ff(x,0)fx2e(xexxxf'(0,0)xxxxx而xxxx故f'x(0,0f(0,y)f(0,y)fy4e(y1e(y21yyyf'y(0,0（B1571、令

f(x,y)

f

f

x

x2

xyx

y0

f'x(0,0) f'y(0,0zf1xf1lim lim

f(x,

(x)2

f(

(x)2 ((

A

x)(

x)2

(x)2

（B）f(x,（A）f(xy)x

y，極限

x

1f(xy在(0,0f(xyxy，在(0,0而極限

f(x,

x2f(xyx2y2)3，在(0,0處可而極限2p

2(x2y2x2xay

(x (xp a(xy)2(xay)2(xy)2(xy)

(x

(x a 158xln x(lnylnx1zeyxe eyx[ (lnylnx)]'eyx (lnylnx)

22

1ln21] 2、zexln(xey

2(ln22

(xey)x[ln(xey)x

xey

(11)1[ln(11)1 12ln2 arctan

11arctan y3 2xy

x(x2y2)

x

arctanx

arctanx

1 x arctan

arctan y2y x(x2y2)y

x 2yarctan

arctanxx

arctanx

1 xdz x[(2xy)dx(2y2

[(2xy)

arctanyxyarctan

arctan y)

x 1 y2 y2xyx2

arctan PAGEPAGE15911x1

f'y

f'

g

f'1x

f'2(222

y(f''y)f'x[f''yf''x]yf''

12y2f''xyf''x2f''f'xyf xxf

xxf

(y)

f'y[f

x

f

x2f''xyf''xyf''y2f''f f(u,v故f f 22g22g

2

(yx)f''11(xy)f''222xyf''122xyf''12

f'2

f(x2y2)(f''f''22

22

f''f 又

22g22g

x2

2x2

f'esinyf'2z

cosyef'1esiny[f''11

cosy

f''1222x(f''excosyf''e2xsinycosyf''2yexsinyf''2xexcosyf 4xyf''excosyf 1

f'

f'[f'

f'

23(23)d3d3

3

2(x)

d(x)

172 f'yf'(yg 2

f'yf''xyf''[g(x)]g'(x)f'yg'(x)[f''xf''

又g(x)可導(dǎo)且在x1處取到列極值g(1) 2 f'(1,1)2

f''12xau

x ,x a

a

a3、 v 1y y a a2 azzuzva2za2 v zzxzy z 1 x y a2 a2 (z (z

(z2z u ux2

u

2 2

2 2 )a2 a2

) a2 a2 a 2z

2z 2z(a2)2

(a (a

(a2)2

2

(2

2

2z00a1621、方程兩邊對x求偏導(dǎo)有F

y

zx' 0

z

2 2

x2FF'1

'

0zF1

xzyz

yF1'zF2'yF1'故選

2（Ⅰ）x

2x'1

z

2y1dz2x'dx2y'1 1

(

)

2z

2) (xy)2

x

2

(2x (2''2x1 1

1x2

(2x

2

1 1

1xy

u2(12x) 33u'(u)up(x)u

p(x) 1u'(u)up(y)u

p( 1zzu

f'(u)

1z

f'(u)

p(y)1'(u)p(y)zp(x)z

p(x)p(y)f'(u)p(y)p(x)f

1 1163163

2x(2y2)11、令 2x2ylny1

得駐點(0,)e2

2(2y2

2

2

2x2y因B2

1(0,e

02(2e2)(0e)eA2(2e20，故f(01e f(0,1)1ln1 x

x2

x

x22

(x)

x22

(1x2)2、令

2 2 f

x

x x

(y)

2

x22

(1x2

x22

(x)

x22

(x322

x2

x22

(x2

( B2

0

2

2(10

2f(1,0)B2

20

2

2(10

2(f(1,0)

2

( 3.2x2zx24x0；2y2zy24y0zx1 z令z y z

z2(x1)z

(z

2(x2

(zz(x (z

(x(z

(z2z

z2(y1)(z

(z2)2(y(zx1y1時，11z2224z10z2z(z (z 又B2 0 1

(z

(z

(zA

zz2A0zz6A0z1641

F(x,y,z,,)x2y2z2(x2y2z)(xyz

2x2x

2y2y

令 2z

x2y2z

xyz4 ②x①y:2xy2xyx2xy2xyy 2(xy)0x 22xz

x

x

2x2

y

，y故(1,12)與(2,2,8u(2,2,8)

z

zumax

165 x2y2

d2x2 L(xyz,,x2y2(x2y22z2(xy L2x2x 2y2y

x

x 4z3

得y

或y

z

z

x2y22z2

xy3z522又d(x,y)(1,1) ，d(x,y)(5,5)222令

yx2

得在D x11Dy

0x

z(x,y) x3x

0y

z(x,y)yxy7(0x7F(xy,)xy

13

y(xy

yx2

x

x令 x1

y

或y

（舍去

xy7 而f(2,5)25 85 7f(x

f(2,5)31671、投入總費(fèi)用Cpxp1 2f(xx,pxpx(2xx12) 1 2

p2x1x

x

2xx1 f2xx12 12 1 2x1x 12 1 2xx ②xpx2xx 2 2③代入⑤有：px122p p1221由⑥⑦x26(p x16(p1221p p1221因此兩要素各投入x26(p ，x16(p1221

2f(xy,)

12

n

)(xy

(l0,nf

nxn1 f

xyn1

yfxyl

[(()](z l[(()]( 22

n(n1) x2

2

x2

()n2

B2AC 故z在 )上取極小值，有2(l)n1

nyn a

b

n11

n

)(a

，且當(dāng)ab 176176sinydxdy1

ysinyy yD

1siny

y 10sinyysincosy11ysin 1cos11(cosy)'ydy1cos1cosyy11cos 11cosydy1siny11 D1D2xD3D4y1(x2y2

]dxdy

1(x2y2

1(x2y2

[y

]dxdy220dyyydx20yxydy

20y2dy2(1y3

0

xr令yrsin1x2

11r22 2dxdy2d2

D1x

01 2

(1

1

) dr222d 2r22ln(1r2)1d2 1 2ln2 (2ln24xr令yr ，x2x24a2x2

4a24a2r2D

d4

4a2sin2r2asin

d4

2a

2a004

4a2sin20

0u1sin2u20204

040

2a2

12

42a2 1 PAGEPAGE177 0yy2y22 2原積分

f(x,

0

f(x,y)dy

f(x,222D:

0x

D:

2x10y1x2

20y

8x2 82原積分0dy282

f(x,

02

xr又令yrr

rr2r

2xyx2r

r2

x2y202故：原積分02

f(x2y22220r

0rcos

0xD: 0

0r

y I0dx0

1x2y211 3 (1x2y2)2 02 1 1(1x2)231

1(1x2)21

x

dx 1

2cos40 1 1D1D2

xy而ln(xyD1故I I,I均大于零，排除B,D1 又令uxsinuC

u II II

I 1

1 xI10dxxycosxdy02x

cos

dx

cosx

cos11cosxdx0

0

I3D2D5yI2 0II 0

D4D5y1xI 1 1x

1 0dx0ycosxdy02I2

cos

dy02

1 1I1I402cosxdx0（A

cosxdx1D如圖所示，則sinx2cosy2d4sinx2cos

1Dxy12I1sinx2cosy2siny2cosxr

sin(x2y2a又令yrsina 2I2d

sinr2 20

1cosr2

0 201

1cosa21 (1cosa2) (1cosa2)24I(1cosa2)sinx2cos2D2f(u,v)dudvADAf(x,y)dxdyxy 1f(xyxy18

08C

xy1821821、D如圖所示DD1D2D3Dyx的偶函數(shù)，故yx2dxdyD

yx2 2122x21x4 x22dx542DIemax(x2,y2)dxdy

emax( ey2dxdyex2 D 1dyyey2dx1dxx 21dyyey2dx21ey2

ydy021ey2ydy0

1e101821823DDD1cos(xy)dxdyD

cos(xy)cos(xy)dxdycos(x D 2

cos(xy)dy

4dy

cos(x 2 2

2sin(x4

dx

4sin(xy)2 2sin2x1dx4

41sin221sin2dxx

cos

4(ax)(x(ax)(xIadxx f'((ax)(x(ax)(x (ax)(xaf'((ax)(x xayay uxayay(ax)(x(ax)(x

ay 2udu2

ay y2

u aa aa 2Iaf'(y)dy(f(y)a)(f(a)

f 1941941、

ln(11

令

ln(11 n 故an與un有相同的斂散性故級數(shù)

ln(11PAGEPAGE1961（A）錯。令

nun

nn錯。令un1)nn

1nn1

21n

錯。令

(1)n

2

2n1

2錯。令

(1)nn

1 1n錯。令

(1)nn

(1)n

1n1n1n1nn1錯。令annn1

anan1

n nan n正確。 n1 (a

2

n3（A）錯。令b(1)n1 an

n（B）錯。令b(1)n,a ab(1)n

n ab正確。limnnlimab 又

收斂，故ab收斂，故ab 錯。令

1

a2b2

n1n2n(n1n2n(n1 n n1 n1、ann n1 nn n

22n n(n1 n

(n1)tan

2、n

ntan

n

2nn

3、an

n) )n

n1

) n p0n

) (n n n1 pn p0n

(n1

n) (n 1 4、

(n333

，而 3收斂1

1 5、anxarctanxdx arctan 2n2 n 0n11

1 2 n n 12又0

1

dx

1

xn1dx n

01 n1

n

n2