1、E? 2012*2013c1?“ ? pAeA Y 1. (1)zx(1,1) = 3, zxy(1,1) = 4. (2) x + 2y + 2z 3 = 0 2x 3y + 5z 4 = 0 x 1 16 = y 1 9 = z 1 1 . (3)2. (4)4 3 . (5)2 3 . (6)2?“ 2.z0(s) = 2 2 . 3.limx0 f(x) f(0) f0(0)x x2 = f00(0) 2 , N0 0n N0: |f( 1 n) f(0) f 0(0)1 n| |f00(0)| + 1 n2 , ? (1)f(0) 6= 0?u (2)f(0) = 0,f0(0) 6
2、= 0? (3)f(0) = f0(0) = 0?“ 4.Pr = px2 + y2, Kzx= 2xf(r2) + 2xr2f0(r2), zxx= 2f(r2) + 2r2+ 8x2f0(r2) + 4x2r2f00(r2), nzyy= 2f(r2) + 2r2+ 8y2f0(r2) + 4y2r2f00(r2), zxx+ zyy= 4f(r2) + 12r2f0(r2) + 4r4f00(r2) = 0. Py(t) = f(et), Ky00+ 2y0+ y = 0, y = (C1+ C2t)et. f(x) = C1+ C2lnx x , “?f(x) = lnx x . 5.
3、Pr = px2 + 4y2+ 4z2, K: x( x r3 ) = 1 r3 3x 2 r5 , y ( y r3 ) = 1 r3 34y 2 r5 , z ( z r3 ) = 1 r3 34z 2 r5 . x( x r3 ) + y ( y r3 ) + z ( z r3 ) = 0. ?= ZZ x2+4y4+4z2=1? xdydz + ydzdx + zdxdy = 3 ZZZ x2+4y4+4z21 dxdydz = . 6(1)a0= 1,an= 0(n 1), bn= 1 + (1)n1 n , 11 f(x) S(x) = 1 2 + X n=1 2 (2n 1) s
4、in(2n 1)x S(7 2 ) = S( 2 ) = 0, S(7) = S() = 1 2. (2) I = Z ?f(x) g(x)?2 + g2(x)?dx = Z ?2?f(x) 2 g(x)?2+ 1 2f 2(x)?dx Ai= ai 2 , i = 0,1,.,10, Bi= bi 2 , i = 1,.,10. q)d Ai I = 0?Ai= ai 2 , i = 0,1,.,10, d Bi I = 0?Bi= bi 2 , i = 1,.,10. 7(1)d y P = xQ)?a = 1, = 1. ?d ?x2+ y2= 1_?= 2, a,X?U?“ (2)d y P = xQ)?a = 4, = 2. dPdx + Qdy = d 2y2 x2+ y2 , ?“ 12
