AP_微积分BC选择题Section1练习

AP-微积分BC选择题Section1练习1) Finda)b)c)d)e)2) Ifwhich of the following is true about y = f (x)?a) f has a local minimum at x = 5 and a point of inflection at x = 10.b) f has a local minimum at x = 5 and a local maximum at x = 10.c) f has a point of inflection at x = 5 and a local minimum at x = 10.d) f has a point of inflection at x = 5 and a local maximum at x = 10.e) f has a local maximum at x = 5 and a local minimum at x = 10.3) A curve is described by parametric equationswhere t > 0. Give an expression fora)b)c)d)e)4) Give the value fora)b)c)d)e)5) Which of the following series converge?I.II.III.a) I and II onlyb) II onlyc) II and III onlyd) I onlye) I, II and III6) If g(f(x)) = x, g(4) = 2 and g'(4) = 11, then f '(2) isa)b)c)d)e)7) If f is a differentiable function and f(0) = -2 and f(5) = 4, then which of the following must be true?I. There exists a c in [0,5] where f(c) = 0.II. There exists a c in [0,5] where f ' (c) = 0.III. There exists a c in [0,5] where f ' (c) = 6/5.a) II onlyb) I onlyc) I and III onlyd) II and III onlye) I, II and III8) Evaluatea)b)c)d)e)9) Find the area enclosed by the graphs ofand the y-axis.a)b)c)d)e)10) What is the minimum value of the functiona)b)c)d)e)11) Give the value ofa)b)c)d)e)12) The side of a cube is expanding at a constant rate of 5 inches per second. What is the rate of change of the surface area, in in2per second, when the volume of the cube is 64 in3?a)b)c)d)e)13) Give the area inside one petal of the polar graph ofa)b)c)d)e)14) Give the solution to the initial value problema)b)c)d)e)15) The position of a particle moving along a horizontal line is given byWhat is the maximum speed of the particle for 0 < t < 10?a)b)c)d)e)16) =a)b)c)d)e)17) Define the functionfor x > 0. Give the interval on which the function is increasing.a)b)c)d)e)18)Which of the following differential equations correspond to the slope field shown in the figure above?a)b)c)d)e)19) Evaluatea)b)c)d)e)20) If then isa)b)c)d)e)21) Find the area of the region enclosed by the graph ofand the linea)b)c)d)e)22) SupposeandWhat is d z/d t when t = 1?a)b)c)d)e)23) Evaluatea)b)c)d)e)24) Give an equation for the tangent line to the parametric curveat t = 0.a)b)c)d)e)25) Evaluatea)b)c)d)e)26) The region bounded byand the x-axis, for 0 < x < , is rotated about the line y = -3. The volume of this solid can be represented by:a)b)c)d)e)27) Give the third degree Taylor polynomial about x = 1 ofa)b)c)d)e)28) Which of the following integrals gives the length of the graph offor x between 0 and 2?a)b)c)d)e)Key1.e2.d3.d4.b5.e6.d7.c8.a9.c10.a11.d12.b13.d14.a15.c16.c17.b18.a19.b20.b21.c22.a23.a24.a25.e26.a27.b28.b。

ap微积分真题答案【篇一：ap 微积分bc 选择题样卷一】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 fis 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 function-1is invertible. give the slope of the normal line to the graph of f atx = 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)【篇二：ap 微积分bc选择题样卷二】17 questions - 50 minutes1) the limit of the sequenceas n approaches is -3. what is the value of c? a)b)c)d)e)2) ifand y = 3 when x = -2, then what is y? a) b) c)d) e)3) the graph of the derivative of fis given below.which of the following is false about the function f?a) f is increasing on [1,4].b) f is concave down on [1,5/2].c) f is concave down on [-3,0).d) f is not differentiable at 0. e) the funciton is constant on (-,-3].4) determinea)b)c) d) e)5) give the area that lies below the x-axis and is contained within theregion bounded by the polar curvea) b) c) d) e)6) give the error that occurs when the area between the curve and the x-axis over the interval [0,1] is approximated by the trapezoid rule with n = 4. a)b)c)d)e)7) letdetermine f(2/3). a)b)c) d) e)8) give the length of the curve determined byfor t from 0 to 2. a)b)c)d)e)9) particles a and b leave the origin at the same time and move along the y-axis. their positions are determined by the functionsfor t between 0 and 8. what is the velocity of particle b when particlea stops for the first time? a) b)c)d) e)10) the base of a solid is the region in the xy plane enclosed by thecurvesover the interval[0,/4]. cross sections of the solid perpendicular to the x-axis are squares. determine the volume of the solid. a)b)c)d)e)11) give the minimum value of the functionfor x 0. a)b)c)d)e)12) select the true statement associated with the function【篇三：ap微积分考试详解】>微积分ap课程包括微积分ab (calculus ab) 和微积分bc(calculus bc)两门课。

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 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 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 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2012_calculusbc_frqAP? Calculus BC2012 Free-Response QuestionsAbout the College BoardThe College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the College Board was created to expand access to higher education. Today, the membership association is made up o f more than 5,900 of the world’s leading educational institutions and is dedicated to promoting excellence and equity in education. Each year, the College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success —including the SAT? and the Advanced Placement Program?. The organization also serves the education community through research and advocacy on behalf of students, educators, and schools.2012 The College Board. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service and inspiring minds are trademarks owned by the College Board. All other products and services may be trademarks of their respective owners. Visit the College Board on the Web: . Permission to use copyrightedCollege 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: .CALCULUS BCSECTION II, Part ATime—30 minutesNumber of problems—2A graphing calculator is required for these problems.t (minut e s) 0 4 9 15 20W t (degrees Fahrenheit) 55.0 57.1 61.8 67.9 71.01. The temperature of water in a tub at time t is modeled bya strictly increasing, twice-differentiable function W ,where W t is measured in degrees Fahrenheit and t is measured in minutes. At time 0,t the temperature of the water is 55F. The water is heated for 30 minutes, beginning at time 0.t Values of W t at selected times t for the first 20 minutes are given in the table above.(a) Use the data in the table to estimate 12.W Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.(b) Use the data in the table to evaluate200.W t dt ? Using correct units, interpret the meaning of200W t dt ? in the context of this problem.(c) For 020,t the average temperature of the water in the tub is 2001.20W t dt ? Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate 2001.20W t dt ? Does this approximation overestimate or underestimate the average temperature of the water over these20 minutes? Explain your reasoning.(d) For 2025,t the function W that models the water temperature has first derivative given by0.06.W t t Based on the model, what is the temperature of the water at time 25?t2. For 0,t a particle is moving along a curve so that its position at time t is ,.x t y t At time 2,t theparticle is at position 1,5. It is known that dx dt eand 2sin .dy t dt (a) Is the horizontal movement of the particle to the left or to the right at time 2?t Explain your answer.Find the slope of the path of the particle at time 2.t(b) Find the x -coordinate of the particle’s position at time4.t(c) Find the speed of the particle at time 4.t Find the acceleration vector of the particle at time 4.t (d) Find the distance traveled by the particle from time 2t to 4.tEND OF PART A OF SECTION IICALCULUS BCSECTION II, Part BTime—60 minutesNumber of problems—4No calculator is allowed for these problems.3. Let f be the continuous function defined on >@4,3 whose graph, consisting of three line segments and asemicircle centered at the origin, is given above. Let g be the function given by 1.x g x f t dt(a) Find the values of 2g and 2.g(b) For each of 3g and 3,g find the value or state that it does not exist.(c) Find the x -coordinate of each point at which the graph of g has a horizontal tangent line. For each of thesepoints, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers.(d) For 43,x find all values of x for which the graph of g has a point of inflection. Explain yourreasoning.x 1 1.1 1.2 1.3 1.4f x 8 10 12 13 14.54. The function f is twice differentiable for 0x ! with 115f and 120.f Values of ,f the derivative off , are given for selected values of x in the table above.(a) Write an equation for the line tangent to the graph of f at1.x Use this line to approximate 1.4.f (b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table toapproximate1.41.f x dx ? Use the approximation for 1.41f x dx ? to estimate the value of 1.4.f Show the computations that lead to your answer.(c) Use Eul er’s method, starting at 1x with two steps of equal size, to approximate 1.4.f Show thecomputations that lead to your answer.(d) Write the second-degree Taylor polynomial for f about1.x Use the Taylor polynomial to approximate1.4.f5. The rate at which a baby bird gains weight is proportional to the difference between its adult weight and itscurrent weight. At time 0,t when the bird is first weighed, its weight is 20 grams. If B t is the weight of the bird, in grams, at time t days after it is first weighed, then 1100.5dB B dt Let y B t be the solution to the differential equation above with initial condition 020.B(a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain yourreasoning. (b) Find 22d B dt in terms of B . Use 22d B dtto explain why the graph of B cannot resemble the following graph.(c) Use separation of variables to find ,y B t the particular solution to the differential equation with initialcondition 020.B6. The function g has derivatives of all orders, and the Maclaurin series for g is 213501.23357n n n x x x x n"(a) Using the ratio test, determine the interval of convergence of the Maclaurin series for g . (b) The Maclaurin series for g e valuate d at 12x is an alternating series whose terms decrease in absolute value to 0. The approximation for 12g using the first two nonzero terms of this series is 17.120Show that this approximation differs from12g by less than 1.200 (c) Write the first three nonzero terms and the general term of the Maclaurin series for .g xSTOPEND OF EXAM。

2021年AP微积分BC选择题是考察学生对微积分知识掌握程度的重要一环。

这份选择题目包含了各个知识点和难度等级的题目，通过分析和解答这些题目，可以全面了解学生对微积分的掌握情况。

本篇文章将按照以下结构进行分析和解答。

一、导言2021年AP微积分BC选择题的重要性和意义本文将通过分析和解答2021年AP微积分BC选择题，全面了解学生对微积分知识的掌握情况。

微积分作为数学中的重要分支，对学生的逻辑思维能力和数学分析能力有着重要的促进作用。

对AP微积分BC选择题进行分析和解答能够帮助学生巩固知识，提高解题能力。

二、分析题目1. 对2021年AP微积分BC选择题的整体印象2. 难度较大的题目分析3. 难度较小的题目分析2021年AP微积分BC选择题的整体难度如何？哪些题目的难度较大，需要特别注意？哪些题目的难度较小，适合作为基础练习？三、解答题目1. 难题的解答方法和步骤2. 基础题的解答方法和步骤通过解答2021年AP微积分BC选择题，指导学生掌握解题方法，培养解决问题的能力，帮助学生更好地掌握微积分知识。

四、总结对学习微积分的启示通过对2021年AP微积分BC选择题的分析和解答，怎样总结对学生学习微积分的启示，如何更好地提高学生的解题能力。

在完成以上结构后，文章将会全面而严谨地分析和解答2021年AP微积分BC选择题，为学生提供深入的学习参考和指导。

五、分析题目1. 对2021年AP微积分BC选择题的整体印象在整体印象方面，2021年AP微积分BC选择题依然保持着较高的难度和严谨的命题水准。

题目涵盖范围广泛，包括了微分和积分的应用、曲线的性质、微分方程和级数等各个方面的知识点。

整体难度确实较高，对考生的数学基础要求较高。

2. 难度较大的题目分析一些考生普遍反映，2021年AP微积分BC选择题中的一些多项式的级数展开以及微分方程的题目难度较大。

这些题目需要考生对级数收敛性和微分方程的解法有着较深入的理解和掌握。

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微积分试题1. 计算 $\displaystyle \lim _{x\rightarrow 2}\frac{x^{2}-4}{x-2}$解答: 当 $x \neq 2$ 时，可以将分子因式分解为 $x^{2}-4=(x-2)(x+2)$，因此原式化简为 $\displaystyle \lim _{x\rightarrow 2}(x+2)=4$2. 计算 $\displaystyle \lim _{x\rightarrow 0}\frac{\sin x}{x}$解答: 当 $x \neq 0$ 时，使用极限定义可以得到 $\displaystyle \lim _{x\rightarrow 0}\frac{\sin x}{x}=1$3. 计算 $\displaystyle \lim _{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^{x}$解答: 将极限中的式子取自然对数，得到 $\ln \left(\lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}\right)$进一步利用对数的性质有 $\ln \left(\lim _{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^{x}\right)=\lim _{x\rightarrow\infty}x\ln \left(1+\frac{1}{x}\right)$再次利用极限性质有 $\lim _{x\rightarrow \infty}\ln\left(1+\frac{1}{x}\right)^{x}=\lim _{x\rightarrow \infty}x\ln\left(1+\frac{1}{x}\right)=\lim _{x\rightarrow \infty}\frac{\ln\left(1+\frac{1}{x}\right)}{\frac{1}{x}}$利用极限的基本性质即可得到 $\ln \left(\lim _{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^{x}\right)=\ln (e)=1$因为 $\ln (e)=1$，所以 $\displaystyle \lim _{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^{x}=e$这些是一些常见的微积分试题，请根据自己的能力和时间来进行适当的练习和复习。