试题答案-微积分复习题

一．单项选择题（每小题345分①③④③③④①①11.二．解答n用比值法判断级数n

n!的敛散性解：由un

n

1 n

得到limun1

n

n

n

根据比值判别法可知，级数1

n的发散n n用根值法判断级数2n的敛散性1 1n n2n解：limn 2n

1

n2nn n∴2n收1 1求幂级数2n1x2n2的和函数1sx2n1x2n213x25x42n1x2n21xSxdxx2n1x2n2dxx2n1x2n2dx

0 =xx3x5 1两边求导得

21x222

x21x21x2S1x2

，x求函数fxex2的幂级数展开式。解：因为fxex的幂级数展开式为0xxx 0xxxx

e1x

n! 把e展开式中的x成xfxe的幂

1x2

xx xx

0

高数复习提要一．积分上限函数及积分上限函数xxfta积分上限函数性质d(xdxftdt

f

dx 解答；1.

limx5sint2dt0 x

x

是0x5sint2x

5sin

3lim3

x3 x解x

5arctantdt0

05arctantdt0型极xlimxx

05arctantdtxx

x0x0

x0x0解

0型极限limxx1cost

x3lim 3 二.分部积分法和变量变换分部积分法:bfxdx uvbb 变量代换令tx,xt,dx aa计算定积

42t=4x1x1t21dx1tdt，当x=0，t=1x=2242t3e4x1dx 2t3

13td 0计算定积

22

2xx12dxtdt，当x=0t=0.x=2t2t2 000tdcos2xdx=tcostdt 2 000td=2sin2-3计算定积3

33 3三.全微分及隐函数偏导数全微分z

fx,y,dzzdxz 隐函数偏fxyz0zFxzFy Fz Fz求由方程sin3xsin3ysin3z1.所确定的函数z=f(x,y)的全微分解：两边微分得：d(sin3xsin3ysin3z)dsin3xdsin3ydsin3z3sin2xcosxdx+3sin2ycosydy+3sin2zdz

sin2xcos2

dx

sin2cos2

zcos

zcos解法2Fx,yzsin3xsin3ysin3xxF3sin2xcosxyF3sin2ycosyzF3sin2zcosz∴

FxFzF

sin2xcossin2zcos2ycos y Fz

2zcossin2xcos

sin2cosdzxdxydy=sin2zcoszdxsin2zcosz求函数ulnx2y2z2的全微分解法1u

x2y2

u

2 x2y2

u

x2y2∴

x2y2

2x2y2

x2y2四.二重积分作出积分区域图并判别其形态（X-型，Y-型把二重点积分列成二次积分计算二次积分1；计算二重积分xexyd,D:0x1,0yDxxyd

xydy

1dx1xyd =

xy

xD

0

0=exx1e0计算二重积分66y2x2dD是由yx2与yx围成的区域1x1x

13y213y22x2解 66y2D

d60dxx26y2

dy=6 0=610

x2

2x3

31x4x2

11550计算二重积分xeyd,D:0x1,0yD五.用比值法及根值法判别级数收敛比值法：如果limun1n①．r＜1.级数un收敛0②．r≥1.级数un发散0判断级数1

7的敛散性7n n用根值法判断级数5n的敛散性1 1判断级数1

n的敛散性1．02．0

n1x 2 n1,x 11n1xn1xx21n1xn10

x

1

nx2n1xx3x51nx2n1 00

1求幂级数2n1x2n2的和函数1sx2n1x2n213x25x42n1x2n21xSxdxx2n1x2n2dxx2n1x2n2dx

0 1 =xx3x5 1 1x22

11x22Sx1x22

x2求幂级数n1xn的和函数x20sxn1xn12x3x2n1xn0两边积分得xSxdxx

n1xndxxn1xndx

xn1xx2xn 0

1两边求导得：Sx 1x

求幂级数1

sxn1xn2x3x2n1