dt, give n thatOy = x A- 4 xandf and the x-axis over theX — COS(3 t)—6 CaS(3Z) sin0a)-3 (2coβ(3:) + 4) sin(3r)b)6 cos (3 t) + 12C)-18 sm(3i)d)3(2: cos(3 ¢)+4) cos(31)e)5) The functionγ(x) = 5?+ 3C(^)is in Vertible. GiVe the slope of the no rmal li ne to the graph of X = 3.130 ÷ 6e6a)-2^1^b)丄~6C)30+6」d)-6e)6) Determ ine广(Sin(6 x) )5 6 (cos(6 x) )7⅛沁(24 x) ÷ C f5 1Tr8 192a) atb)* x + 命Sm(12 x) + C b)1217) GiVe the polar representatiOn for the CirCIe Of radius 2Centered at (0,2 ).r = 2 sm(θ) + 2 cos(θ)a)r = 4 cos(θ)b)r = 4 sm(θ)C)r ≡m(θ) = 2d)r = 4 sin(θ) — cos(θ)e)8) Determ ine Ξa)1 b)16C)C)时心心+Cd)e) τx ~ Sin(12 x) + C12 1 丄7 d)32e)9) Determ ine'.χΞa)1b)d)1-Tπe)10) GiVe the radius Of COnV erge nce for the SerieSβCfc + 3) 2⅛= ι7⅛e Seri&s d^r^esfor all x.a)1b)C)丄Td)3e)11) Determi neMGM2+÷) ^kιc2j B R)a)2b)Σ C)CId) e)12) The POSitiOn Of a PartiCIe moving along the x-axis at time t is given byX (f) = (Sin(4 Tr Z) )2At WhiCh of the followi ng VaIUeS of t will the PartiCIeCha nge directi on?I) t = 1/8II) t = 1/6Hl) t = 1IV) t = 2a) I, II and IIIb) I and IIC) I, III and IVd) II, III and IVe) III and IV13) Determi ne肚 X cos (τ) ∙iτ4 a) b)-2C)d) -⅜÷y-i ntercept of the tangent line to the CUrVeJ ∕ = √Z 2 + 33at X = 4.45_〒a)4÷1 3^πe)14) Determine the66 4?b)-3349C)135^49^d)33〒e)15) The function f is graphed below.GiVe the nu mber of VaIUeS of C that SatiSfy the COn clusi On Of the Mean Value TheOrem for derivatives on the interval [2,5].3a)Ξb)C)d)3.2e)16) GiVe the average value Of the fun Cti On on the in terval [1,3].a)—Z 牡7)+ Z 亡(T)3 3b)ZJT)3C)_」-引+心d)-2e(_3)+ 2√-υe)17) A recta ngle has both a Cha nging height and a Cha nging width, but the height and Width Cha nge so that the area of the recta ngle is always 20 SqUare feet. GiVe the rate of Cha nge of the Width (in ft/sec) Whe n the height is 5 feet, if the height is decreas ing at that mome nt at the rateof 1/2 ft/sec.2_Ta)-2^3^b)1C)1莎d)Ξ05e)18) The graph of the derivative of f is shown below.GiVe the number Of VaIueS Of X in the interval [-3,3] Where the graph Of f has in flect ion.1a)2b)OC)3d)There is not enough Iyifbrjnatio^.e)19) A rectangle has its base On the x-axis and its VertiCeS On the POSitiVe porti On Of the ParabOIaWhat is the maximum POSSibIe area of this recta ngle?a)b)1b) C)d)⅜√^√ie)20) COmPutee (3X) (tan(√2^))2⅛*(应(亡S)))S Ua)It an (e<2^)e c2x5 ÷cb) "4tan(e t2 X)) (sec(e (2 Jt)))3 e x5 + C C)*u n(邛 “))_+』")+cd) J ^丄 tan(e (2X)) + ±e C2x)+c2 、 7 2e)21) Determi ne36 + X1 ∑π a)1Ξ3 TrC)d)6 Tre)22) Determi ne(4Λ + 7j)11~Γa)1b)C)d)11e)23) GiVe the exact value Offfi?3)a)sm(5)b)coε(5)C)d)-sm(5)e)24) Determi ne—2)IIIn ---------------------------------x^0 I 1 —CoS(X) Ja) b)ΞC)Iindefinedd)3_~2e)25) GiVe the derivative Of-2xιc^2υ - Ξx t-3X) In(X)a)-2xx t-2,^υ +z t~2J) In(X)b)“(-"-I) _ 2邛-2町I n(X)C)-2XX(^X-^ -2x(~2X) In(X)d)-≡xx t^2τ-I)e)26) GiVe the first 3 non zero terms in the Taylor SerieS expa nsion about X = 0 for the fun Cti onf(x) = CoS(2 x)1 -2 Λ3 ÷4 z4a)1 -2 x2 ÷—x4 3b)C)1 +2 Λ^ + 2 Qd)1-2/e)27) Determi ne(孟+ 4)) +Ua)1 2^ln(IX-2∣) +-∣-In(IX+ 4∣) + Cb)2 1-I-In(IX-2∣)- -In(IX+ 4∣)÷C fC)2 1 -yta(μ + 4∣)- ^-k(∣r-2∣)+Cd) _yln(∣(x-2) (x + 4)∣) +Ue)28) WhiCh Of the follow ing SerieS COn Verge(s)?a) B onlyb) A, B and C C) B and Cd) A and Be) A and C1) d)2) e)3) b)4) b)5) e)6) a)7) C)8) C)9) b)10) d)11) C)12) C)13) c)14) e)15) a)16) d)17) a)18) b)19) a)20) d)21) b)22) b)23) a)24) c)25) d)26) b)27) b)28) c)。
AP 微积分BC选择题Section2练习
29） Find the average value of the functionover the interval [0， 4］.a)b)c)d)e)30) What is the y—intercept of the line tangent to the curve y = x2 + 7 at x = 3？a）b)c)d)e）31) Which of the following function(s） is continuous and differentiable？I。
II.III。
a） I and III onlyb） III onlyc) I and II onlyd) I onlye) II only32) Find ma)b）c）d)e)33) The graph of the derivative of f is shown below。
Which of the following must be true?a) f is concave down on [0， 4］。
b) f is increasing on ［—2， 2]。
c) f has a local maximum at x = 0.d）f has a local minimum at x = —2.e）f has a point of inflection at x = 4.34） The sum of two positive integers x and y is 60。
Find the value of x that minimizesa)b)c)d)e）35) A particle moves on the curvefind the speed of the particle at time t = .a） 7。
1414b） 6.7082c) 7。
2801d） 3。
3166e) 3.000036) The function f is defined asx 6Which of the following is false？a) f has a horizontal asymptote at y = 1。
ap微积分bc考试原题
AP微积分BC考试原题及答案一、选择题1.下列函数中，在区间(0, +∞)上是减函数的是( ) A. y = x^2 B. y = 1/x C. y =x^3 D. y = 2/x 答案：D2.若f(x) = ∫ (x^2 + 2x - 5) dx，则f'(x) = ( ) A. x^2 + 2x - 5 B. x^2 + 2x - 4 C.x^2 + 2x - 3 D. x^2 + 2x - 6 答案：D3.已知f(x) = sin x + cos x，则f'(x) = ( ) A. -cos x - sin x B. cos x - sin x C. sinx + cos x D. cos x + sin x 答案：B二、填空题4.若f(x) = (x - 1)/(x^2 + 1)，则f'(x) = _______．答案：f'(x) = \frac{x^2 +1}{(x^2 + 1)^2}5.设f(x) = x^3 + 4x^2 + x，则f'(x) = _______．答案：f'(x) = 3x^2 + 8x + 1三、解答题6.求函数f(x) = (sin x + cos x)^5 的导数．答案：f'(x) = (5\sin{x} \cdot (\sin{x}+ \cos{x})^4 \cdot (\cos{x} - \sin{x}) - 5\cos{x} \cdot (\sin{x} + \cos{x})^4 \cdot (\sin{x} - \cos{x})) / (\sin{x} + \cos{x})^2$7.求函数f(x) = x^3 - 3x^2 在区间(-∞, a) 上的最小值．答案：f'(x) = 3x^2- 6x = 3x(x - 2)，令，令f'(x) > 0，解得，解得x < 0或或0 < x < 2，因此，函数，因此，函数f(x)在在( - \infty,0)上单调递增，在上单调递增，在(0,2)上单调递减，在上单调递减，在(2, + \infty)上单调递增，又上单调递增，又f(0) = 0,f( - 1) = 4,f(1) = -2,f(4) = 16，故当，故当a < 0时，函数时，函数f(x)在区间在区间( - \infty,a)上的最小值为上的最小值为0；当；当0 \leqslant a < 1时，函数时，函数f(x)在区间在区间( - \infty,a)上的最小值为上的最小值为f(a)；当；当a > 1时，函数时，函数f(x)在区间在区间( - \infty,a)上的最小值为上的最小值为- 2$．。
AP考试模拟试题与答案1-微积分BC- AP Calculus-BC
1
14
(A) (B) 1 (C)
(D) 4 (E) 5
2
16
15. Which of the following is an equation of the line tangent to the curve with parametric equations x = 3t2 − 2, y = 2t3 + 2 at the point when t = 1?
−
d 3)
x
=
n
5x
(A) lim n→0
−3
(x
+
2)(x
−
dx 3)
−2
5x
(B) lim n→− 3 + n
(x
+
2)(x
−
dx 3)
n
5x
(C) lim n→− 2−
− 3 (x + 2)(x − 3) d x
n
5x
(D) lim n→− 3
ห้องสมุดไป่ตู้
−3
(x
+
2)(x
−
dx 3)
−2
5x
(E) lim n→0 n
III. f < 0 on (0, b)
(A) III only (B) I and II only
(C) II and III only (D) I and III only (E) I, II, and III
y f
a
0
x b
Figure 1T-3
7.
∞
1
=
n = 1 (2n − 1)(2n + 1)
AP考试模拟试题与答案1-微积分BC- AP Calculus-BC
6. The graph of the function f is shown in Figure 1T-3. Which of the following statements is/are true?
I. f (0) = 0
II. f has an absolute maximum value on [a , b]
Use the answer sheet provided on the previous page. All questions are given equal weight. Points are not deducted for incorrect answers and no points are given to unanswered questions. Unless otherwise indicated, the domain of a function f is the set of all real numbers. The use of a calculator is not permitted in this part of the exam.
GO ON TO THE NEXT PAGE
(C) e 2
x
a
0
b
(A)
y
Figure 1T-1 A possible graph of f is (see Figure 1T-2):
(B)
y
(C)
y
a
b
x
a0
b
xa
0
bx
(D)
y
(E)
y
a0
b
x
a
0
bx
Figure 1T-2
AP微积分章节练习
Chapter 2 Limits and ContinuityExercise 11.Evaluate each of the following limits.2.3.4.5.6.7.Chapter 4 Differentiating&Chapter 5 Advanced topics in differentiation 19.puting the derivative of the following functions.Find the first two derivatives of each of the following functionsChapter 6 Applications of the derivativesChapter 7 Integration&Chapter 9 Applications of integrationEvaluating the indefinite integral of the following functions.Chapter 10 Differential Equations1.If the differential equation is given as,rewrite it in the form,where2.Find the integrating factor.3.Evaluate the integral4.Write down the general solution.5.If you are given an IVP, use the initial condition to find the constant C.6.Consider the autonomous equationwith parameter a.(1). Draw the bifurcation diagram for this differential equation.(2).Find the bifurcation values and describe how the behavior of the solutions changes close to each bifurcation value.7. Solve the equationknowing that y1 = 2 is a particular solution.8. Find all the solutions of9. Find all the solutions to10. Consider the autonomous differential equation with the initial condition .(1).Find .(2).Find the first five terms of the Euler Approximation when .(3).Is there a contradiction between the results of 1 and 2 ? If yes, explain what happened.11. Find the approximated sequence , for the IVP.12. Find the solution of13.Let be the solution to the IVPand be the solution to the IVPFind the Wronskian of . Deduce the general solution to 14.Find the general solution to the Legendre equation,using the fact that is a solution.15. Find the solution to the IVP16. Find a particular solution to the equation17. Find the particular solution to。
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29) Find the average value of the function
over the interval [0, 4].
a)
b)
c)
d)
e)
30) What is the y-intercept of the line tangent to the curve y = x2 + 7 at x = 3?
a)
b)
c)
d)
e)
31) Which of the following function(s) is continuous and differentiable?
I.
II.
III.
a) I and III only
b) III only
c) I and II only
d) I only
e) II only
32) Find m
a)
b)
c)
d)
e)
33) The graph of the derivative of f is shown below. Which of the following must be true?
a) f is concave down on [0, 4].
b) f is increasing on [-2, 2].
c) f has a local maximum at x = 0.
d) f has a local minimum at x = -2.
e) f has a point of inflection at x = 4.
34) The sum of two positive integers x and y is 60. Find the value of x that minimizes
a)
b)
c)
d)
e)
35) A particle moves on the curve
find the speed of the particle at time t = .
a) 7.1414
b) 6.7082
c) 7.2801
d) 3.3166
e) 3.0000
36) The function f is defined as
x 6
Which of the following is false?
a) f has a horizontal asymptote at y = 1.
b) f has a vertical asymptote at x = 6.
c) f is decreasing on [3, 6].
d) f has a local maximum at x = 3.
e) f is concave up for x > 6.
37) A particle is moving along the x-axis and its position at time t > 0 is given by
Which of the following is (are) true?
I. The particle changes direction at x = 2 and x = 6.
II. The particle is slowing down on [0, 2].
III. The particle is speeding up on [2, 6].
a) II and III only
b) I only
c) II only
d) I and III only
e) I, II and III
38) f(x) is a differentiable function and it is decreasing on (,). If
then g has a local maximum at
a)
b)
c)
d)
e)
39) The rate at which a bacteria population grows is proportional to the number of bacteria present. Initially, there were 1000 bacteria present and the population doubled in 5 hours. Roughly how many hours does it take for the population to reach 10000?
a)
b)
c)
d)
e)
40) Find the radius of convergence of the series
a)
b)
c)
d)
e)
41)
for 0 < x < 3
The graph of g is shown above. Which of the following must be true? I.
II.
III.
a) II and III only
b) II only
c) I and III only
d) I and II only
e) I only
42) If the region bounded by y= tan-1(x), y= and the y-axis is rotated about the y-axis, the volume of the solid formed is
a) 0.674
b) 0.215
c) 1.348
d) 0.430
e) 0.413
43) f (x) is represented by the Maclaurin series
What is the slope of the line normal to the graph of f at x =?
a)
b)
c)
d)
e)
44) What are all values of h for which
converge?
a) h > 1
b) h < 1
c) h > 1
d) h < 1
e) -1 < h < 1
45) The base of a solid is the region bounded by
the x-axis, and
the line x = 7
Each cross-section of the solid perpendicular to the x-axis is a square, with one side on the xy-plane. Which of the following expressions represents the volume of the solid?
a)
b)
c)
d)
e)
29.e
30.d
31.c
32.b
33.e
34.d
35.c
36.a
37.c
38.a
39.d
40.b
41.d
42.a
43.c
44.c
45.b。