(2 )
2 < > ( ) f x (x, y) 2x 3+y f y (x, y) x 2y +2 f(3, 2) f x (3, 2) 5 f y (3, 2) L y 2 z 5x 5 ` x 3 z y 2
2 2 < > (2 ) f(, 2) 7 f x (x, y) 2x y f x (, 2),f y (x, y) x +4y,f y (, 2) 7 z (x ) + 7(y 2) + 7 ( ) z 7y 7
2 3 < 2 > (3 ) () f x (x, y) 3x 2 y 2,f y (x, y) 2x 3 y 3 µ µ 3a (a + x) 3 (b + y) 2 2 2a 3 ; x y + a3 b 2 b 3 b 2 (2) f x (x, y) (sin x) y, f y (x, y) cos x 2 y cos(a + x) p b + y ; (sin a) b x + cos a 2 b y +(cosa) b (3) f x (x, y) y,f y (x, y) x y 2 a + x b + y ; µ ³ a x y + a b b 2 b
2 4 < 2 > (4 ) d dt f (x(t),y(t)) f x(x, y) dx dt + f y(x, y) dy dt
2 5 < 2 2 > (5 ) () 2f x ( + 2t, 3 t) f y ( + 2t, 3 t) (2) (cos θ)f x (r cos θ, rsin θ)+(sinθ)f y (r cos θ, rsin θ)
2 6 < > (6 ) () dz 9(3x +2y 2 ) 2 dx +2y (3x +2y 2 ) 2 dy (2) dz sin(2y)dx +2xcos(2y)dy 2 () dx du dv (2) dy e u du + v dv
2 7 < > (7 ) () J 3 5 5 ( 3) 8 (2) J cos θ sin θ r sin θ r cos θ r cos2 θ + r sin 2 θ r
2 8 < > (8 ) () f(x, y) 6y 4 (2) f(x, y) 5sin(2x y)
2 < 3 > ( ) () z r 2cos(2r), z rr 4sin(2r), z θ, z θθ z z rr + r z r 4sin(2r)+ 2 r cos(2r) 4sin ³ 2 p x 2 + y 2 + 2cos ³2 p 2 x 2 + y p x2 + y 2 (2) z r, z rr, z θ sin θ +cosθ, z θθ cos θ sin θ z r 2 z θθ cos θ sin θ r 2 x + y (x 2 + y 2 ) p x 2 + y 2
2 < > ( ) S Z 2 Z 2 6 3 ( x 2 +4)dx ( 2x 2 +4)dx Z 2 x 2 dx +8 6 + 24 3 2 2 3 x3 +4x 8 3
2 2 < 2 > (2 ) S Z b a Z b a 2 S Z 3 Z 3 Z b {f(x)+c} dx {g(x)+c} dx a {f(x) g(x)} dx ( x 2 +4x) (x 2 2x) ª dx ( 2x 2 +6x)dx 23 2 3 x3 +3x 2 27 3 +3 9 9
2 3 < > (3 ) µ 5 2 7 V lim V n lim + µ 2+ n n 2 n n 52 7 2 2 75 6
2 4 < 2 > (4 ) V Z 7 f(x)dx Z 7 7 25 25 98 x2 dx 98 x3 3 25 98 73 3 75 6 µ ; 29.6
2 5 < 3 > (5 ) () f(x) 2 DE DF sin 3 2 x 3 4x 9 2 x2 27 (2) V Z 9 f(x)dx Z 9 x 2 27 dx x 3 8 9 9
2 6 < 4 > (6 ) f(x) πx2 4 V Z 6 πx 2 πx 3 4 dx 2 6 π 63 2 8π 2 f(x) V Z 6 µ 2x 2 4x2 3 9 4x 2 4x 3 9 dx 27 6 4 63 27 32
2 7 < 5 > (7 ) V Z r r ³ Z 2 r π r2 x 2 dx π(r 2 x 2 )dx r π µr 2 x 3 r x3 π ½µr 3 3 r3 µ r 3 + 3 ¾ r3 r π ½2r 3 23 ¾ r3 4 3 πr3
2 8 < 6 > (8 ) V Z r cos θ x π tan 2 3 θ 3 π {(tan θ) x} 2 dx + r cos θ π tan 2 θ r 3 cos 3 θ 3 πr3 3 Z r r cos θ + π r 2 3 x x3 r r cos θ π n r2 x 2 o 2 dx ½ ¾ 2r 3 + π µr 3 3 cos θ r3 3 cos3 θ 2+ +tan 2 θ cos 3 θ 3cosθ ª 2πr3 3 ( cos θ)
2 9 < 7 > (9 ) a + x 3 2 b 2+ x 3 3 S(x) ³ 2 (a + b) 42a +2b 2 + x ³ +2 2+ x 3 3 6+ 4 3 x 4 V Z 3 S(x)dx Z 3 8+ 2 3 32 24 µ6+ 43 x 23 dx 6x + x2 3
2 2 < 8 > (2 ) S(x) Z 3 (5 x +.2y)dy (5 x)y +.y 2 y3 y (5 x) 3+.9 5.9 3x Z 2 V S(x)dx 5.9x 32 x2 2 Z 2 3.8 625.8 (5.9 3x)dx
2 2 < 9 > (2 ) S(x) V Z 3 µ3 x2 2 + xy y3 +2y y2 dy 3y x2 2 2 y + xy2 4 + y2 y3 3 y 9 3 2 x2 + 9x 4 +9 99 3 2 x2 + 9 4 x Z 2 S(x)dx Z 2 µ9 32 x2 + 94 x dx 9x 2 98 x3 + x2 2 8 2 8+9 8 437 2
2 22 < > (22 ) S(y) V Z 3 µ 3 x 4 y +2 dx 6 x2 x3 yx +2x 4 x 9 6 3 4 y +6 5 2 3 4 y Z 4 S(y)dy 3 624 Z 4 µ 5 2 3 5 4 y dy 2 y 3 4 8 y2
2 23 < > (23 ) Z 2 ½Z 3 ¾ Z 2 ½ h (x 2 xy )dy dx x 2 y x 2 y2 y i y3 y ¾ dx Z 2 ½µ3x 2 92 x 3 µx 2 2 ¾ Z 2 x dx 2x 2 4x 2 dx 2 x2 µ µ 6 2 3 x3 2x 2 2x x 3 8 4 3 2 2 3
2 24 < 2 > (24 ) Z 3 ½Z 2 ¾ Z ( 3 x (x 2 3 xy )dx dy 3 x2 2 y x Z 3 Z 3 x2 x ) dy ½µ 8 3 4 µ 2 y 2 3 ¾ Z 3 µ 7 2 y dy 3 3 2 y dy µ 4 3 3 4 2 y dy 3 y 3 y3 µ 4 y2 4 27 µ 4 y 4 3 3 4 3
2 25 < 2 > (25 ) ZZ Z 2 Z 2 (2x 3y 2 )dxdy D ½Z ¾ (2x 3y 2 )dx dy n x 2 3y 2 x o Z 2 x dy ª x 6y 2 dy 2y 3 y2 6 y ( ) ZZ Z Z ½Z 2 ¾ (2x 3y 2 )dxdy (2x 3y 2 )dy dx D n 2xy y 3 y2 y o dx Z (4x 8)dy 2x 2 8x x x 6
2 26 < 2 2 > (26 ) () (2) ZZ D ( ZZ µz 2 x 3 sin(2y)dxdy 4 x4 D ( x2 x ) ZZ e 2x y dxdy 2 e2x x x 2 (e2 ) ) ( 2 cos(2y) D Ã Z π! x 3 2 dx sin(2y)dy π y 2 y µz e 2x e y dxdy ) ½ ¾ 2 4 ½ 2 4 cos π + 2 ¾ cos 4 4 µz e 2x dx e y dy n e o ½ y y y 2 e2 ¾ e ( ) ª 2 µ ½ e 2 e + ¾ e 2 e
2 27 < 2 > (27 ) ZZ Z 2 Z 2 (x + y)dxdy D ½Z ¾ Z 3 (x + y)dy dx + 2 µ x + Z 3 µ dx + x + 3 dx 2+46 2 2 2 ½Z 2 ¾ (x + y)dy dx ( ) ZZ Z Z ½Z 2 ¾ Z 2 ½Z 3 ¾ (x + y)dxdy (x + y)dx dy + (x + y)dx dy D 2 µ Z 3 2 µ 5 2 + y dy + 2 + y dy 2+46
2 28 < 2 2 > (28 ) ZZ Z Z D (6xy 2 )dxdy Z ½Z x+ n 2xy 3 y o y x+ dx y Z 2x 4 +6x 3 6x 2 +3x ª dx ¾ (6xy 2 )dy dx 2x( x +) 3 ( x +) ª dx 2 5 x5 + 3 2 x4 2x 3 + 3 x 2 x2 x x 2 5 + 3 2 2+3 2 2 5
2 29 < 2 3 > (29 ) ZZ Z D (xy y)dxdy ½ 2 x2 y xy xy x Z ½Z y ¾ (xy y)dx dy ¾ Z ½ ¾ dy 2 y3 y 2 dy 8 y4 y 3 y3 y 8 3 3 8 24 5 24
2 3 < > (3 ) () (u, v) 2 24, (x, y) 6 4 2 8, (x, y) (u, v) 4 4 (2) (x, y) (u, v) 3 2 4 2 2
2 3 < > (3 ) ZZ ZZ (y x)dxdy D Z 2 ½Z 2 ¾ Z 2 ( 2u +2v)du dv Ω {(u +4v) (3u +2v)} dudv n u 2 +2uv o u2 dv u Z 2 { 4+4v} dv 4v +2v 2 2 ( 8+8)
2 32 < 2 > (32 ) n Ω (r, θ) :5 r 5 R, 5 θ 5 π o 2 ZZ Z π 2 Z π 2 D ZZ e x2 y 2 dxdy ½Z R Ω e r2 rdrdθ ¾ Z ( π e r2 2 rdr dθ 2 e r2 µ + dθ π n e R2o 2 e R2 2 4 rr r ) dθ
2 33 < > (33 ) g M {m x + m 2 x 2 + + m n x n }
2 35 < 3 > (35 ) M g M Z 6 Z f(x)dx Z xf(x)dx µ 4 + 3 ( x 2 + x)dx 3 x3 + 6 Z 6 2 2 2 x2 3 + 2 6 ( x 3 + x 2 )dx 6 4 3 x4 + x3
2 36 < 4 > (36 ) g y M 3 3 3 ZZ D Z ½Z 2x Z Z ( y 2 yf(x, y)dxdy 3 2 ZZ D ydxdy + 3 ZZ ¾ ydy dx + Z 3 ½Z x+3 ¾ ydy dx 3 y2x y ) dx + 3 Z 3 ( 2 y2 y x+3 y ) dx D 2 ydxdy 2x 2 dx + Z 3 3 2 ( x +3)2 dx x 2 3 3 x3 + x 3 µ+ 6 23 6 ( x +3)3 x3 x 3 2 3 + 3 2 3
2 37 < > (37 ) () Z e λx dx lim b Z b e λx λ dx lim e λx b lim µ λ e λb + λ e b b λ (2) (3) Z Z Z b µ 2 dx lim dx lim x 2 b lim x3 b x3 b b 2b + 2 2 2 Z b dx lim xr b dx lim xr b r + x r+ µ lim b (r )b + r r b r
2 38 < 2 > (38 ) Z Z ZZ e x2 y 2 dxdy lim R ½ π x2 y2 e dxdy lim ¾ π D R R 4 e R2 4
2 4 < 4 > (4 ) () x 2 t Z e x2 4 dx Z Z e t2 2dt 2 e t2 dt 2 π (2) x 2λt Z e x2 2λ dx Z Z e t2 2λdt 2λ e t2 2λπ