mugensho.dvi
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- みいか いのら
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1 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) lim = lim(t 1) = 0 (t 1) 3 (t 1) 2 t 1 (t 1) 2 t 1 (t 1) 3 = o((t 1) 2 ) (t 1) 1.3 f (x) x = a lim f (x) f (a) f (a)(x a) lim x a x a x a = 0 f (x) = f (a)(x a) + f (a) + o(x a) (x a) f (x) f (a) x a = f (a) f (x) f (a)(x a)+ f (a) f (x) { f (a)(x a)+ f (a)} x a y = f (a)(x a) + f (a) x = a y = f (x) x a y = f (x)
2 2 1.4 lim h 0 f (x) x = a lim f (a + h) f (a) f (a)h h = 0 h 0 f (a + h) f (a) h = f (a) h 0 f (a + h) f (a) f (a)h h f (a + h) = f (a) + f (a)h + o(h) (h 0) f (a + h) f (a) Ah = o(h) (h 0) h A 0 = lim h 0 f (a + h) f (a) Ah h A = f (a) f (x) x = a f (a + h) = f (a) + Ah + o(h) (h 0) ( f (a + h) f (a) = lim h 0 h ) A A A = f (a) y = f (x) x x y f (x + x) f (x) y y = f (x + ) f (x) ( 1 2)(3) a x h x f (x + x) = f (x) + f (x) x + o( x) ( x 0) y = f (x) x + o( x) ( x 0) y f (x) x y f (x) x x 1 y f (x) x f (x) (differential) dy, d f, d f (x) dy = d f = d f (x) = f (x) x f (x) = x f (x) = 1 dx = d f (x) = 1 x = x dx = x dy = d f = d f (x) = f (x)dx 1 x
3 3 y = f (x) dy = f (x)dx y f (x) x y f (x) x x lim x y f (x) x x = 0 f (x) = x 2 (differential) d f (x) = 2xdx 2 2 z = F(x, y) 1.4 F(x, y) (a, b) h, k A, B F(a + h, b + k) = F(a, b) + Ah + Bk + o( h 2 + k 2 ) ((h, k) (0, 0)) 1 A 2 A, B k = 0 F(a + h, b) = F(a, b) + Ah + o(h) (h 0) f (x) = F(x, b) f (a + h) = f (a) + Ah + o(h) (h 0) A = f (a) A F(x, y) y y = b F(x, b) x x = a (a, b) F(x, y) x F (a, b), F x(a, b) F(x, y) x x = a F(a, y) y y = b B (a, b) F(x, y) y F (a, b), F y(a, b) A = F (a, b), B = F (a, b) a x b y h x k y z = F(x + x, y + y) F(x, y) z = F x + F y + o(s) (s 0)
4 4 s = ( x) 2 + ( y) 2 F F = F (x, y), F = F (x, y) F(x, y) y x F(x, y) x x y F 2.1 y F(x, y) = x 2 y + 3x y F = 2xy + 3, F 1 z F dz, df, df(x, y) dz = df = df(x, y) = F F x + y = x2 1 x + F y z = F(x, y) 2 F(x, y) = x F = 1, F = 0 dx = df(x, y) = 1 x + 0 y = x dy = y dz = df = df(x, y) = F F dx + dy 2.2 z = F(x, y) = x 2 y dz = 2xydx + x 2 dy F(x, y) 3 dz = F F dx+ dy z F 3 2 x+ F y 3 F(x, y) (a, b) F x (x, y), F y (x, y) (2 ) (a, b) F(x, y)
5 5 u = F(x, y, z) du = df = df(x, y, z) = F F F dx + dy + z dz 2.1 x = ϕ(t) z = F(x, y) y = y(t) dz = z dx + z dy 3 u = F(x, y, z) du = u dx + u dy + u z z = F(ϕ(t), y(t)) t t x, y, z x, y, z s = ( x) 2 + ( y) 2 z = f (x + x, y + y) f (x, y) = f x (x, y) x + f y (x, y) y + o( s) t 0 x 0, y 0 s 0 o( s) t dz = o( s) s s t ( = o( s) ) 2 ( ) 2 ( ) 2 ( x y dx dy s t t = 0 ( t 0) z = f x (x, y) x + f y (x, y) y t t t f dx + f dy + o( s) t ) 2 t 0 dz =
6 6 3 x = x(t) y = y(t) xy C : F(x, y) = 0 1 x (t 0 ) y 0 P C (t 0 ) a = x(t 0 ), b = y(t 0 ) P(a, b) P P P C df = F dx + F dy 1 t F x (a, b)x (t 0 ) + F y (a, b)y (t 0 ) = 0 F x(a, b) (a, b) F y (a, b) F x(a, b) F y (a, b) F x (a, b)(x a) + F y (a, b)(y b) = 0 C : y = f (x) F(x, y) = y f (x) F x (x, y) = f (x), F y (x, y) = 1 f (a)(x a) + 1 (y b) = 0 y b = f (a)(x a) 3.1 F(x, y) = x 2 + 2xy + 2y 2 5 F x (x, y) = 2x + 2y, F y (x, y) = 2x + 4y C : x 2 + 2xy + 2y 2 5 = 0 (x 0, y 0 ) C (2x 0 + 2y 0 )(x x 0 ) + (2x 0 + 4y 0 )(y y 0 ) = 0 (x 0 + y 0 )(x x 0 ) + (x 0 + 2y 0 )(y y 0 ) = 0
7 7 xyz S : F(x, y, z) = 0 2 F(x, y, z) x x(t) P(a, b, c) S P S ( ) l : y = y(t) z z(t) a = x(t 0 ), b = y(t 0 ), c = z(t 0 ) df = F dx + F dy + F z 2 t dz F x (a, b, c)x (t 0 ) + F y (a, b, c)y (t 0 ) + F z (a, b, c)z (t 0 ) = 0 F x (a, b, c) x (t 0 ) x (t 0 ) F y (a, b, c) y (t 0 ) y (t 0 ) P l F z (a, b, c) z (t 0 ) z (t 0 ) F x (a, b, c) F y (a, b, c) P π l F z (a, b, c) l P S π P S P π P S F x (a, b, c)(x a) + F y (a, b, c)(y b) + F z (a, b, c)(z c) = 0 S : z = f (x, y) F(x, y, z) = z f (x, y) F x (x, y, z) = f x (x, y), F y (x, y, z) = f y (x, y), F z (x, y, z) = 1 f x (a, b)(x a) f y (a, b)(y b) + 1 (z c) = 0 z c = f x (a, b)(x a) + f y (a, b)(y b) 3.2 K : z = x 2 + y 2 A(x 0, y 0, z 0 ) K f (x, y) = x 2 + y 2 f x (x, y) = 2x, f y (x, y) = 2y z z 0 = 2x 0 (x x 0 ) + 2y 0 (y y 0 )
8 8 z = f (x, y) dz = f x (x, y)dx + f y (x, y)dy z c = f x (a, b)(x a) + f y (a, b)(y b) + o( (x a) 2 + (y b) 2 ) ((x, y) (a, b) c = f (a, b) (x, y) (a, b) z f x (a, b)(x a) + f y (a, b)(y b) + f (a, b) z = f x (a, b)(x a) + f y (a, b)(y b) + f (a, b) (a, b, f (a, b)) (x, y, z) (a, b, f (a, b)) z = f (x, y) 4 xy C t : f (x, y, t) = 0 E {C t } (i) E C t (ii) E C t E C t : f (x, y, t) = 0 f (x, y, t) = 0 1 E (x, y) f t (x, y, t) = 0 2 E C t P(x, y) C t 1 x = x(t), y = y(t) t P E E 1 f (x(t), y(t), t) = 0 t f x (x(t), y(t), t) x (t) + f y (x(t), y(t), t) y (t) + f t (x(t), y(t), t) 1 = 0 3 x = x(t) E P E y = y(t) x (t) C y t f (x, y, t) = 0 (t ) (t)
9 9 f x (x(t), y(t), t) (x x(t)) + f y (x(t), y(t), t) (y y(t)) = 0 f x(x(t), y(t), t) f y (x(t), y(t), t) x (t) f x(x(t), y(t), t) y (t) f y (x(t), y(t), t) f x (x(t), y(t), t) x (t) + f y (x(t), y(t), t) y (t) = f t (x(t), y(t), t) = 0 2 {C t } t t 1 2 C : f (x, y) = 0 (a, b) f x (a, b) = f y (a, b) = 0 (a, b) C 4.2 f (x, y, t) = 0 1 C t : f (x, y, t) = 0 (x, y) x(t), y(t) f t (x, y, t) = 0 2 x (t) 2 + y (t) 2 0 x = x(t), y = y(t) 3 x = x(t) C : {C t } C t y = y(t) C t C 3 1 f (x(t), y(t), t) = 0 4 C P(x(t), y(t)) C t 4 f x (x(t), y(t), t)x (t) + f y (x(t), y(t), t)y (t) + f t (x(t), y(t), t) = f t (x(t), y(t), t) = 0 6
10 f x (x(t), y(t), t)x (t) + f y (x(t), y(t), t)y (t) = 0 f x(x(t), y(t), t) x (t) f y (x(t), y(t), t) y 7 (t) x (t) 0 C P y (t) f x(x(t), y(t), t) = 0 P C t f y (x(t), y(t), t) f x(x(t), y(t), t) 0 C t P f y (x(t), y(t), t) 7 C C t P C C t 4.1 l θ : (cos θ)x + (sin θ)y = 1 (0 < = θ < = π) 1 C 1 θ (sin θ)x + (cos θ)y = cos θ 2 sin θ x = cos θ 1 sin θ + 2 cos θ y = sin θ l θ x = cos θ C : (0 < y = sin θ = θ < = π)
11 C t : x 4 x 2 +(y t) 2 = 0 f (x, y, t) = x 4 x 2 +(y t) 2 f t (x, y, t) = 2(y t) f (x, y, t) = 0 f t (x, y, t) = 0 x(t) = 0 x(t) = 1 x(t) = 1 t y(t) = t y(t) = t y(t) = t x = 0, x = 1, x = 1 x = 1, x = 1 f x (x, y, t) = 4x 3 2x f y (x, y, t) = 2(y t) f x (0, t, t) = 0 f y (0, t, t) = 0 (0, t) C t x = f (x, y, t) = 0 1 f t (x, y, t) = 0 2 (x(t), y(t)) C x (t) = 0 y (t) = 0 t ( ) 4.3 C t : y = t(x 1) f (x, y, t) = t(x 1) y f t (x, y, t) = x(t) = 1 x 1 f (x, y, t) = 0 f t (x, y, t) = 0 y(t) = 0 1 (1, 0) C t
.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,
.1.1 Y K L Y = K 1 3 L 3 L K K (K + ) 1 1 3 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 1 3 3 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,
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i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
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
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
v er.1/ c /(21)
12 -- 1 1 2009 1 17 1-1 1-2 1-3 1-4 2 2 2 1-5 1 1-6 1 1-7 1-1 1-2 1-3 1-4 1-5 1-6 1-7 c 2011 1/(21) 12 -- 1 -- 1 1--1 1--1--1 1 2009 1 n n α { n } α α { n } lim n = α, n α n n ε n > N n α < ε N {1, 1,
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
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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
DVIOUT
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)
B 38 1 (x, y), (x, y, z) (x 1, x 2 ) (x 1, x 2, x 3 ) 2 : x 2 + y 2 = 1. (parameter) x = cos t, y = sin t. y = f(x) r(t) = (x(t), y(t), z(t)), a t b.
2009 7 9 1 2 2 2 3 6 4 9 5 14 6 18 7 23 8 25 9 26 10 29 11 32 12 35 A 37 1 B 38 1 (x, y), (x, y, z) (x 1, x 2 ) (x 1, x 2, x 3 ) 2 : x 2 + y 2 = 1. (parameter) x = cos t, y = sin t. y = f(x) r(t) = (x(t),
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) = (, )
d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r
![d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r](/thumbs/85/92098187.jpg "d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r")
2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t)
() 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 ()
x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)
2011 I 2 II III 17, 18, 19 7 7 1 2 2 2 1 2 1 1 1.1.............................. 2 1.2 : 1.................... 4 1.2.1 2............................... 5 1.3 : 2.................... 5 1.3.1 2.....................................
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..........................................
1/1 lim f(x, y) (x,y) (a,b) ( ) ( ) lim limf(x, y) lim lim f(x, y) x a y b y b x a ( ) ( ) xy x lim lim lim lim x y x y x + y y x x + y x x lim x x 1
1/5 ( ) Taylor ( 7.1) (x, y) f(x, y) f(x, y) x + y, xy, e x y,... 1 R {(x, y) x, y R} f(x, y) x y,xy e y log x,... R {(x, y, z) (x, y),z f(x, y)} R 3 z 1 (x + y ) z ax + by + c x 1 z ax + by + c y x +
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,
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
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 ),
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
(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)
Microsoft Word - 触ってみよう、Maximaに2.doc
i i e! ( x +1) 2 3 ( 2x + 3)! ( x + 1) 3 ( a + b) 5 2 2 2 2! 3! 5! 7 2 x! 3x! 1 = 0 ",! " >!!! # 2x + 4y = 30 "! x + y = 12 sin x lim x!0 x x n! # $ & 1 lim 1 + ('% " n 1 1 lim lim x!+0 x x"!0 x log x
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
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
.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(
06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin
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n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
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II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2
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