習(xí)題試題庫(kù)高數(shù)注釋做模板之前注意查看

//說(shuō)明10：盡量避免直接使用{}[]<>，如果要用的話最好寫(xiě)到編輯器里面zx3yx2y2xy3的偏導(dǎo)數(shù)zzx 3 x3xy2xyyyx2xyzxy2y3dzy2dx2xy3y2dyx求函數(shù)zxy 的全微分dzxydzy

dx(xx zzx5xyy的一階偏導(dǎo)數(shù)xy x2x5yy5x2zx23xyy2的一階偏導(dǎo)數(shù)zzxz2x3yz3x2 zxy2z

2

2,y2

2.2

22

y

y(y

,y2

(lnx)

xy

(1yln zz

sin(xy，求xy

x

[ysin(xy)cos(xy)],y

[xsin(xy)cos(xzyx

2.

2

(1xlnzylnx

2.2z

ln

x

(1lnxln2已知zxln(xy), 2

x2zz(xyx2y2z2yez確定，求dz

dz

dx

2 yez yez

2xy

4z0,

z

2

2,

(2z)2(2xlnzzz xz

xz,yy(xzx2yz

xyxz2xyz xz2xyz

x xyzxy,y已知ezxyz0zzx

xezxy,yezf(xye2x(xy21 f(,0 f(xy)2xy3x22y210答案：極大值f(xye2yyx21 f(0,) f(xye2x(xy22y1f(,11 f(xy6xx24yy2的極值.答案：極大值，f(32)36分?jǐn)?shù)難度求

難度n求n的斂散性n難度求

2

難度求

2nsin

難度3求n的斂散性3難度判斷級(jí)數(shù)

(1)n1n難度n判斷級(jí)數(shù) n難度判斷級(jí)數(shù)

n難度cos判斷級(jí)數(shù) n3難度判斷級(jí)數(shù)

(1)n1 ln(n難度計(jì)算級(jí)數(shù)nx2nn1

2,(xn計(jì)算級(jí)數(shù) n計(jì)算級(jí)數(shù)

(x2n

答案2,[1計(jì)算級(jí)數(shù)答案：12

n1(xn

2n

2計(jì)算級(jí)數(shù) n2

2,f(x1展成(x3xn 答案：11n n( ,(0, 3 分?jǐn)?shù)難度f(wàn)(x)

x24x

展成(x1)的冪級(jí)數(shù)答案：

(1)n(11)(x1)n,(1,x24x

f(x)

x23x

展成(x4答案：

(11)(x4)n,(6,x23x

f(xln(1x2x2xln(1x

)

(1)n12nn

xn,x(1,12exe將函數(shù)f(x) 展成x的冪級(jí)數(shù)2exe答案

,x(,2

n1(2n A22,2B130AB答案：2,cos1,cos1,cos 2,2,, a2,a2,b3,(a,b)3已 ，求a2b 已知(a,b) ,61

2k，使得2abakb2x4z3和2xy5z1的交線平行且過(guò)點(diǎn)(325

x

y2

z

求過(guò)點(diǎn)(2,13x1y1

z

x

y1

z2

3xy2z7求過(guò)點(diǎn)(235且平行于直線x3y2z303xy2z7答案x3y2z21xyz1求直線xyz10xyz1yz1xyz求直線

2x4yz

在平面4xyz13xy2z9

x31y37z11704xyz10x2y2z214在點(diǎn)(123 x2y3z14

x22y23z221x4y6z0

x4y6z21x2y2z23x求曲線2x3y5z40在點(diǎn)(1,1,1答案：法平面方程：16x9yz240x1y1z

x

ty1tzt2在對(duì)應(yīng)于t11 x 2y2z1，法平面方程：2x8y16z1 求曲面ezzxy3在點(diǎn)(2,10)x2yx2y40

z2x7yz8求過(guò)點(diǎn)(25,1及直線x2x7yz88x13y11z38求過(guò)點(diǎn)(43,1且平行于下列兩直線l:xyz與

x2yz1 1 答案：11x30y2z136

22xz2

xyz

x

y

x2y2z22xy2z3求直線x2yz1

4x3y505x3z705y4z1求過(guò)點(diǎn)(12,1與兩直線x2yz10和2xyz0xyz1 xyzxyz 2 交換積分次序0dy

f(x,y)dx4x04x2

f(x,y)dy42交換積分次序242

f(x,y)dy44404

f(x,y)dx0

f(x,1

f(x,y)dx

f(x, 1dx1yedyx 答案 1dy1yedx 1交換積分次序1

xsinydy 1答案：

ysinydx (x2y2D

，Dy2yx及y2x2答案：原式

y(x2y2x)dx66

2(xD

D(xy)x2y2x

d

12(rcosrsin1)rdr 計(jì)算x2ydxdyDy0x1yxD1dxxx2ydy 計(jì)算二重積分D

yx2)dxdy，D(x,y)0x1,0y1

0dxx2y

)dy0dx0

y)dy(xD

，Dy1yx2及y4x2102012

ydx利用極坐標(biāo)計(jì)算D

x2y2

，D(x,y)2x2y242.2d2rsinrdr6 利用極坐標(biāo)計(jì)算x2y24D

，D(x,y)x2y2 答案：原式=2d2r(4r2dr2d4r(r24)dr 利用極坐標(biāo)計(jì)算

x2y2

，D(x,y)a2x2y2b2答案：原式=

darlnrdr2arlnrdr

lnab 2a2 2a2利用極坐標(biāo)計(jì)算D

，D(x,y)x2y2x8

22

rrcosdr利用極坐標(biāo)計(jì)算e(x2y2D

，D(x,y)x2y2a2 答案：2daer2rdr2aer2rdre

a(1ea2)計(jì)算xydxdydz，其中zxyxy1z0 0

xydz計(jì)算zdxdydz，其中x0y0z0xyz1

1x 0

zdz計(jì)算xdxdydz，其中x2yz111

2

1x2

xdz

(xyz)dxdydz，其中

b(1x

c(1xy

答案：xdxdydz a

abdz

，原式 (ab

ydxdydz，其中z0yz1yx2 答案：原式=1dxx2dy

ydz計(jì)算(x22xy)dxy22xy)dyLyx2上從點(diǎn)(1,1到點(diǎn)(1,1L計(jì)算[exsinyb(xy)]dx[excosyax]dyLA(2a0L2axy 到點(diǎn)O(00)的弧ab2axb(xy)dxaxdy2a2b1a2b1 計(jì)算(xy)dxyx)dyLy2x上從點(diǎn)(1,1到點(diǎn)(42)L3計(jì)算(exsiny2y)dxexcosy2x6)dyLx2y29L2ydx2xL2x計(jì)算(x2y)dxxsin2y)dy2xLsin

上從點(diǎn)(00)到點(diǎn)(1,1)答案 計(jì)算xdS，其中x2y2z2R2 r2 答案：ds dxdy，原式=2cosd dr

R2x2

R2r2 (x2y2dS，其中z23(x2y2z0z3