2004-2005中央财经大学微积分期末试卷B

微积分期末考试题2005

0 0 0 0 0 0 1 1 1 1 1 1
4
2005 级第二学期微积分学期中考试试题解答与评分标准
2006-4-22 考试一、1．24
2 y
2． 3 x + 2 y + z − 10 = 0
4 2
3． 2edx + (e + 2)dy
4．
1 6
5． ∫ dy ∫ y f ( x, y )dx + ∫ dy ∫ y f ( x, y )dx
0 2 2 2
2 x0 2 y 0 1 = = ⇒ x0 = 1, y 0 = 1, z 0 = 2 2 2 1
∴所求切平面 π 0 : 2( x − 1) + 2( y − 1) + ( z − 2) = 0 即 2 x + 2 y + z − 6 = 0 （4 分）在 π 0 上取点 P0 (3, 0, 0) ，则 P0 到 π 的距离 d = 五、解法一取球坐标， I = ∫
令 F ( x, y ) =
4 Fx = 13 (2 x + 3 y − 6) + 2λx = 0 6 解方程组 Fy = (2 x + 3 y − 6) + 8λy = 0 13 2 x + 4 y 2 − 4 = 0 8 3 8 3 得 P1 ( , ), P2 (− ,− ) （4 分）于是 d1 5 5 5 5 短距离存在，且为 d1 =
π
π 1 = ⋅ 6 cos 3 ϕ
=
π (2 2 − 1) （5 分） 6
1 1 r 0
解法二取柱坐标， I = ∫
dθ ∫ rdr ∫
1 z 0 0
r 2 + z 2 dz π (2 2 − 1) 6

混合班04-05微积分(下)期末考试试卷解答

3x2
x
∫ = lim
sin 3tdt
0
=
lim
sin 3x
=
lim
3x
=
1
.
x→0 3x2
x→0 6x
x→0 6x 2
∫ 3.
+∞ 0
x4 ex
dx
=
24
.
∫ 解原式= +∞ x4e−x dx = Γ(5) = 4!= 24 . 0
4.
已知
z = xy
,则
∂z ∂y
(e, 1) = e
.
解 z(e , y) = e y , z y (e , 1) = (e y )′ |y=1 = e .
2 cos xdx = 2sin
0
x |02
=
2
.
2
2
x
∫ 2.
lim
x→0
(x − t) sin 3tdt
0
x3
=
1 2
.
x
x
x
∫ ∫ ∫ x sin 3tdt − t sin 3tdt
sin 3tdt + x sin 3x − x sin 3x
解原式= lim 0 x→0
0
x3
= lim 0 x→0
( C ) 动点 (x , y) 沿过定点 (x0 , y0 ) 的任意直线趋于定点 (x0 , y0 ) 时, f (x , y) 的极限均存在且相等
第2页共8页
浙江工商大学章乃器学院 04/05 微积分(下)期末考试试卷解答
( D ) f (x , y) 在点 (x0 , y0 ) 处连续 3. 下列二元函数中,在点 (0, 0) 处取得极小值的是( A ).

《微积分》课程期末考试试卷(B)及参考答案

二.
单项选择题（每题 2 分，共 12 分） 2. A 3. B 4. A 5. C 6C .
1. B 三. 1. 2.
求偏导数（每题 6 分，共 24 分）
z 1 z 1 ; （6 分）； x x y y z x 2z x 2y ln x y (6分) （3 分）； 2 x x y x ( x y) 2 y x2 y2
六、求方程 y
y 1 的通解.（6 分） x
七、判别级数 2 n sin
n 1

33
的收敛性.（6 分）
《微积分》课程期末考试试卷（B）参考答案一. 填空题. （每题 3 分，共 36 分） 1. x y 2 x y 2 2. 0 3. 2 4. 1 5. 1,1,2 6. x, y x y 2 0 7. 1 8. 2 9. e xy y 2 xy dx e xy x x 2 dy 10. 1 11. 发散 12. 10
1 1 ，则 f ( ,0) ______. cos xy 2
3. y '' ( y ' ) 3 2 xy 是______阶微分方程. 4. 方程 F ( x, y, y ' ) 0 的通解中含______个任意常数. 5. 点 (1,1,2) 关于 xoy 平面的对称点是______. 6. 函数 Z lnx y 2 的定义域是______. 7. 设 f ( x, y ) x 2 y 2 ，则 f x1 2,0 ______. 8. 设 f x, y x 2 y 2 ，则 f y1 1,1 ______. 9. 设 Z e xy yx 2 ，则 dz ______. 10. 11. 12. 设积分区域 D : 1 x 2,2 y 3 ，则 d ______.

2003-2004学年第一学期微积分(B)Ⅰ期末考试试卷答案

sin
x
+x x
cos
x
⋅
lim
x→0
sin
x
−x x3
cos
x
=
lim
x→0
cos x
+
cos x 1
−
x sin
x
⋅ lim x→0
cos x
−
cos x 3x 2
+
x sin
x
=
2 ⋅ lim x→0
x sin x 3x2
=
2 lim
3 x→0
sin x
x
=
2 3
∫ 2．求导数 d
0
x2 sin
1
x2dx =
π
．积分
40
12
1
π
0
x2 x2 + 1 2 dx
作变换
x = tan t
，有
π
π
π
π
∫ ∫ ∫ ∫ π
1 0
tan2 t sec2 sec4 t
tdt
=π
4 0
tan2 t sec2 t
dt
=π
4
sin2 tdt
0
=π
4 1 − cos 2t 02
dt
=
π 2
⎜⎛ t ⎝
−
1 sin 2t ⎟⎞ 4
a
dx
=
1
∫
0
x (a
−
x)dx
=
a 2
−
1 3
．
∫ ∫ ∫ ⑶
1
当 0 < a < 1时， x
x − a dx = a x (a − x)dx + 1 x (x − a)dx = 1 − a + a3

AP微积分2005B真题

满分网——AP真题 AP® Calculus BC2005 Free-Response QuestionsForm 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 4,700 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three and a half 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.Copyright © 2005 by 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 Entrance Examination 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 Entrance Examination Board.PSAT/NMSQT is a registered trademark of the College Entrance Examination Board and National Merit Scholarship Corporation. 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 and Pre-AP: .CALCULUS BC SECTION II, Part ATime—45 minutes Number of problems—3A graphing calculator is required for some problems or parts of problems.1. An object moving along a curve in the xy -plane has position ()()(),x t y t at time t 0≥ with2123dx t t dt =- and ()()4ln 14.dy t dt=+- At timet 0,= the object is at position ()13,5.- At time 2,t = the object is at point P with x -coordinate 3.(a) Find the acceleration vector at time 2t = and the speed at time t 2.= (b) Find the y -coordinate of P .(c) Write an equation for the line tangent to the curve at P .(d) For what value of t , if any, is the object at rest? Explain your reasoning.WRITE ALL WORK IN THE TEST BOOKLET.2. A water tank at Camp Newton holds 1200 gallons of water at time 0.t = During the time interval 018t ££hours, water is pumped into the tank at the rate()(26tW t = gallons per hour. During the same time interval, water is removed from the tank at the rate()(2275sin 3t R t = gallons per hour.(a) Is the amount of water in the tank increasing at time 15?t = Why or why not? (b) To the nearest whole number, how many gallons of water are in the tank at time 18?t =(c) At what time t , for t 018,££ is the amount of water in the tank at an absolute minimum? Show the workthat leads to your conclusion. (d) For 18,t > no water is pumped into the tank, but water continues to be removed at the rate ()R t until thetank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k .WRITE ALL WORK IN THE TEST BOOKLET.3. The Taylor series about 0x = for a certain function f converges to ()f x for all x in the interval ofconvergence. The n th derivative of f at 0x = is given by()()()()()1211051n n nn f n +-+!=- for 2.n ≥ The graph of f has a horizontal tangent line at 0,x = and ()0 6.f =(a) Determine whether f has a relative maximum, a relative minimum, or neither at 0.x = Justify your answer. (b) Write the third-degree Taylor polynomial for f about 0.x =(c) Find the radius of convergence of the Taylor series for f about 0.x = Show the work that leads to youranswer.WRITE ALL WORK IN THE TEST BOOKLET.END OF PART A OF SECTION IICALCULUS BC SECTION II, Part BTime—45 minutes Number of problems—3No calculator is allowed for these problems.4. The graph of the function f above consists of three line segments.(a) Let g be the function given by ()()4.xg x f t dt -=Ú For each of ()1,g - ()1,g -¢ and ()1,g -¢¢ find the valueor state that it does not exist.(b) For the function g defined in part (a), find the x -coordinate of each point of inflection of the graph of g onthe open interval 4 3.x -<< Explain your reasoning. (c) Let h be the function given by ()()3.x h x f t dt =Ú Find all values of x in the closed interval 43x -££ forwhich ()0.h x =(d) For the function h defined in part (c), find all intervals on which h is decreasing. Explain your reasoning.WRITE ALL WORK IN THE TEST BOOKLET.5. Consider the curve given by 22.y xy =+(a) Show that .2dy ydx y x=-(b) Find all points (),x y on the curve where the line tangent to the curve has slope 1.2(c) Show that there are no points (),x y on the curve where the line tangent to the curve is horizontal. (d) Let x and y be functions of time t that are related by the equation 22.y xy =+ At time 5,t = the valueof y is 3 and 6.dy dt = Find the value of dxdt at time 5.t =6. Consider the graph of the function f given by ()12f x x =+ for 0,x ≥ as shown in the figure above. Let R be the region bounded by the graph of f , the x - and y -axes, and the vertical line ,x k = where 0.k ≥(a) Find the area of R in terms of k .(b) Find the volume of the solid generated when R is revolved about the x -axis in terms of k .(c) Let S be the unbounded region in the first quadrant to the right of the vertical line x k = and below thegraph of f , as shown in the figure above. Find all values of k such that the volume of the solid generated when S is revolved about the x -axis is equal to the volume of the solid found in part (b).WRITE ALL WORK IN THE TEST BOOKLET.END OF EXAM。

2004-2005(2)期末考试试卷(B)参考答案(高等数学)

5
1
sin
y(1
y)dy
0
8
1 sin1
10
1
5.原式 2 d 2 er2 rdr
0
0
(e4 1)
5 10
6.
P ye
x ， Q 2x cos y 2 ，
Q x
P y
211
2
由格林公式
I d x d y
D
=
6 10
ydy
7.
1
y2
dx x
两边积分得
ln(1 2
y2)
ln
x
高等数学期末试卷（B）参考答案
05.6 一、填空题(每小题 2 分, 共 12 分)
1. 5
2. 4x y 2
3. a 2 。
4、 9 2 。
5、
x3 6
C1x C2 .
6、 y C1 y1 C2 y2 .
二、解答下列各题(每小题 8 分,，总计 72 分)
1. 所求平面垂直于平面 3x 2 y z 4 0 和 2x 3y 4z 5 0 ，
zx
e z e y cos x 2z xe z
，
4
同理 z y
e y sin x 2z xe z
7dzຫໍສະໝຸດ z x dxz y dy
e z e y cos x 2z xe z
dx +
e y sin x 2z xe z
dy
10
4.
sin y dxdy
1
dy
y sin y dx
Dy
0
y y 2
8
x
又 y x
1，所求特解为 y
1 cos x x

微积分下期末考试

z (0,0) = 0,
z (3,0) = −9,
z (0,2) = −4,
z (0,3) = −3,
z (2,1) = −11.
所以 z min = z ( 2,1) = −11, z max = z (0,0) = 0. 评分：解 2：求 D 内嫌疑点 3 分，求 y=0，x=0，x+y=3 上嫌疑点分别为 1、1、3 分. 结论 1 分. 13．计算
π /4
4 ⎡15a 4 ⎤ ⎡ 1 1 π 1 ⎤ 15πa = 2π ⎢ (2 + π ) ⎥ ⎢ + ( − ) − (0 + 0)⎥ = 16 ⎦ ⎣ 4 ⎦⎣ 2 2 4 2
评分:得第一行 3 分, 得第二行再加 4 分(每对积分限各 1 分,被积函数 1 分),最后结果再加 2分 14．求幂级数
∫∫ zxdydz + yzdzdx + z
Σ
x 2 + y 2 dxdy ，
其中
Σ是由 a 2 ≤ x 2 + y 2 + z 2 ≤ 4a 2 , z ≥ x 2 + y 2 所确定的立体的表面的外侧(a>0).
解:原式=
∫∫∫ ( z + z +
Ω
x 2 + y 2 )dxdydz
2a
(用球坐标)
评分：对应以上各等号分别给到 1、2、4、5 分。 6．交换二次积分 I = dx
0
∫
2
4− x 2
2 x− x2
∫ f ( x, y)dy 的积分次序.
1 4− y 2 2 4− y 2
解： I = dy
0
∫
1
1− 1− y 2

04-05高等数学试卷B答案

04-05高等数学试卷B答案高等数学试卷（B 卷）第 2 页共 14 页广州大学2004-2005学年第二学期考试卷答案与评分标准课程：高等数学（90学时）考试形式：闭卷考试题号一二三四五六七总分分数 15 15 20 20 15 7 8 100 评分评卷人一．填空题（本题共5小题，每小题3分，满分15分）1．设y x xy z +=，则=dz dy yx x dx y y ⎪⎪⎭⎫ ⎝⎛-+⎪⎪⎭⎫ ⎝⎛+21 2．设),(v u f z =具有一阶连续偏导数，y x u +=2,┋┋┋┋┋装 ┋┋┋┋┋┋┋订┋┋┋┋┋┋┋┋线┋┋┋┋┋┋┋┋┋装┋┋┋┋┋┋┋┋┋订┋┋┋┋┋┋线┋┋┋┋┋┋┋学院领导审批并签名B 卷高等数学试卷（B 卷）第 3 页共 14 页xyv = , 则=∂∂xzvuf y f+23．L 为圆周122=+y x，则2Lx ds =⎰π4．若级数∑∞=1n nu 收敛，则=∞→nn ulim 05．微分方程02=-ydx xdy 的通解是2y c x =二．单项选择题（本题共5小题，每小题3分，满分15分）1．函数),(y x f z =在点),(y x 处可微是),(y x f 在该点偏导数x z∂∂及y z ∂∂存在的【 A 】（A ）充分非必要条件（B ）必要非充分条件（C ）充分必要条件（D ）无关条件.2.曲线2t x =，12+=t y ，3t z =在点)1,1,1(--处的法平面方程为【 B 】（A ）3322-=++z y x （B ）7322=--z y x高等数学试卷（B卷）第 4 页共 14 页（C）当10≤<p时，级数∑∞=--11)1(npnn绝对收敛（D）当10≤<p时，级数∑∞=--11)1(npnn条件收敛高等数学试卷（B卷）第 5 页共 14 页高等数学试卷（B 卷）第 6 页共 14 页三．解答下列各题（本题共3小题，第1、2小题6分，第3小题8分，满分20分） 1．求函数2221)ln(y x x y z --+-= 的定义域，并画出其区域图解：要使函数有意义，须满足⎪⎩⎪⎨⎧≥-->-010222y x x y 即⎪⎩⎪⎨⎧≤+>1222y x x y所求定义域为}1|),{(222≤+>=y x x y y x D 且 ┉┉┉┉┉ 3分区域D 的图形如左图阴影部分┉┉┉┉┉┉┉┉┉ 6分2．函数),(y x z z =是由方程0=+-xy yz e z确定，求xz ∂∂及22x z ∂∂ 解：令=),,(z y x F xyyz ez+- 则 yFx=， ye Fz z-=┋┋┋┋┋ 装┋┋┋┋┋┋┋订┋┋┋┋┋┋┋┋线┋┋┋┋┋┋┋┋┋装┋┋┋┋┋┋┋┋┋订┋┋┋┋┋┋线┋┋┋┋┋┋┋高等数学试卷（B 卷）第 7 页共 14 页zyx e y yFF x z-=-=∂∂ ┉┉┉┉┉┉┉┉┉┉┉┉ 3分22x z ∂∂2)(z z e y x z e y -⎪⎭⎫ ⎝⎛∂∂--= ┉┉┉┉┉┉┉┉┉┉┉┉┉ 5分 32)(z z e y e y -= ┉┉┉┉┉┉┉┉┉┉┉┉┉┉ 6分3．求表面积为36而体积最大的长方体解：设长方体的三棱长为z y x ,,，则体积xyz V =，且 18=++xz yz xy令)18(),,(-+++=xz yz xy xyz z y x L λ ┉┉┉┉┉┉┉┉┉ 3分由⎪⎪⎩⎪⎪⎨⎧=++=++==++==++=180)(0)(0)(xz yz xy y x xy L z x xz L z y yz L z y x λλλ ┉┉┉┉┉┉┉┉┉┉┉┉┉┉ 5分得6===z y x ┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉ 7分由实际问题可知，当棱长为6的正方体时体积最大 ┉┉┉┉ 8分高等数学试卷（B 卷）第 8 页共 14 页四．计算下列积分（本题共3小题，第1、2小题6分，第3小题8分，满分20分）1．计算dxdy y x D⎰⎰，其中D 由直线x y =，1=y 及0=x 围成的闭区域解：dxdy y x D⎰⎰⎰⎰=101xdyxy dx ┉┉┉┉┉┉┉┉┉┉┉┉┉ 3分dx y x x ⎰=1012|21 ┉┉┉┉┉┉┉┉┉┉┉┉┉┉ 4分dx x x ⎰-=13)(21 ┉┉┉┉┉┉┉┉┉┉┉┉┉ 5分81= ┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉ 6分2．计算⎰⎰⎰Ωdz dy dx z ，其中Ω是由平面1=++z y x 及三个坐标面所围成的闭区域高等数学试卷（B 卷）第 9 页共 14 页解：⎰⎰⎰Ωdz dy dx z ⎰⎰⎰---=y x x dzz dy dx 10101┉┉┉┉┉┉┉┉ 3分 dy y x dx x ⎰⎰---=10102)1(21 ┉┉┉┉┉┉┉ 4分⎰--=103)1(61dx x ┉┉┉┉┉┉┉┉┉┉┉ 5分=241┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉ 6分3．利用格林公式计算22()()yxLI xy e dy x y e dx =+-+⎰，其中L 为圆周422=+y x ，取逆时针方向解：记4:22≤+yx D ，由格林公式⎰⎰+=Ddydx y x I )(22 ┉┉┉┉┉┉┉┉┉┉┉ 3分 ⎰⎰⋅=πρρρθ20202d d ┉┉┉┉┉┉┉┉┉┉┉6分高等数学试卷（B 卷）第 10 页共 14 页420|2πρ=┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉ 7分π8= ┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉┉ 8分五．解答下列级数（本题共3小题，第1小题5分，第2小题10分，满分15分） 1．判别级数∑∞=123n nn 的敛散性解：nn n nn n n n uu 33)1(lim lim 2)1(21+∞→+∞→+= ┅┅┅┅┅┅┅┅┅┅ 2分211lim 31⎪⎭⎫⎝⎛+=∞→nn131<=┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅ 4分该级数收敛 ┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅ 5分2．求幂级数∑∞=+1)1(n nx n n 的收敛域及其和函数解：nn n aa 1lim +∞→=ρ)1()2)(1(lim+++=∞→n n n n n ⎪⎭⎫⎝⎛+=∞→n n 21lim 1= ┅┅ 2分故11==ρR ┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅ 3分当1-=x 时，级数∑∞=+-1)1()1(n nn n 发散 ┅┅┅┅┅┅┅┅┅ 4分当1=x 时，级数∑∞=+1)1(n n n 发散 ┅┅┅┅┅┅┋┋┋┋┋ 装┋┋┋┋┋┋┋订┋┋┋┋┋┋┋┋线┋┋┋┋┋┋┋┋┋装┋┋┋┋┋┋┋┋┋订┋┋┋┋┋┋线┋┋┋┋┋┋┋┅┅┅┅┅┅ 5分幂级数的收敛域为)1,1(- ┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅ 6分记=)(x S ∑∞=+1)1(n nx n n 11<<-x=⎰x dx x S 0)(∑∞=+11n n nx2x=∑∞=-11n n nx又设=)(x g ∑∞=-11n n nx ，11<<-x ，=⎰xdx x g 0)(∑∞=1n nx=xx-1 ┅┅ 8分知2)1(11)(x x x x g -='⎪⎭⎫⎝⎛-=()3222)1(2)1()()(x xx x x g x x S -='⎪⎪⎭⎫ ⎝⎛-='= （11<<-x ）┉┉┅┅ 10分六．（本题满分7分）设有连结点(0,0)O 和点(1,1)A 的一段向上凸的曲线弧OA ，对于OA 上任一点(,)P x y ，曲线弧OP 与直线段OP 所围成的图形的面积为2x ，求曲线弧OA 的方程解：设曲线弧OA 的方程为()y y x =，依题意21()2xy t dt xy x -=⎰ ┅┅┅┅┅┅┅┅┅┅┅┅ 2分两边关于x 求导，得1()()22y x y xy x '-+= 即14y y x '-=- ┅┅┅┅┅┅┅┅┅┅ 3分该方程为一阶线性微分方程，由常数变易公式得(4)dxdx xxy e e dx C -⎡⎤⎰⎰=-+⎢⎥⎣⎦⎰┅┅┅┅┅┅┅┅┅┅┅ 4分14x dx C x ⎡⎤=-+⎢⎥⎣⎦⎰(4ln )x x C =-+ ┅┅┅┅┅┅┅┅┅┅┅┅┅ 6分由1|1x y ==得，1C =所求方程为4ln y x x x =-+┅┅┅┅┅┅┅┅┅┅┅┅ 7分七．（本题满分8分）求微分方程2xy y y xe '''--=的通解解：该方程为二阶常系数非齐次线性微分方程，且()f x 为()xmP x e λ型（其中()mP x x =，1λ=）与所给方程对应的齐次方程为20y y y '''--= 它的特征方程 220r r --=┅┅┅┅┅┅┅┅┅┅┅┅ 2分特征根11r =-，22r =齐次方程的通解为212xxY C e C e -=+┅┅┅┅┅┅┅┅┅ 4分由于1λ=不是特征根，设()xy ax b e *=+ ┅┅┅┅┅┅ 5分代入原方程得 22ax a b x -+-=由比较系数法得2120a ab -=⎧⎨-=⎩，解得11,24a b =-=-， 1(21)4xy x e *=-+，┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅ 7分所求通解为2121(21)4xx xy C eC e x e -=+-+┅┅┅┅┅┅8分。

05年经济数学B(微积分)修改后考题学会(1)

经济数学基础(一) 微积分统考试题(B)(120分钟)一、填空题（20102=⨯分）1、设()⎩⎨⎧≥-<=0202x x x x x f ，则()[]=-1f f 。

2、()=--+∞→x x xx 2lim。

3、为使()xx x x f 111⎪⎭⎫⎝⎛-+=在0=x 处连续，需补充定义()=0f 。

4、若()()x f x f =-，且()21'=-f ，则()=1'f 。

5、==)0(,2sin )5(y x y 则设。

6、设)(x y y =由y y x =所确定，则=dy 。

7、设某商品的需求函数为p Q 2.010-=，则需求弹性分析()=10E 。

8、设()⎩⎨⎧>+≤=01x ax x e x f x，且()x f 在0=x 处可导，则=a 。

9、=⎥⎦⎤⎢⎣⎡=ξππ上满足罗尔定理的在65,6sin ln )(x x f 。

10、='---=)0(),99()2)(1()(f x x x x x f 则设。

二、单项选择（1052=⨯分）1、若0→x 时，k x x x ~2sin sin 2-，则=k （）。

A 、1 B 、2 C 、3 D 、42、若(),20'-=x f 则()()=--→0002limx f x x f xx （）。

A 、41 B 、41- C 、1 D 、1- 3、设 ⎪⎩⎪⎨⎧=≠=001)(x x xxarctgx f 则)(x f 在0=x 处（）。

A 、不连续B 、极限不存在C 、连续且可导D 、连续但不可导 4、12-=x x y 有（）条渐近线。

A 、 1B 、 2C 、 3D 、 45、下列函数中，（）不能用洛必达法则。

A 、x x xx x sin sin lim +-∞→ B 、()x x x 101lim +→ C 、x x x cos 1lim0-→ D 、⎪⎭⎫ ⎝⎛--→111lim 0x x e x 三、计算题（一）（1535=⨯分）1、求()xx x 3sin 21ln lim0-→。

2004～2005 学年第一学期《高等数学》期末考试试题B卷及答案

x ♦ 2004～2005 学年第一学期《高等数学》期末考试试题 B 卷（216 学时）专业班级学号姓名一、填空题：（4×4 分）1、设 f (x + 1 ) = x 2+1-1，则 f (x ) =。

2、lim xx 2sin(1 - x ) = 。

x →1(x -1)(x + 2)3、设 f '(x ) = -2 ，则limf (x 0 - h ) - f (x 0 + h )= 。

h →0h4、 ⎰[ f (x ) + xf '(x )]dx = 。

dx 2sin t5、dx ⎰01 + cos2 t dt = 。

二、选择题：（5×3 分）1、 x = 2 是函数 f (x ) = arctan12 - x的（）A 、连续点；B 、可去间断点；C 、第一类不可去间断点；D 、第二类间断点；♣1 - cos x 2、设 f (x ) = ♠, x > 0 ，其中 g (x ) 是有介函数，则 f (x ) 在 x = 0 处（）♠♥x 2 g (x ) , x ≤ 0 A 、极限不存在； B 、极限存在，但不连续； C 、连续，但不可导； D 、可导；3、在区间(a ,b ) 内，f (x ) 的一阶导数 f '(x ) >0，二阶导数 f '(x ) <0，则 f (x ) 在区间(a ,b )内是（）A 、单增且凸； C 、单增且凹；4、下列命题中正确的是B 、单减且凸； D 、单减且凹；（）A 、 f ''(x 0 ) = 0 ，则(x 0 , f (x 0 )) 一定是由曲线 y = f (x ) 的拐点；B 、若 f '(x 0 ) = 0 ，则 f (x ) 在 x 0 处一定取极值；C 、 f (x ) 可导，且在 x = x 0 上取得极值，则 f '(x 0 ) = 0 ；D 、 f (x ) 在[a , b ] 上取得最大值，则该最大值一定是 f (x ) 在(a , b ) 内的极大值。

2005VB期末考试A与答案

2004~2005学年第二学期VB期末考试（A卷）（2004级全校）班级学号姓名一、选择题(用铅笔把选中的方格涂黑40×1=40分)1 2 3 45 6 7 89 11131721252933371、下列控件中，控件不论其属性如何改变，它在程序运行时永远是不可见的。

A. LabelB. ImageC. CommandD. Timer2、向一个控件数组中动态地添加数组元素，应该使用。

A.Insert语句B.Load语句C.Append语句D.AddItem语句3、下面的程序段中，Print语句执行的次数为____ _____。

Dim B(-5 To 5) As Integer, x As VariantFor Each x In Bx = Int(10 * Rnd)Print xNext xA. 10B. 11C.9D.124、应用程序窗体Name属性是Frm1，窗体上有一个命令按钮，其Name属性为Cmd1，窗体和命令按钮的Click事件过程名分别是___ _。

A. Form_Click Cmd1_ClickB. Form1_Click Cmd1_ClickC. Frm1_Click Command1_ClickD. Frm1_Click Cmd1_Click5、有变量定义语句 Dim a,b As Integer ，变量 a 的类型和初值是。

(A) Integer ，0 (B) Variant ，空值 (C) String ，"" (D) Long ，0.06、在VB 中，要使一个窗体不可见，但不从内存中释放，应使用的语句是______。

A ．End B. Load C. Hide D. Unload7、VB 有三种工作状态，其中不包含__ _。

A. 调试态B. 设计态C. 运行态D. 中断态8、数学算式x e m d y x x sin 51223⋅--+-的VB 算术表达式是________。

微积分二同济期末考试B答案

对外经济贸易大学 2004─2005学年第二学期《微积分（二）》期末考试试卷（B 卷）课程代码及课序号：CMP101-1-14学号：姓名：成绩：班级：课序号：任课教师：一、选择题：（每题2分，共10分）得分 1．设级数1nn a ∞=∑绝对收敛，则11(1)n n n a n ∞=+∑（）. A ．发散 B ．条件收敛 C ．敛散性不能判定 D ．绝对收敛答案：D 答案：C 答案：B 答案：B5．设)(1x y 是方程)()(x Q y x P y =+'的一个特解，c 是任意常数，则该方程的通解是（）．A.()1P x dt y y e -⎰=+B.()1P x dxy y ce -⎰=+ C.()1P x dxy y e c -⎰=++D. ()1P x dx y y ce ⎰=+ 答案：B二、填空题：（每空2分，共10分）得分 1．部分和数列{}n s 有界是正项级数∑∞=1n nu收敛的条件。

答案：充要2． ._________)2sin 1(1lim 010=+⎰→x t x dt t x答案： 2e3．函数z =的定义域是 .答案：2{(,)|0,0}x y x y x y ≥≥≥， 4．),(y x f Z =的偏导数x z ∂∂及yz ∂∂在点),(y x 存在且连续是),(y x f 在该点可微分的条件。

答案：充分5．微分方程20y y '''-=的通解为。

答案：212xc c e +⋅三、判断题：（每题2分，共10分,正确的划√，错误的划╳）得分00005.(,)(,)(,)(,).().z f x y x y z f x y x y ==若在点偏导数存在，则在点一定连续答案：1 . √ 2. × 3. √ 4. × 5. ×四、计算题（每题6分，共48分）得分1．设(,),w f x y z x yz =++2,.w w f x x z∂∂∂∂∂具有二阶连续导数，求解：,,(,)u x y z v x yz w f u v =++==令则1211(,)wf f yz f x y z x yz x∂'''=⋅+⋅=++∂2(,)yz f x y z x yz '+++ -------2分 21112222()f y x z f x y z f y f '''''''=++++ -------4分 2.方程(,)0(,),z zF x y z f x y F y x++==确定了函数其中为可微函数，求z z x y ∂∂∂∂，解：122F zF F x x∂=-∂， ------- 1分 122F zF F y y ∂=-+∂， ------- 1分 1211F F F z y x∂=+∂ ------- 2分所以2212121212,()()x yF yzF xzF xy F z z x x xF yF y y xF yF -+-∂∂==∂+∂+， ------- 2分 3．计算二重积分d ,Dx y σ⎰⎰其中D 为抛物线2y x =及直线2y x =-所围成的闭区域。

2004～2005学年第一学期《高等数学》期末考试试题B卷(.

2004~2005学年第一学期《高等数学》期末考试试题B 卷答案一、填空题(4×4分1、32−x ; 2、31−; 3、4; 4、(xf x c +; 5、2222sin 1cos x x x + 二、单项选择题(5×3分1、C;2、D;3、A;4、C;5、B三、试解下列各题解:1、0000→→→→x 2、66sin 31ln(2lim sin 20lim 31(lim 00e e e x x x x x x x x x ===+→→+→ 3、xdx dx x x x x x erc dy arctan 11tan 22=⎦⎤⎢⎣⎡+−++= 4、两边对x 求导(10x y dy dy e y x dx dx++−−= x y x y dy e y dx x e ++−=−5、22sin dx t t dt =−222222(cos 2sin cos 2sin dy t t t t dt t tdt =−−=−2222sin 2sin dy t t t dx t t == dy d dt dx = 22212sin d y dx t t =− 6、2c ==+ 7、22204 4044sin sin sin 111x x xx x x dx dx dx e e e ππππ−−−−−−=++++∫∫∫ 220404sin sin 11x t x t dxx t dt e e ππ−−=−++∫∫ 22444004sin 1sin (1cos 221xx dx xdx x dx e ππππ−−==−+∫∫∫ 40111(sin 2(2228x x ππ=−=− 8、2201arctan(1arctan (1td t ′∫+− ∫+−−−=2122121(arctan 1(21dt tt t t 125/2arctan −+=u四、解:例如广义积分∫10d 1x x 收敛时,但广义积分∫10d 1x x 发散。

中央财经大学2007-2008学年微积分(上 A类)试卷

中央财经大学2007-2008学年第一学期期末试卷姓名专业学号一、单项选择题（每小题3分，共18分）1. 设函数()2; 1;1x x f x ax b x ⎧≤=⎨+>⎩在1x =处可导，则（ b ）A. 0,1a b ==B. 2,1a b ==-C. 3,2a b ==-D.1,2a b =-=2. 已知()f x 在0x =的某邻域内连续，且()()000,lim 21cos x f x f x→==-，则在0x =处()f x 满足（ b ）A. 不可导B. 可导C. 取极大值D. 取极小值 3. 若广义积分()2ln kdx x x +∞⎰收敛，则（ b ）A. 1k >B. 1k ≥C. 1k <D. 1k ≤4. )(lim 111=+-→x x e cA ． 0 B.∞ C.不存在 D.以上都不对 5. 当0→x 时，x cos 1-是关于2x 的（）.aA ．同阶无穷小.B ．低阶无穷小.C ．高阶无穷小.D ．等价无穷小. 6.函数)(x f 具有下列特征：0)0(,1)0(='=f f ，当0≠x 时，⎩⎨⎧>><<''>'0,00,0)(,0)(x x x f x f则)(x f 的图形为（ b ）。

(A)(B) (C) (D)二、填空（每小题3分，共18分）1.sin limx x x→∞= 0 。

2. 1-=⎰0 。

3. 已知0()f x '存在，则000()()limh f x h f x h h→+--= 2f ’(x.) 。

4．设ln(1)y x =+，那么()()n y x = 。

5．220txd e dt dx=⎰。

6．某商品的需求函数275Q P =-，则在P ＝4时，需求价格弹性为4P η== ，收入对价格的弹性是4P ER EP== 。

三、计算（前四小题每题5分，后四小题每题6共44分） 1．arctan limx x tdt →∞2． xx x x 21lim ⎪⎭⎫⎝⎛+∞→3．1ln ex xdx ⎰4．6(1)dx x x +⎰5．求由 0 0cos 0yx t e dt tdt +=⎰⎰所决定的隐函数()x y y =的导数.dxdy6．已知sin x x是()f x 的原函数，求()xf x dx '⎰。

已解锁-AP微积分2005B真题

满分网——AP真题 AP® Calculus BC2005 Free-Response QuestionsForm 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 4,700 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three and a half 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.Copyright © 2005 by 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 Entrance Examination 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 Entrance Examination Board.PSAT/NMSQT is a registered trademark of the College Entrance Examination Board and National Merit Scholarship Corporation. 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 and Pre-AP: .CALCULUS BC SECTION II, Part ATime—45 minutes Number of problems—3A graphing calculator is required for some problems or parts of problems.1. An object moving along a curve in the xy -plane has position ()()(),x t y t at time t 0≥ with2123dx t t dt =- and ()()4ln 14.dy t dt=+- At timet 0,= the object is at position ()13,5.- At time 2,t = the object is at point P with x -coordinate 3.(a) Find the acceleration vector at time 2t = and the speed at time t 2.= (b) Find the y -coordinate of P .(c) Write an equation for the line tangent to the curve at P .(d) For what value of t , if any, is the object at rest? Explain your reasoning.WRITE ALL WORK IN THE TEST BOOKLET.2. A water tank at Camp Newton holds 1200 gallons of water at time 0.t = During the time interval 018t ££hours, water is pumped into the tank at the rate()(26tW t = gallons per hour. During the same time interval, water is removed from the tank at the rate()(2275sin 3t R t = gallons per hour.(a) Is the amount of water in the tank increasing at time 15?t = Why or why not? (b) To the nearest whole number, how many gallons of water are in the tank at time 18?t =(c) At what time t , for t 018,££ is the amount of water in the tank at an absolute minimum? Show the workthat leads to your conclusion. (d) For 18,t > no water is pumped into the tank, but water continues to be removed at the rate ()R t until thetank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k .WRITE ALL WORK IN THE TEST BOOKLET.3. The Taylor series about 0x = for a certain function f converges to ()f x for all x in the interval ofconvergence. The n th derivative of f at 0x = is given by()()()()()1211051n n nn f n +-+!=- for 2.n ≥ The graph of f has a horizontal tangent line at 0,x = and ()0 6.f =(a) Determine whether f has a relative maximum, a relative minimum, or neither at 0.x = Justify your answer. (b) Write the third-degree Taylor polynomial for f about 0.x =(c) Find the radius of convergence of the Taylor series for f about 0.x = Show the work that leads to youranswer.WRITE ALL WORK IN THE TEST BOOKLET.END OF PART A OF SECTION IICALCULUS BC SECTION II, Part BTime—45 minutes Number of problems—3No calculator is allowed for these problems.4. The graph of the function f above consists of three line segments.(a) Let g be the function given by ()()4.xg x f t dt -=Ú For each of ()1,g - ()1,g -¢ and ()1,g -¢¢ find the valueor state that it does not exist.(b) For the function g defined in part (a), find the x -coordinate of each point of inflection of the graph of g onthe open interval 4 3.x -<< Explain your reasoning. (c) Let h be the function given by ()()3.x h x f t dt =Ú Find all values of x in the closed interval 43x -££ forwhich ()0.h x =(d) For the function h defined in part (c), find all intervals on which h is decreasing. Explain your reasoning.WRITE ALL WORK IN THE TEST BOOKLET.5. Consider the curve given by 22.y xy =+(a) Show that .2dy ydx y x=-(b) Find all points (),x y on the curve where the line tangent to the curve has slope 1.2(c) Show that there are no points (),x y on the curve where the line tangent to the curve is horizontal. (d) Let x and y be functions of time t that are related by the equation 22.y xy =+ At time 5,t = the valueof y is 3 and 6.dy dt = Find the value of dxdt at time 5.t =6. Consider the graph of the function f given by ()12f x x =+ for 0,x ≥ as shown in the figure above. Let R be the region bounded by the graph of f , the x - and y -axes, and the vertical line ,x k = where 0.k ≥(a) Find the area of R in terms of k .(b) Find the volume of the solid generated when R is revolved about the x -axis in terms of k .(c) Let S be the unbounded region in the first quadrant to the right of the vertical line x k = and below thegraph of f , as shown in the figure above. Find all values of k such that the volume of the solid generated when S is revolved about the x -axis is equal to the volume of the solid found in part (b).WRITE ALL WORK IN THE TEST BOOKLET.END OF EXAM。