2006级微积分(上)A理工课程试题及其参考答案

满分网——AP真题 AP® Calculus BC2006 Free-Response QuestionsThe 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.© 2006 The College Board. All rights reserved. College Board, AP Central, APCD, Advanced Placement Program, AP, AP Vertical Teams, Pre-AP, SAT, and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service, CollegeEd, connect to college success, MyRoad, SAT Professional Development, SAT Readiness Program, and Setting the Cornerstones are trademarks owned by the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. 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: .© 2006 The College Board. All rights reserved.CALCULUS BC SECTION II, Part ATime—45 minutes Number of problems—3A graphing calculator is required for some problems or parts of problems.1. Let R be the shaded region bounded by the graph of ln y x = and the line 2,y x =- as shown above.(a) Find the area of R .(b) Find the volume of the solid generated when R is rotated about the horizontal line 3.y =-(c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generatedwhen R is rotated about the y -axis.WRITE ALL WORK IN THE PINK EXAM BOOKLET.© 2006 The College Board. All rights reserved.2. At an intersection in Thomasville, Oregon, cars turn left at the rate ()()23tL t = cars per hour over the time interval 018t ££ hours. The graph of ()y L t = is shown above.(a) To the nearest whole number, find the total number of cars turning left at the intersection over the timeinterval 018t ££ hours.(b) Traffic engineers will consider turn restrictions when ()150L t ≥ cars per hour. Find all values of t forwhich ()150L t ≥ and compute the average value of L over this time interval. Indicate units of measure. (c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of thetotal number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.WRITE ALL WORK IN THE PINK EXAM BOOKLET.© 2006 The College Board. All rights reserved.3. An object moving along a curve in the xy -plane is at position ()()(),x t y t at time t , where()1sin 12t dx e dt --=- and 341dy tdt t =+ for0.t ≥ At time 2,t = the object is at the point ()6,3.- (Note: 1sin arcsin x x -=) (a) Find the acceleration vector and the speed of the object at time 2.t =(b) The curve has a vertical tangent line at one point. At what time t is the object at this point?(c) Let ()m t denote the slope of the line tangent to the curve at the point ()()(),.x t y t Write an expression for()m t in terms of t and use it to evaluate ()lim .t m t Æ•(d) The graph of the curve has a horizontal asymptote .y c = Write, but do not evaluate, an expressioninvolving an improper integral that represents this value c .WRITE ALL WORK IN THE PINK EXAM BOOKLET.END OF PART A OF SECTION II© 2006 The College Board. All rights reserved.CALCULUS BC SECTION II, Part BTime—45 minutes Number of problems—3No calculator is allowed for these problems.t (seconds)0 10 20 30 40 50 60 70 80 ()v t(feet per second)5 14 22 29 35 40 44 47 494. Rocket A has positive velocity ()v t after being launched upward from an initial height of 0 feet at time 0t =seconds. The velocity of the rocket is recorded for selected values of t over the interval 080t ££ seconds, as shown in the table above.(a) Find the average acceleration of rocket A over the time interval 080t ££ seconds. Indicate units ofmeasure. (b) Using correct units, explain the meaning of()7010v t dt Ú in terms of the rocket’s flight. Use a midpointRiemann sum with 3 subintervals of equal length to approximate ()7010.v t dt Ú(c) R ocket B is launched upward with an acceleration of ()a t =0t = seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time 80t = seconds? Explain your answer.WRITE ALL WORK IN THE PINK EXAM BOOKLET.© 2006 The College Board. All rights reserved.5. Consider the differential equation2652dy x dx y =-- for 2.y π Let ()y f x = be the particular solution to thisdifferential equation with the initial condition ()1 4.f -=-(a) Evaluate dy dx and 22d ydxat ()1,4.--(b) Is it possible for the x -axis to be tangent to the graph of f at some point? Explain why or why not. (c) Find the second-degree Taylor polynomial for f about 1.x =-(d) Use Euler’s method, starting at 1x =- with two steps of equal size, to approximate ()0.f Show the workthat leads to your answer.6. The function f is defined by the power series()()231232341nn nx x x x f x n -=-+-++++""for all real numbers x for which the series converges. The function g is defined by the power series()()()23112!4!6!2!nn x x x x g x n -=-+-+++"" for all real numbers x for which the series converges.(a) Find the interval of convergence of the power series for f . Justify your answer.(b) The graph of ()()y f x g x =- passes through the point ()0,1.- Find ()0y ¢ and ()0.y ¢¢ Determinewhether y has a relative minimum, a relative maximum, or neither at 0.x = Give a reason for your answer.WRITE ALL WORK IN THE PINK EXAM BOOKLET.END OF EXAM更多AP资料下载：/thread-9638-1-1.html。

微积分（上）期中测验专业班级 学号 姓名 得分 一、填空：（每格3分，共33分）1. 因为0>∀ε，取==)(εδδ ，使得对于一切满足δ<-0x x 的x ，都有ε<-02cos 2cos x x 成立，故0 2cos 2cos lim 0x x x x =→。

2. 设0→x 时，x x e e sin tan -与n x 为同阶无穷小，则=n 。

3. 211)(+-+=x ex f 在区间______________上是上凹的。

4. 设⎪⎪⎩⎪⎪⎨⎧≤+>-+=0 , 0,1)1()(2x bxa x x x x f 在),(∞+-∞内连续可导，则⎪⎩⎪⎨⎧==。

，____________b a 5. 24)(x e x f x --=在)2,0(内的零点个数为________。

6. 13)(2-++=x x x x f 的斜渐近线为 。

7. 若1)1(-='f ,则=--+→kk f k f k )1()21(lim 0。

8. 若1])sh(4[lim 22=---∞→b xa x x ，则=a ，和=b 。

9. 设)()2)(1()(n x x x x x f +++= , 则=)0()(n f______________。

二、计算与证明：（共78分）1. )1ln()cos 1(2lim22x x xx e x x ---→ （8分）2．xx x x e cot arc 1lim ⎪⎪⎭⎫⎝⎛-+∞→ （8分）3．设()e xxe f xy π++=)(arcsin cot 13, 求y '及dy （8分）4．设⎪⎩⎪⎨⎧-=+=ty tx arctan 1ln 2π, 求22,dx y d dx dy 以及在1=t 处的曲率半径 （8分） 5. 求由方程)ln()(2x y x y y x --=-所确定的函数)(x y y =的y '以及在)2,1(处的切线与法线方程 （8分）。

微积分试题 (A 卷)一. 填空题 (每空2分,共20分)1.已知则对于，总存在δ>0，使得当,)(lim 1A x f x =+→0>∀ε时，恒有│ƒ(x )─A│< ε。

2.已知，则a = ，b =2235lim 2=-++∞→n bn an n 。

3.若当时，α与β 是等价无穷小量，则 。

0x x →=-→ββα0limx x 4.若f (x )在点x = a 处连续，则 。

=→)(lim x f ax 5.的连续区间是 。

)ln(arcsin )(x x f =6.设函数y =ƒ(x )在x 0点可导，则______________。

=-+→hx f h x f h )()3(lim0007.曲线y = x 2＋2x －5上点M 处的切线斜率为6，则点M 的坐标为 。

8. 。

='⎰))((dx x f x d 9.设总收益函数和总成本函数分别为，，则当利润最大时产2224Q Q R -=52+=Q C 量是。

Q 二. 单项选择题 (每小题2分,共18分)1.若数列{x n }在a 的ε 邻域（a -ε,a +ε）内有无穷多个点，则（）。

(A) 数列{x n }必有极限，但不一定等于a (B) 数列{x n }极限存在，且一定等于a(C) 数列{x n }的极限不一定存在 (D) 数列{x n }的极限一定不存在2.设则为函数的（ ）。

11)(-=x arctg x f 1=x )(x f(A) 可去间断点(B) 跳跃间断点 (C) 无穷型间断点(D) 连续点3.（ ）。

=+-∞→13)11(lim x x x(A) 1 (B) ∞(C)(D) 2e 3e4.对需求函数，需求价格弹性。

当价格（ ）时，5p eQ -=5pE d -==p 需求量减少的幅度小于价格提高的幅度。

(A) 3 (B) 5 (C) 6(D) 105.假设在点的某邻域内(可以除外)存)(),(0)(lim ,0)(lim 0x g x f x g x f x x x x ''==→→得0x 0x 在，又a 是常数，则下列结论正确的是（ ）。

2002级 微积分(理工)课程试题答案（A 卷）一、(5分)的敛散性讨论级数∑∞=1n n22n11lim12n n nu u +→∞=< 所以2nn 1n 2∞=∑级数收敛二、（5分）du ,yxzu 22求全微分设+=22d u d x d dz= 三、（5分）求球面x 2+y 2+z 2=9上点(3,3,3)处的切平面方程解：F （x,y,z ）= x2+y 2+z 2 -9 n==(F x F y F z)=()切平面方程()(3)3)0x y z -++= 四、（5分） 设函数u （x,y ），v （x,y ）由下述方程组1022=-=+⎩⎨⎧yv xu yv xu 所确定,求偏导数u x ，v x两端对x 求导解，22''2,2222x x u u v uu v u x vxu y vy+==++五、（5分）设函数22,),,(xz f yx x f z∂∂=求具有连续的二阶偏导数其中2"""1211122222121''z z f f f f f xyxyy∂∂=+=++∂∂六、（5分）21321D37x d x d y 12xxxd x d y --+==⎰⎰⎰⎰计算七、（10分）333Sx d yd z y d zd x z d x d y ,S z ++=⎰⎰计算其中为上半球面取上侧解：补充平面221:0,(1)S z x y =+≤取下侧-11S S S I +=-⎰⎰⎰⎰2223()0xyD x y zd xd yd z d xd y Ω=++-⎰⎰⎰⎰⎰21222063sin 5d d r r d r πππθϕϕ==⎰⎰⎰八、（10分）2Sz d S,S z z 0z 1===⎰⎰计算其中为锥面介于平面和之间的部分解：由圆锥面方程y xz 22+=，得：yxyxyyz xxz 2222,+=∂∂+=∂∂.2122d x d y d x d y dS y z x z =++=⎪⎪⎭⎫⎝⎛∂∂⎪⎭⎫⎝⎛∂∂所以⎰⎰=SzdS 22212312xd y d d r yxyxrπθ+≤+==⎰⎰⎰九、（10分）的势函数是保守场，求验证设向量场→→→+++=f f ),12yy 8x x,8xyy (3xf 2232212f f y x∂∂=∂∂所以是保守场3223(,)44u x y x y x y y C=+++十、（10分）f(x)1,f(1))(0,f(x)xf(x)ydy xdx y 0x 2求数，还满足内有定义且有连续的导在与积分路径无关，其中内曲线积分设在右半平面=+∞+>⎰l因12f f yx∂∂=∂∂ 所以xyf ’(x)+yf(x)=2xy21'()()2,()(2)dx xd x exf x f x xx C f x eed x C x⎰-+=+⎰=+=⎰ f(1)=1 C=0 f(x)=x十一、（10分）n nn 1n 1x1(1).(2).nn 3∞∞==∑∑求幂级数的收敛半径；求级数的和数R=1 s ’(x)=11x- s(x)=-ln(1-x) n n 11 n 3∞=∑12()ln33S ==-十二、（10分）0(0,0)f (0,0)f (0,0)y)f(x,)0,0(),(, 0)0,0(),(,)(y)f(x,y x 22==⎪⎩⎪⎨⎧=≠-+=点不连续，但在证明设y x y x y x x xy22(,)(0,0)(,)(0,0)0lim(,)0lim1x y x y x f x y y x x→→+=→=所以(,)(0,0)lim(,)(0,0)x y f x y f →≠ f(x,y)(0,0)在点不连续 '(,0)(0,0)00(0,0)limlim0,x x x f x f f xx∆→∆→∆--===∆∆即/(0,0)0;x f =同理/(0,0)0.y f =十三、（10分）(最优价格问题)假设某电视机厂的生产处于平衡状态：即生产量等于销售量。