III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y
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1 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 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)},
2 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
3 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
4 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
5 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
6 (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/ /
7 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
8 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 sin θ cos θ 2 z x y + sin2 θ 2 z y 2. 8
9 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
10 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
11 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 = (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
12 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 (v) f(x, y) = 1 x 2 + y (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
13 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)
14 (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
15 (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
16 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
17 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
18 III No f(x, y)dxdy (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) [a,b] [c,d] d b c a 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/ π π π 0 2 π 0 0 2y log xdydx (cos(x + t)dxdt (sin x + cos y)dxdy cos xdydx 18
19 (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
20 (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) 2 x 2 0 x 2 x 0 0 π x x 2 0 x 2 1/y 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
21 (i) 2/3. (ii) 2. (iii) π 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
22 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
23 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 = dx /2 2 x x 2 y x2 dy (2 x x 2 ) 3/2 dx ( ( 3 2 )2 (x )2 ) 3/2 dx = 2 ( ( 3 ) 3/2 3 3/2 2 )2 t 2 dt (t = x ) = 2 ( ) (1 s 2 ) 3/2 ds = 2 ( ) (1 s 2 ) 3/2 ds ( = 4 ( ) 4 3 π/2 cos 4 θdθ (s = sin θ) 3 2 = 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)
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
40 6 y mx x, y 0, 0 x 0. x,y 0,0 y x + y x 0 mx x + mx m + m m 7 sin y x, x x sin y x x. x sin y x,y 0,0 x 0. 8 x r cos θ y r sin θ x, y 0, 0, r 0. x,
9.. x + y + 0. x,y, x,y, x r cos θ y r sin θ xy x y x,y 0,0 4. x, y 0, 0, r 0. xy x + y r 0 r cos θ sin θ r cos θ sin θ θ 4 y mx x, y 0, 0 x 0. x,y 0,0 x x + y x 0 x x + mx + m m x r cos θ 5 x, y 0, 0,
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
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1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y
No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)
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3............................... 3................................ 5.3 Fubini........................... 7.4.............................5..........................6.............................. 3.7..............................
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20010916 22;1017;23;20020108;15;20; 1 N = {1, 2, } Z + = {0, 1, 2, } Z = {0, ±1, ±2, } Q = { p p Z, q N} R = { lim a q n n a n Q, n N; sup a n < } R + = {x R x 0} n = {a + b 1 a, b R} u, v 1 R 2 2 R 3
1... 1 2... 1 1... 1 2... 2 3... 2 4... 4 5... 4 6... 4 7... 22 8... 22 3... 22 1... 22 2... 23 3... 23 4... 24 5... 24 6... 25 7... 31 8... 32 9... 3
3 2620149 3 6 3 2 198812 21/ 198812 21 1 3 4 5 JISJIS X 0208 : 1997 JIS 4 JIS X 0213:2004 http://www.pref.hiroshima.lg.jp/site/monjokan/ 1... 1 2... 1 1... 1 2... 2 3... 2 4... 4 5... 4 6... 4 7... 22
() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
![() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y](/thumbs/85/92979039.jpg "() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y")
5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
f (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x >
5.1 1. x = a f (x) a x h f (a + h) f (a) h (5.1) h 0 f (x) x = a f +(a) f (a + h) f (a) = lim h +0 h (5.2) x h h 0 f (a) f (a + h) f (a) f (a h) f (a) = lim = lim h 0 h h 0 h (5.3) f (x) x = a f (a) =
2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) Minoru TANAKA (Osaka Univ.) I(2011), Sec p. 1/30
2.4 ( ) 2.4.1 ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) I(2011), Sec. 2. 4 p. 1/30 (2) Γ f dr lim f i r i. r i 0 i f i i f r i i i+1 (1) n i r i (3) F dr = lim F i n i r i. Γ r i 0 i n i
f : R R f(x, y) = x + y axy f = 0, x + y axy = 0 y 直線 x+y+a=0 に漸近し 原点で交叉する美しい形をしている x +y axy=0 X+Y+a=0 o x t x = at 1 + t, y = at (a > 0) 1 + t f(x, y
017 8 10 f : R R f(x) = x n + x n 1 + 1, f(x) = sin 1, log x x n m :f : R n R m z = f(x, y) R R R R, R R R n R m R n R m R n R m f : R R f (x) = lim h 0 f(x + h) f(x) h f : R n R m m n M Jacobi( ) m n
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39 3 3.1 2 Ax 1,y 1 Bx 2,y 2 x y fx, y z fx, y x 1,y 1, 0 x 1,y 1,fx 1,y 1 x 2,y 2, 0 x 2,y 2,fx 2,y 2 A s I fx, yds lim fx i,y i Δs. 3.1.1 Δs 0 x i,y i N Δs 1 I lim Δx 2 +Δy 2 0 x 1 fx i,y i Δx i 2 +Δy
I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n
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i 18 2H 2 + O 2 2H 2 + ( ) 3K ii 1 1 1.1.................................. 1 1.2........................................ 3 1.3......................................... 3 1.4....................................
1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ
1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c
1 2 1 No p. 111 p , 4, 2, f (x, y) = x2 y x 4 + y. 2 (1) y = mx (x, y) (0, 0) f (x, y). m. (2) y = ax 2 (x, y) (0, 0) f (x,
No... p. p. 3, 4,, 5.... f (, y) y 4 + y. () y m (, y) (, ) f (, y). m. () y a (, y) (, ) f (, y). a. (3) lim f (, y). (,y) (,)... (, y) (, ). () f (, y) a + by, a, b. + y () f (, y) 4 + y + y 3 + y..3.
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i ii iii iv v vi vii viii ix - 1 - - 2 - - 3 - - 4 - - 5 - - 6 - - 7 - - 8 - - 9 - 玢 - 10 - - 11 - - 12 - - 13 - - 14 - - 15 - - 16 - - 17 - - 18 - - 19 - - 20 - - 21 - - 22 - - 23 - - 24 - - 25 - -
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a
9 203 6 7 WWW http://www.math.meiji.ac.jp/~mk/lectue/tahensuu-203/ 2 8 8 7. 7 7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa,
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +