.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0,
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1 .1.1 Y K L Y = K 1 3 L 3 L K K (K + ) L 3 K 3 L 3 K 0 (K + K) 1 3 L 3 K 1 3 L 3 lim K 0 K = L (K + K) 1 3 K lim K 0 K = 1 3 K 3 L 3 z = f(x, y) x y z x-y-z.1 z = e x +xy y 3 x-y ( ) z 0 f(x, y) = z 0 (x, y) ( ) y = f(x, y) x-y 1. z = e x +xy y.1 z = x +y z = 1 z = f(x, y)
2 .1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0, 0, 0) (1, 0, ) (0, 1, 1) n f(x 1, x,..., x n ) n x 1, x,..., x n n n 3 () 3 n (hyperplane) a 1, a,..., a n, b a 1 x 1 + a x + + a n x n = b (.) 0
3 . z = e x +xy y y x 1 1 f(x) df(x) dx = lim f(x + h) f(x) h 0 h f(x, y) (x 0, y 0 ) (λ, µ) t f(x, y) g(t) g(t) f(x 0 + λt, y 0 + µt) g(t) t = 0 g(0 + h) g(0) f(x 0 + λh, y 0 + µh) f(x 0, y 0 ) lim = lim h 0 h h 0 h f(x, y) (x 0, y 0 ) (λ, µ) (x 0, y 0 ) (λ, µ) ( ).1 f(x, y) = x + y (, ) (1, 1) ( + h) + ( + h) ( + ) h + 8h lim = lim h 0 h h 0 h = lim h + 8 = 8 h 0..1 (1, 1) (1, 1) 1
4 (1, 0) (x 0, y 0 ) f(x 0 + h, y 0 ) f(x 0, y 0 ) lim h 0 h f(x, y) (x 0, y 0 ) x x f(x, y) x f(x, y) x z = f(x, y) x f(x, y) x z f(x, y) x x f x (x, y) (0, 1) y.1 F K F L Y K F K(K, L) Y L F L(K, L).3 f(x, y) = x y f(x, y) x y.3 f x = 4xy f y = x i) x + 3y + 4 ii) (x + y) 3 iii) e x +xy+y iv) x y
5 1 n 1 f(x) x = x 0 f(x 0 + h) f(x 0 ) lim = c h 0 h c f (x 0 ) = c 9 h 0 o(h) h o(h) f(x 0 + h) f(x 0 ) ch (.3) 0 c f(x) x = x 0 f(x) = f(x 0 ) + f (x 0 )(x x 0 ) + 1 f (x 0 )(x 0 x) + x = x 0 + h f(x 0 + h) = f(x 0 ) + f (x 0 )h + 1 f (x 0 )h + (.3) o(h) h h 0 o(h) h 0 f(x, y) (x 0, y 0 ) (u, v) a = (h, k) o(h, k) o(h, k) f(x 0 + h, y 0 + k) f(x 0, y 0 ) (uh + vk) (.4) o(h, k) a 0 0 a a = h + k f(x 0 + h, y 0 + k) f(x 0, y 0 ) (uh + vk) lim = 0 (.5) a 0 a a = (h, k) k = 0 ( u = f x (x 0, y 0 ) f ) x (x 0, y 0 ) v = f y (x 0, y 0 ) ( f ) y (x 0, y 0 ) 3
6 f(x, y) (x 0, y 0 ) (x 0, y 0 ) f(x, y) z = f(x, y) (x 0, y 0 ) z = ux + vy + α = f x (x 0, y 0 )x + f y (x 0, y 0 )y + α α = f(x 0, y 0 ) f x (x 0, y 0 )x 0 f y (x 0, y 0 )y 0 (.6) (.6) z = f(x, y) (x 0, y 0 ).4 f(x, y) = 0.5x + y 1 (, 1) f x (, 1) = 0.5 f y (1, 1) = 1 (.6) α = = 1 z = 0.5x + y 1.5 f(x, y) = 1 8 x + 1 y z = f(x, y) (, 1) f x (, 1) = 1 8 = 0.5 f y (, 1) = 1 1 = 1 α = = 1 z = 0.5x + y (1, 1).1 f(x, y) i) (λ, µ) f(x, y) (λ, µ) ii) f(x, y) x y f(x 0 + λh, y 0 + µh) f(x 0, y 0 ) lim = λu + µv h 0 h u = x f(x 0, y 0 ) v = y f(x 0, y 0 ) 4
7 .3 z y 1 x z y 1 x 5
8 .1 f(x, y) x y.6 f(x, y) = { xy, x +y (x, y) (0, 0) 0, (x, y) = (0, 0) f(x, 0) = 0 f(x, y) x 0 f x (x, 0) = 0 f y (0, y) = 0 f x (0, 0) = f y (0, 0) = 0 (0, 0) (λ, µ) ( ) λµh f(0 + λh, 0 + µh) f(0, 0) h = 1 h (λ + µ f(0, 0) )h λµ = 1 h (λ + µ ) λ µ 0 h 0 h = x k = y f = f(x 0 + h, y 0 + k) f(x 0, y 0 ) (.4) f = u x + v y + o(h, k) = f x (x 0, y 0 ) x + f y (x 0, y 0 ) y + o(h, k) (.7) (.7) x x y y f f x y 1 a = (h, k) a 0 o(h,k) a 0 df = f x (x 0, y 0 )dx + f y (x 0, y 0 )dy (x 0, y 0 ) f.7.1 df = f x (x, y)dx + f y (x, y)dy (.8) Y K = 1 3 K 3 L 3 dy = 1 3 K 3 L 3 dk + 3 K 1 3 L 1 3 dl Y L = 3 K 1 3 L 1 3 6
9 c 1 c U(c 1, c ) = 1 γ cγ 1 + α 1 γ cγ f(x, y) x f(x,y) x x y f(x, y) x x f(x,y) f x xx f(x, y) x x ( ) f(x, y) f(x, y) x y f(x,y) x y ( ) f(x, y) f(x,y) y x f(x, y) x y y x x f yx, f(x,y) f y yy.8 f(x, y) = x e y 1 f(x, y) x = xe y f(x, y) y = x e y f(x, y) x = e y f(x, y) x y = xe y f(x, y) y = x e y f(x, y) = xe y y x f xy. f(x, y) f(x,y) y f(x,y) x y f(x,y) x y f(x,y) y x f(x, y) y x = f(x, y) x y..6 f(x, y) = x 4 y 3 + xy f xy = f yx.7 f(x, y) = x e x y 7
10 n f(x 1, x,..., x n ) (.9) (a 1, a,..., a n ) x i lim h 0 f(a 1, a,..., a i + h,..., a n ) f(a 1, a,..., a i,..., a n ), i = 1,..., n (.9) h y = f(x 1,..., x n ) x i f(x 1,..., x n ) x i, x i f(x 1,..., x n ), y x i, f xi (x 1,..., x n ), f i (x 1,..., x n ), (.10) y = f(x 1, x,..., x n ) (a 1, a,..., a n ) y = f 1 (a 1,..., a n )x 1 + f (a 1,..., a n )x + + f n (a 1,..., a n )x n + c c = f(a 1,..., a n ) f 1 (a 1,..., a n )a 1 f (a 1,..., a n )a f n (a 1,..., a n )a n (.11) f j (a 1,..., a n ) = f x j (a 1,..., a n ) df = f 1 (x 1,..., x n )dx 1 + f (x 1,..., x n )dx + + f n (x 1,..., x n )dx n (.1) f(x 1,..., x n ) x i x j i j f(x 1,..., x n ) ( ) f(x1,..., x n ) x i x j x j x i i = j f(x 1,..., x n ) x i f xi x j k k f(x 1,...,x n) x k 1 1 xk xk n n ( ) f(x1,..., x n ) x i x i k 1 + k + + k n = k.8 f(x, y, z) = x + xy + xz + yz 8
11 . z x y z = f(x, y) x y t x = g(t) y = h(t) z t f(x, y) g(t) h(t) t f(g(t), h(t)) t dz dt = f dg x dt + f dh y dt (.13).9 z = x + xy y = e x dz dx z x y = e x z = x + xe x dz dx = x + e x + ( 1)xe x = x + ( x + 1)e x f(x, y) = x + xy, h(x) = e x dz dx = f x + f dh y dx = x + y + x( e x ) = x + ( x + 1)e x.10 G 1 G p 1 p G 1 p 1 p I c(p 1, p, I) I M(p 1, p, u) p 1 h(p 1, p, u) = c ( p 1, p, M(p 1, p, u) ) h(p 1, p, u) p 1 h p 1 h p 1 = c p 1 + c I M p 1.9 f(x, y) = (x 4 + x y + xy) 3 x = t + 1 y = t + 4 df dt 9
12 . K L F(K, L) F(K, L) = 5K 0.4 L 0.6 (.14) L L + L F(K + K, L + L) = F(K, L) K K L Y = F(K, L) K L dk dl dy = F K dk + F L dl dy = 0 dk dl = F L F K F K (K, L) = K 0.6 L 0.6, F L (K, L) = 3K 0.4 L 0.4 dk dl = F L(K, L) F K (K, L) = 1.5K L L ɛ K 1.5 K L ɛ x y f y = f(x) ( ) dy dx x y F(x, y) = 0 x + y 1 = 0 (.15) x + y e xy = 0 (.15') y = f(x) (.15) (.15') F(x, y) dy dx 30
13 F(K,L)= K 4 3 F(K,L)=.5 1 F(K,L)= L.3 ( ) F(x, y) (x 0, y 0 ) F(x 0, y 0 ) = 0 F y (x 0, y 0 ) 0 x = x 0 i) F(x, f(x)) = 0 ii) y 0 = f(x 0 ) f(x) y = f(x) dy dx = F x(x, y) F y (x, y).11 F(x, y) = x + y e xy dy dx (1, 0) F(1, 0) = 0 F x (x, y) = 1 ye xy F y (x, y) = 1 xe xy dy yexy = 1 dx 1 xe xy 31
14 .10 F(x, y) = 3x + 4xy + y 9 F(x, y) = 0 y = f(x) dy dx.3 y = e x t + t 1 x t t f(x, y, t) t f(x, y, t) = 0 (x, y) x{y t.1 f(x, y, t) = x (t 1)x y+t t 0, 1,, 3, 4 f(x, y, t) = x (t 1)x y + t = y 8 t=4 6 t=3 4 t= t=1 0 t= x t C C 3
15 C C.6.6 x (t 1)x y + t = 0 C y 4 0 C x t t t (x(t), y(t)) (.16) t f(x(t), y(t), t) = 0 (.16) d dt f(x(t), y(t), t) = f x(x(t), y(t), t) dx(t) + f y (x(t), y(t), t) dy(t) + f t (x(t), y(t), t) (.17) dt dt (x(t), y(t)) C dy(t) dt / dx(t) dt dy dx = lim y x 0 x = lim y(t + t) y(t) t 0 x(t + t) x(t) y(t + t) y(t) t = lim t 0 t x(t + t) x(t) = dy(t) dt /dx(t) dt (.18) f(x, y, t) = 0 x y (x(t), y(t)) dy dx = f x(x(t), y(t), t) f y (x(t), y(t), t) (.19) 33
16 C C (.18) (.19) f x (x(t), y(t), t) dx(t) dt (.17) (.0) + f y (x(t), y(t), t) dy(t) dt (.16) (.1) = 0 (.0) f t (x(t), y(t), t) = 0 (.1).3 f(x, y, t) = e x t + t y (.16) (.1) e x(t) t + t y(t) 1 = 0 e x(t) t + 1 = 0 t y = x C
mugensho.dvi
1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -
M3............................................................................................ 3.3................................................... 3 6........................................... 6..........................................
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
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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
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 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 0 f(a + h, b) f(a, b) h............................................................... ( ) f(x, y) (a, b) x A (a, b) x
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
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B p.1/14 f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) B p.1/14 f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) f(x 1,...,x n ) (x 1 x 0,...,x n 0), (x 1,...,x n ) i x i f xi
1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
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 ),
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)
I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
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i 6 3 ii 3 7 8 9 3 6 iii 5 8 5 3 7 8 v...................................................... 5.3....................... 7 3........................ 3.................3.......................... 8 3 35
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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....................................
CALCULUS II (Hiroshi SUZUKI ) f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b)
CALCULUS II (Hiroshi SUZUKI ) 16 1 1 1.1 1.1 f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b) lim f(x, y) = lim f(x, y) = lim f(x, y) = c. x a, y b
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A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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x ( ) x dx = ax 1 dx = a x log x = at + c x(t) = e at C (C = e c ) a > 0 t a < 0 t 0 (at + b ) h dx = lim x(t + h) x(t) h 0 h x(t + h) x(t) h x(t) t x(t + h) x(t) ax(t) h x(t + h) x(t) + ahx(t) 0, h, 2h,
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7 b f n f} d = b f n f d,. 5,. [ ] ɛ >, n ɛ + + n < ɛ. m. n m log + < n m. n lim sin kπ sin kπ } k π sin = n n n. k= 4 f, y = r + s, y = rs f rs = f + r + sf y + rsf yy + f y. f = f =, f = sin. 5 f f =.
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
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