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Transcription:

II

1 3 2 5 3 7 4 8 5 11 6 13 7 16 8 18 2

1 1. x 2 + xy x y (1 lim (x,y (1,1 x 1 x 3 + y 3 (2 lim (x,y (, x 2 + y 2 x 2 (3 lim (x,y (, x 2 + y 2 xy (4 lim (x,y (, x 2 + y 2 x y (5 lim (x,y (, x + y x 3y (6 lim (x,y (, x + 2y xy (7 lim (x,y (, x 2 + 2y 2 x 2 y (8 lim (x,y (, x 2 + 2y 2 x 2 y (9 lim (x,y (, x 2 + y 2 x 3 y 3 (1 lim (x,y (, x2 + y 2 3

1. x y (1 lim (x,y (1,1 sin π(x y sin(xy (2 lim (x,y (, x 2 + y 2 2. xy (x, y (, f(x, y = x2 + 2y 2 1 (x, y = (, 4

2 1. f(x, y f x (x, y f y (x, y (1 f(x, y = x 3 + 2xy 2 y 3 (2 f(x, y = 3 sin(x + y 2 cos(x y (3 f(x, y = sin(x + y cos(x y (4 f(x, y = x y x + y (5 f(x, y = e x sin y (6 f(x, y = xy cos(xy (7 f(x, y = sin(x 2 + xy (8 f(x, y = e x log(1 + y 5

1. (1 z = f(x, y = xe y (2 z = f(x, y = x y x + y (3 z = f(x, y = e (x2 +y 2 (4 z = f(x, y = (x ye x+y 2. z = f(x, y (1 z = f(x, y = x 2 + 3y 2 (1, 1, 4 (2 z = f(x, y = x 2 2xy + 2y 2 (1, 2, 5 (3 z = f(x, y = log x 2 + y 2 ((x, y (, (1, 1, log 2 3. z = f(x, y = (x y 2 (1,, f(1, yz 4. z = f(x, y = e 2x y (1, 1, f(1, 1 xz 1. z = f(x, y = x 2 xy + 2y 2 (1, 1, 2 2. z = f(x, y (1 f(x, y = log x 2 + y 2 ((x, y (, (2 f(x, y = x 2 + y 2 6

3 1. z u, z v (1 z = xy, x = u + v, y = 5u + 6v (2 z = e x cos y, x = u 2 v 2, y = 2uv 2. z = xy, x = sin θ, y = cos θ dz dθ 3. g (t, z(x, y z x (x, y (1 g (t = e t2, z(x, y = g(xy (2 g (t = et t, z(x, y = g(x2 + y 2 4. f x (x, y = y x 2 + y 2, f y(x, y = x x 2 + y 2 (1 x(t = sin t, y(t = cos t, z(t = f(x(t, y(t z (t (2 x(s, t = e s sin t, y(s, t = e s cos t, z(s, t = f(x(s, t, y(s, t z s (s, t z t (s, t 1. z = f(x, y x, y r, θ x = r sin θ, y = r cos θ ( 2 ( 2 ( 2 z z z + = + 1 ( 2 z x y r r 2 θ 7

4 1. f(x, y = x 3 + y 3 + 5x 2 y + 6xy 2 + xy f xx, f xy, f yxx, f yxy 2. 2 2 f x 2, 2 f x y, 2 f y 2 (1 f(x, y = e x2 +2y (2 f(x, y = e x log(1 + y (3 f(x, y = e x cos y (4 f(x, y = e x+y sin(x y (5 f(x, y = log(x 2 + y 2 3. x 3 + y 3 = 3xy dy dx x y 8

1. f(x, y 2 (1 f(x, y = 1 1 x + y (2 f(x, y = log(1 + x + y 2 (3 f(x, y = e xy cos(x + y (4 f(x, y = e x cos y (5 f(x, y = cos(x + 2y (6 f(x, y = ey 1 x (7 f(x, y = e x log(1 + y 2. f xxxyy (, = 2 f(x, y x 3 y 2 3. dy dx (1 x 2 + y 2 e xy = x y (2 x 2 3 + y 2 3 = 1 (3 x + log(y x = 1 4. x 2 + 4y 2 8 = y = φ(x (1 φ (x x, y (2 (2, 1 5. x 2 y 2 = 1 9

1. f(x, y = sin x cos y ( π 2 < y < π 2 (1 f xxx (x, y, f xyy (x, y (2 f(x, y x 3, xy 2 2. (1 x 3 2x 2 + x y 2 = (2 x + y 2 2x 2 y = 3. f(x, y = x 3 + y 3 6xy = (1 (a, b = ( 4 3, 8 (a, b f(x, y = 3 (2 (a, b f(x, y = f x (x, y = f y (x, y = (3 (a, b f(x, y = 1

5 1. (1 f(x, y = x 2 + y 2 (2 f(x, y = x 2 y 2 (3 f(x, y = x 2 y 2 2. x 2 + y 2 = 1 f(x, y = 2x y 11

1. (1 f(x, y = xy + x + y (2 f(x, y = x 3 + y 3 3xy (3 f(x, y = x 3 3x y 2 (4 f(x, y = x 4 + 2xy + y 2 2. (1 2x 2 + y 2 = 4 f(x, y = x 2 + y 2 (2 x 2 + y 2 = 4 f(x, y = x 2 2xy + y 2 (3 x 2 + y 2 = 1 f(x, y = 2x 2 y (4 x 2 + 9y 2 = 1 f(x, y = xy (5 x 2 + y 2 = 1 f(x, y = x 2 + 4xy + 4y 2 1. (1 x 2 y 2 + 1 =, 1 x 1, y > f(x, y = x + 2y (2 2x 4 + y 2 = 1 f(x, y = xy 2. = {(x, y 2x 2 + y 2 4} f(x, y = x 2 + y 2 12

6 1. (1 : x 1, y 1 xy dxdy (2 = {(x, y x 1, 2 y 1} (x 2 + y 2 dxdy (3 = {(x, y 1 x 3, y 1} (x 2 y dxdy 2. xy ( 2 y 2 (1 f(x, y dxdy = f(x, y dx dy (2 (3 (4 f(x, y dxdy = f(x, y dxdy = f(x, y dxdy = ( f(x, y dy dx 1 x 2 ( 2 y 1 ( x+1 1 f(x, y dx dy f(x, y dy dx 3. xy f(x, y dxdy (1 = {(x, y x 1, 2x y 2} (2 = {(x, y 1 x 1, y 1 + x} (3 = {(x, y y 1 x 2 } 13

1. ( y (1 x 2 y dx dy y (2 ( 1 y y 4 2 + 1 dx dy 2. xy f(x, y dxdy (1 = {(x, y y x 2, x y 2 } (2 y = e x y = 2 y (3 4y = x 2 x 2y + 4 = 3. ( 1 x (1 f(x, y dy dx x (2 (3 ( 1 x 2 f(x, ydy dx 2 ( e 2 f(x, y dy dx e x 14

4. ( y y 2 (1 x 2 + x + 1 dx dy (2 (3 (4 (5 (6 π 2 ( y ( e x2 dx dy y e x3 dx ( 4 y ( x ( x dy 1 x 2 + 1 dx dy sin x sin 3 y dy dx 1 y2 dy dx 5. x (1 y dxdy, = {(x, y 1 y 2, x y2 } (2 xy dxdy, = {(x, y x 2 y 2x} 15

7 1. (1 x = au + bv, y = cu + dv J(u, v (2 x = r cos θ, y = r sin θ J(r, θ (3 x = uv, y = u + v J(u, v (4 u = e x e y, v = e x + e y J(x, y 2. (1 (x + y dxdy, = {(x, y 2x + y 1, x + 2y 1} (2 (3 (4 (3x + 5y dxdy, = {(x, y x + y 1, x y 1} (3x + 2y dxdy, = {(x, y x y 1, 1 2x + 3y 2} (x + 2y(y x dxdy, = {(x, y 1 x + 2y 4, x y 1} 16

1. (1 (x 2 y 2 e x y dxdy, = {(x, y x y 1, x + y 1} (2 (3 (4 (5 (6 (4x 2 y 2 dxdy, {(x, y x, 1 xy 2, 2 2x y 3} 2xy dxdy, = {(x, y 1 x 2 + y 2 4, x, y } y 2 dxdy, = {(x, y y, x 2 + y 2 1} (x 2 + y 2 + xy dxdy, = {(x, y 1 x 2 + y 2 9} x2 + y 2 dxdy, = {(x, y y 3x, x 2 + y 2 1} 17

8 1. xyz,. (1 z = x 2 + 3y 2 2, z = 4x + 1 V 2,. { V = dxdy, = (x, y } (2 = { (x, y x 1, y 1 } z = x 2 y 3 S, 2,. S = dxdy 2. (1 z = 1 x 2 y =, y = 2, z = (2 z = 1 x 2 y 2 z = (3 z = 2 x 2 + y 2 xy (4 z = x 2 + y 2 x 2 + y 2 = 1 3. = {(x, y x 4 + y 4 1} z = 1 x 4 y 4 S 18

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