III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2. (x,y) (1,0) x 2 + y 2 5x 2 y x 2 + y 2. xy x2 + y 2. 2x + y 3 x 2 + y 2 + 5. sin(x 2 + y 2 ). x 2 + y 2 sin(x 2 y + xy 2 ). xy (i) (ii) (iii) 2xy x 2 + y 2. xy 2 x 2 + y 4. x + y x2 + y 2. f(x, y) { f(x, y)} x 0 y 0 (i) f(x, y) = y2 x 2 y 2 + x 2. x 2 + y 2 (ii) f(x, y) = xy + (x y). 2 sin x sin y (iii) f(x, y) = x 2 + y. 2 (iv) f(x, y) = x3 y 3 x 2 + y 2. 1 f(x, y), y 0 { x 0 f(x, y)},
III No (i) 3. x = r cos θ, y = r sin θ f(x, y) (ii) f(x, y) 5r sin θ cos 2 θ 5r 0 (r 0) 0. (iii) f(x, y) = r2 cos θ sin θ r 0 (r 0). r (iv) 3/5. sin r (v) 2 r 0 = 1. r 2 sin xy(x+y) (vi) (x + y) 0 (x, y) (0, 0). xy(x+y) (i) y = mx (0, 0) 2m/(1 + m 2 ). (ii) y 2 = mx (0, 0) m/(1 + m 2 ). (ii) y = mx (0, 0) (1 + m)x/ 1 + m 2 x. (i) y = mx (0, 0) (m 2 1)/(m 2 + 1). f(x, y) { f(x, y)} = 1, { f(x, y)} = y 0 x 0 x 0 y 0 1. (ii) y = mx (0, 0) (1 + m 2 )/(m + (1 m) 2 ). f(x, y) { f(x, y)} = 1, { f(x, y)} = y 0 x 0 x 0 y 0 1. (iii) y = mx (0, 0) 0. (iv) 0. sin x sin y x 2 +y 2 = ( sin x x )( sin mx mx ) m 1+m 2. f(x, y) y 0 { x 0 f(x, y)} = 0, x 0 { y 0 f(x, y)} = f(x, y) = 0. { f(x, y)} = 0, { f(x, y)} = y 0 x 0 x 0 y 0 2
III No f(x, y) (a, b) 1 (i) f(x, y) =, (a, b) = (2, 3). x 2 + y2 (ii) f(x, y) = log(x + y), (a, b) = (2, 5). (iii) f(x, y) = e x2 +y 2, (a, b) = ( 1.1). (iv) f(x, y) = { sin x x 2 +y 2 (x, y) (0, 0), 0 (x, y) = (0, 0), (a, b) = (0, 0) (v) f(x, y) = xy, (a, b) = (0, 0). (vi) f(x, y) = x 2 + y 2, (a, b) = (0, 0). (vii) f(x, y) = { xy x 2 +y 2 (x, y) (0, 0), 0 (x, y) = (0, 0), (a, b) = (0, 0). (i) f(x, y) = (x 3 1)(y + 3). (ii) f(x, y) = log y x. (iii) f(x, y) = x y. (iv) f(x, y) = e y (cos(x + y)). (v) f(x, y) = x y y x. (vi) f(x, y, z) = x+y. x z (vii) f(x, y, z) = log x 2 + y 2 + z 2. (viii) f(x, y, z) = sin 1 (xyz). 3
III No (i) f x (2, 3) = 4/169, f y (2, 3) = 6/169. (ii) f x (2, 5) = 1/7, f y (2, 5) = 1/7. (iii) f x ( 1, 1) = 2e 2, f y ( 1, 1) = 2e 2. f(h,0) f(0,0) (iv) f x (0, 0) = h 0 h f y (0, 0) = 0. (v) f x (0, 0) = 0, f y (0, 0) = 0. (vi) f x (0, 0) = h 0 f(h,0) f(0,0) h f y (0, 0) = k 0 f(0,k) f(0,0) k = h 0 h h 3 = h 0 h h = k 0 k k (vii) f x (0, 0) = h 0 f(h,0) f(0,0) h = h 0 0 f y (0, 0) = k 0 f(0,k) f(0,0) k = k 0 0 = 0. h = 0. k (i) f x (x, y) = 3x 2 (y + 3), f y (x, y) = x 3 1. (ii) f x (x, y) = 1, f x log y y(x, y) = log x. y(log y) 2 (iii) f x (x, y) = x y 1 y, f y (x, y) = x y log x. (iv) f x (x, y) = e y sin(x + y), f y (x, y) = e y cos(x + y) e y sin(x + y). (v) f x (x, y) = yx y 1 y x + x y y x log y, f y (x, y) = x y xy x 1 + y x x y log x. (vi) f x (x, y, z) = y+z, f (x z) 2 y (x, y, z) = 1, f x z z(x, y, z) = x+y. (x z) 2 x y z (vii) f x (x, y, z) =, f x 2 +y 2 +z 2 y (x, y, z) =, f x 2 +y 2 +z 2 z (x, y, z) =. x 2 +y 2 +z 2 yz (viii) f x (x, y, z) = 1 x, f xz y(x, y, z) = 2 y 2 z 1 x f xy z(x, y, z) = 2 2 y 2 z 2 1 x 2 y 2 z 2. 4
III No f(x, y) (a, b) (i) f(x, y) = x 3 4xy + 2y 5 (a, b) = (1, 1). (ii) f(x, y) = sin(x + 2y) (a, b) = (2, 3). (iii) f(x, y) = e x2 y 2 (a, b) = (0, 1). f(x, y.z) (a, b, c) (i) f(x, y, z) = 2x + 3y 2 4z 3 + 5xyz (a, b, c) = (1, 1, 1). (ii) f(x, y, z) = cos(x 2y + z 2 ) (a, b, c) = (π, π/2, 0). (iii) f(x, y, z) = log(x 2 + y + 2z + 4) (a, b, c) = (1, 2, 3). (a, b) (a, b, c) e (i) f(x, y) = 2x + 3y 1, (a, b) = (1, 2), e = (1/ 2, 1/ 2). (ii) f(x, y) = x 2 + y 2 4, (a, b) = (c, d), e = ( 1/ 2, 1/ 2). (iii) f(x, y, z) = xy + yz + zx, (a, b, c) = (1, 2, 1), e = (1, 2, 3)/ 14. (iv) f(x, y, z) = sin(xyz), (a, b, c) = (π, 1/2, 1/3), e = (1/ 3, 1/ 3, 1/ 3). (v) f(x, y, z) = x y 2 + z 2, (a, b, c) = ( 2, 1, 3), e = (2, 1, 4)/ 21. 5
(i) (7, 6). (ii) (cos 8, 2 cos 8). (iii) (0, 2e 1 ). III No (i) (7, 11, 7). (ii) (0, 0, 0). (iii) (2/13, 1/13, 2/13). (i) 5/ 2. (ii) ( 2c + 2d)/ 2. (iii) 8/ 14. (iv) (1 + π)/12. (v) 2/ 21 + 13/ 210. 6
III No z = f(x, y) x = a + ut, y = b + vt dz dt = uf d 2 z x + vf y, dt = 2 u2 f xx + 2uvf xy + v 2 f yy a, b, u, v R z = f(x, y) x = r cos θ, y = r sin θ (i) x z x + y z y = r z r. ( ) 2 ( ) 2 ( ) 2 z z z (ii) + = + 1 ( ) 2 z. x y r r 2 θ (iii) 2 z x + 2 z 2 y = 2 z 2 r + 1 ( z 2 r r + 1 ) 2 z. r θ 2 7
III No (i) (ii) dz dt = f dx x dt + f dy y dt = uf x + vf y. d 2 z = (uf x + vf y ) dx dt 2 x dt + (uf x + vf y ) dy y dt = (uf xx + vf yx )u + (uf xy + vf yy )v = u 2 f xx + 2uvf xy + v 2 f yy. (i) (ii) z r = z x x r + z y y r r z r z z = cos θ + sin θ x y. z z = r cos θ + r sin θ x y = x z x + y z y. 1 r 2 ( ) 2 z = cos 2 θ r ( ) 2 z + sin 2 θ x z θ = z x x θ + z y y θ ( ) 2 z = sin 2 θ θ ( ) 2 z + cos 2 θ x ( ) 2 z + 2 cos θ sin θ z y x z = ( r sin θ) + (r cos θ) z x y. ( ) 2 z 2 cos θ sin θ z y x z y, z y. (iii) ( ) 2 z + 1 r r 2 ( ) z = θ ( ) 2 z + x ( ) 2 z. y 2 z = cos θ r 2 x + cos θ y z x x r + sin θ z x x y r z y x r + sin θ z y y y r = cos 2 θ 2 z x 2 + 2 sin θ cos θ 2 z x y + sin2 θ 2 z y 2. 8
2 z = ( r cos θ) z θ 2 x + ( r sin θ) θ = ( r cos θ) z x z + ( r sin θ) z x y + (r cos θ) θ ( ) 2 z x + ( r sin θ) x 2 θ + 2 z y y x θ ( ) 2 z x y + 2 z y y 2 θ 2 +( r sin θ) z + (r cos θ) y = (r 2 sin 2 θ) 2 z x + 2 ( 2r2 sin θ cos θ) 2 z x y ( +(r 2 cos 2 θ) 2 z y r cos θ z ) z + sin θ. 2 x y z y 2 z r + 1 ( z 2 r r + 1 ) 2 z = 2 z r θ 2 x + 2 z 2 y. 2 9
III No ( h x + k ) 2 f(x, y) y (i) f(x, y) = sin x sin y. (ii) f(x, y) = (x y)/(x + y). ( h x + k y + l ) 2 f(x, y, z) z (i) f(x, y, z) = x 3 + y 3 + z 3. (ii) f(x, y, z) = xze x2 y 2. (iii) f(x, y, z) = cos(x yz). (0, 0) P 2 R (i) f(x, y) = cos(x + y). (ii) f(x, y) = cos x sin y. (iii) f(x, y) = ye x y. (iv) f(x, y) = sin(x + y). (v) f(x, y) = sin x + cos y. 10
III No (i) h 2 ( sin x sin y) + 2hk cos x cos y + k 2 ( sin x sin y). (ii) h 2 ( 4y) + 2hk 2(x y) + k 2 4x. (x+y) 3 (x+y) 3 (x+y) 3 (i) 6(h 2 x + k 2 y + l 2 z). (ii) (iii) 2xz(3 + 2x 2 )e x2 y 2 h 2 4yz(1 + 2x 2 )e x2 y 2 hk + 2xz(2y 2 1)e x2 y 2 k 2 + 2(1 + 2x 2 )e x2 y 2 hl 4xye x2 y 2 kl. cos(x yz)h 2 + 2z cos(x yz)hk z 2 cos(x yz)k 2 + 2y cos(x yz)hl y 2 cos(x yz)l 2 + 2(sin(x yz) yz cos(x yz)kl. (i) P 2 = 1 1 2 (x2 +2xy+y 2 ). R = 1 6 (x2 +3x 2 y+3xy 2 +y 3 ) sin(θx+θy). (ii) P 2 = y. R = 1 6 (x3 sin θx sin θy 3x 2 y cos θx cos θy+3xy 2 sin θx sin θy y 3 cos θx cos θy). (iii) P 2 = y + xy y 2. R = 1 6 {x3 θy + 3x 2 y(1 θy) + 3xy 2 ( 2 + θy) + y 3 (3 θy)}e θx θy. (iv) P 2 = x + y. R = 1 6 (x3 + 3x 2 y + 3xy 2 + y 3 ) cos(θx + θy). (v) P 2 = 1 + x y2 2. R = 1 6 ( x3 cos θx + y 3 sin θy) 11
III No R 2 (i) f(x, y) = 5 + x + y x 2 y 2. (ii) f(x, y) = x 2 + xy. (iii) f(x, y) = x 2 xy + y 2 + 3x y + 4. (iv) f(x, y) = 3xy x 4 y 4 + 2. (v) f(x, y) = 1 x 2 + y 2 + 1. (vi) f(x, y) = 1 x + 2xy + 1 y. (vii) f(x, y) = y sin x. (viii) f(x, y) = sin(x + y) + sin x + sin y. (ix) f(x, y) = e x2 +y 2 2y. (x) f(x, y) = xy log(x 2 + y 2 ). 12
III No (i) f x = 1 2x = 0, f y = 1 2y = 0 x = y = 1/2. f xx = 2, f xy = 0, f yy = 2. H f (1/2, 1/2) = f xx (1/2, 1/2)f yy (1/2, 1/2) f xy (1/2, 1/2) 2 = 4 > 0. f xx (1/2, 1/2) = 2 < 0 (x, y) = (1/2, 1/2) 11/2. (ii) f x = 2x + y = 0, f y = x = 0 x = y = 0. f xx = 2, f xy = 1, f yy = 0. H f (0, 0) = f xx (0, 0)f yy (0, 0) f xy (0, 0) 2 = 1 < 0 (x, y) = (0, 0) (iii) f x = 2x y +3 = 0, f y = x+2y 1 = 0 x = 5/3, y = 1/3. f xx = 2, f xy = 1, f yy = 2. H f ( 5/3, 1/3) = 3 > 0. f xx ( 5/3, 1/3) = 2 > 0 (x, y) = ( 5/3, 1/3) f( 5/3, 1/3) = 5/3. (iv) f x = 3y 4x 3 = 0, f y = 3x 4y 3 = 0 (x, y) = (0, 0), ( 3/2, 3/3), ( 3/2, 3/2). f xx = 12x 2, f xy = 3, f yy = 12y 2. H f (0, 0) = 9 < 0 (x, y) = (0, 0) H f ( 3/2, 3/2) = 72 > 0. f xx ( 3/2, 3/2) = 9 < 0 ( 3/2, 3/2) H f ( 3/2, 3/2) = 72 > 0. f xx ( 3/2, 3/2) = 9 < 0 ( 3/2, 3/2) (v) f x = ( 1)(x 2 + y 2 + 1) 2 2x = 0, f y = ( 1)(x 2 + y 2 + 1) 2 2y = 0 (x, y) = (0, 0). f xx = 2(x 2 + y 2 + 1) 3 4x 2 2(x 2 + y 2 + 1) 2, f yy = 2(x 2 + y 2 + 1) 3 4y 2 2(x 2 + y 2 + 1) 2, f xy = 2(x 2 + y 2 + 1) 3 4xy. H f (0, 0) = 4 > 0, f xx (0, 0) = 2 < 0 (x, y) = (0, 0) f(0, 0) = 1. (vi) f x = 1 x 2 +2y = 0, f y = 2x+ 1 y 2 = 0 2x 2 y = 1, 2xy 2 = 1. x, y > 0 x = y = 1/ 3 2. f xx = 2/x 3, f yy = 2/y 3, f xy = 2. H f (1/ 3 2, 1/ 3 2) = 12 > 0. f xx (1/ 3 2, 1/ 3 2) = 4 > 0 (x, y) = (1/ 3 2, 1/ 3 2) 3 3 2. 13
(vii) f x = y cos x = 0, f y = sin x = 0 x = nπ(n = 0, ±1, ±2,...), y = 0. f xx = y sin x, f yy = 0, f xy = cos x. H f (nπ, 0) = (cos nπ) 2 < 0 (x, y) = (nπ, 0) (viii) f x = cos(x + y) + cos x = 0, f y = cos(x + y) + cos y = 0 cos x = cos y = cos(x + y). 0 x 2π, 0 y 2π cos x = cos y x = y x = 2π y. x = y cos x = cos(x + y), cos 2x + cos x = 2 cos 2 x + cos x 1 = 0, (2 cos x 1)(cos x + 1) = 0. cos x = 1/2, 1. x = π/3, π, 5π/3. x = 2π y cos x cos(x + y) cos x = 1. x = π. y = π. f x = f y = 0 0 x 2π, 0 y 2π (x, y) = (π/3, π/3), (π, π), (5π/3, 5π/3). f xx = sin(x + y) sin x, f yy = sin(x + y) sin y, f xy = sin(x + y). H f (π/3, π/3) = 9/4 > 0. f xx (π/3, π/3) = 3 < 0 (π/3, π/3) H f (π, π) = 0 h > 0 f(π + h, π + h) = 2 sin(π + h){1 + cos(π + h)} < 0. h < 0 f(π + h, π + h) < 0. f(π, π) = 0. (π, π) H f (5π/3, 5π/3) = 9/4 > 0. f xx (5π/3, 5π/3) = 3 > 0 (π/3, π/3) < x <, < y < (2mπ + π/3, 2nπ + π/3) (2mπ + 5π/3, 2nπ + 5π/3) (ix) f x = 2xe x2 +y 2 2y = 0, f y = (2y 2)e x2 +y 2 2y = 0 x = 0, y = 1. f xx = 2e x2 +y 2 2y + 4x 2 e x2 +y 2 2y, f yy = 2e x2 +y 2 2y + (2y 2) 2 e x2 +y 2 2y, f xy = 2x(2y 2)e x2 +y 2 2y. H f (0, 1) = 4e 2 < 0. (0, 1) 14
(x) f x = y log(x 2 + y 2 ) + 2x2 y x 2 +y 2 = 0, f y = x log(x 2 + y 2 ) + 2xy2 x 2 +y 2 (x, y) = (±1, 0), (0, ±1), (1/ 2e, 1/ 2e), (1/ 2e, 1/ 2e), = 0 ( 1/ 2e, 1/ 2e), ( 1/ 2e, / 2e). f xx = 2x3 y+6xy 3 (x 2 +y 2 ) 2, f yy = 6x3 y+2xy 3 (x 2 +y 2 ) 2, f xy = log(x 2 + y 2 ) + 2(x4 +y 4 (x 2 +y 2 ) 2. H f (±1, 0) = 4 < 0, H f (0, ±1) = 4 < 0. (±1, 0), (0, ±1) H f (±1/ 2e, ±1/ 2e) = 4 > 0. f xx (±1/ 2e, ±1/ 2e) = 2 > 0 (±1/ 2e, ±1/ 2e) H f (±1/ 2e, 1/ 2e) = 4 > 0. f xx (±1/ 2e, 1/ 2e) = 2 < 0 (±1/ 2e, 1/ 2e) 15
III No y x y (i) xy 2 x 2 y = 16. (ii) x 3 + y 3 x y = 0. (iii) x 3 3axy + y 3 = 0 (a > 0). 16
III No (i) F = xy 2 x 2 y 16 = 0, F x = y 2 2xy = 0 (x, y) = (2, 4) y = f(x) f (2) = F xx(2, 4) F y (2, 4) = 2 3 > 0 y = f(x) x = 2 4 (ii) F = x 3 +y 3 x y = (x+y)(x 2 xy+y 2 1) = 0, F x = 3x 2 1 = 0 (x, y) = (1/ 3, 1/ 3), (1/ 3, 2/ 3), ( 1/ 3, 1/ 3), ( 1/ 3, 2/ 3) (1/ 3, 3), ( 1/ 3, 1/ 3) F y = 0 f (1/ 3) = F xx(1/ 3, 2/ 3) F y (1/ 3, 2/ 3) = 2/ 3 < 0 y x = 1/ 3 2/ 3 f ( 3) = F xx( 1/ 3, 2/ 3) F y ( 1/ 3, 2/ 3) = 2/ 3 > 0 y x = 1/ 3 2/ 3 (iii) F = x 3 3axy + y 3 = 0, F x = 3x 2 3ay = 0 (x, y) = (0, 0), (2 1/3 a, 2 2/3 a). F y = 0 F xx(2 1/3, 2 2/3 ) F y (2 1/3, 2 2/3 ) = 2 a < 0. x = 2 1/3 2 2/3 17
III No f(x, y)dxdy (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) [a,b] [c,d] d b c a 2 1 1 0 1 2 0 1 2 1 1 0 7 1 4 3 2 1 1 1 1 2 1 1 3 2 f(x, y)dxdy (1 + xy)dxdy (1 + xy)dydx dxdy dydx e u+v dudv e u+v dudv 1 1 π/2 π/2 0 0 2π π π 0 2 π 0 0 2y log xdydx (cos(x + t)dxdt (sin x + cos y)dxdy cos xdydx 18
(i) 7/4. (ii) 7/4. (iii) 1. (iv) 12. (v) e 3 e 2 e + 1. (vi) e 3 e 2 e + 1. (vii) 9 log 3 6. (viii) 0. (ix) 2π. (x) π sin 2. III No 19
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) 2 x 2 0 x 2 x 0 0 π x 0 0 1 x 2 0 x 2 1/y 1 0 1 y 0 1 y dydx. xydydx. x sin ydydx. (2x y)dydx. xe xy dxdy. y 2 xydxdy. 0 0 e log y 1 1 π cos y 0 0 x y 2 x 2 dxdy. e x dxdy. x sin ydxdy. III No 20
(i) 2/3. (ii) 2. (iii) π 2 /2 + 2. (iv) 1/10. (v) 1/2. (vi) 2/27. (vii) 1/12. (viii) e 2 /2 + e 1/2. (ix) 1/3. III No 21
III No f(x, y) = (x + y) p G = {(x, y); a x b, a y b} a > 0, p 2, p 1 G x + y = 2 y = x 2 I = y x2 dxdy π/2 0 cos 4 θdθ = 3 16 π G p > 2, p 1, G = {(x, y); 0 < x 1, 0 < y 1} (x + y) p dxdy G 22
III No (x + y) p dxdy = G = = = b a b a dx b a (x + y) p dy [ (x + y) p+1 b p + 1 ] y=b dx y=a 1 {(x + b) p+1 (x + a) p+1 }dx p + 1 a 1 (p + 1)(p + 2) {(2b)p+2 + (2a) p+2 2(a + b) p+2 }. I = = 2 3 = 2 3 1 dx 2 1 2 1 2 3/2 2 x x 2 y x2 dy (2 x x 2 ) 3/2 dx ( ( 3 2 )2 (x + 1 2 )2 ) 3/2 dx = 2 ( ( 3 ) 3/2 3 3/2 2 )2 t 2 dt (t = x + 1 2 ) = 2 ( ) 4 3 1 (1 s 2 ) 3/2 ds 3 2 1 = 2 ( ) 4 3 1 2 (1 s 2 ) 3/2 ds ( 3 2 0 = 4 ( ) 4 3 π/2 cos 4 θdθ (s = sin θ) 3 2 = 81 64 π. 0 G n = {(x, y); 1/n x 1, 1/n y 1} G (x + y) p 1 dxdy = G n (p + 1)(p + 2) n G (x + y)p dxdy = 23 { 2 p+2 + ( 2 n )p+2 2( n + 1 } n )p+2. 2p+2 2. (p+1)(p+2)