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______。

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)。

2003 AP 微积分BC 真题17题．223,4102/3/24Let x t t t dy dy dt t dx dx dt t =−−+=⇒===−此时发现 导数不存在，即切线为垂直与x 轴的直线。

答案为A2t =82题气球海拔变化速率函数为32()46(08)r t t t t =−+≤≤题目问在海拔减小时 海拔的变化。

海拔减小即：32()460r t t t =−+<此时做出图像（Ti-83）从图像上看到有三个根。

在y=输入另外一条直线y=0点击graph 得接着按2nd 、trace 选择intersec（交点）出下面对话框。

让输入第一条曲线的x值，点击右导航键，移到零点附近，此时可以看到x=1.36……，按enter ，接着让选第二条曲线值，同样按enter。

Guess？（让你猜零点），按enter。

计算器自动算出x=1.36……附近的零点x=1.572。

同样的方法算出右边的零点x=3.514。

然后积分即可。

2008 AP 微积分BC 真题22题[]1111000011001100'()()()()()()()()'()()(1)(1)(0)(0)()'()()'()(1)(1)(0)(0)'()()15f xg x dx g x df x g x f x f x dg x f x g x dx g f g f f x g x dx f x g x dx g f g f f x g x dx ==−=−−⇒=−−∫∫∫∫∫∫∫= 88题()f x 单调递减大于0，所以图像只能在x 轴上方。

根据定积分几何意义求面积，有： 1221()()0f x dx f x dx =−<∫∫，所以答案CDE 都错。

2312()()0f x dx f x dx >∫∫>，所以答案A 对。

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$7.求函数f(x) = x^3 - 3x^2 在区间(-∞, a) 上的最小值．答案：f'(x) = 3x^2- 6x = 3x(x - 2)，令，令f'(x) > 0，解得，解得x < 0或或0 < x < 2，因此，函数，因此，函数f(x)在在( - \infty,0)上单调递增，在上单调递增，在(0,2)上单调递减，在上单调递减，在(2, + \infty)上单调递增，又上单调递增，又f(0) = 0,f( - 1) = 4,f(1) = -2,f(4) = 16，故当，故当a < 0时，函数时，函数f(x)在区间在区间( - \infty,a)上的最小值为上的最小值为0；当；当0 \leqslant a < 1时，函数时，函数f(x)在区间在区间( - \infty,a)上的最小值为上的最小值为f(a)；当；当a > 1时，函数时，函数f(x)在区间在区间( - \infty,a)上的最小值为上的最小值为- 2$．。