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
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1 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 = axy < 0 dy = ay(x b a ) 1 = 100 dy > 0 y x < b a = 100
2 dy < 0 y x x = 50 y 0 x = 1000 x = 100 y 0 dy = 0 x = y = 0 x = 0 dx = No.3 4 B 1 x (1 + x) n. = 1 + nx T dt m dm k = 4π 2 m T 2 k + dk = 4π 2 m + dm (T + dt ) 2 = 4π 2 m T dm m (1 + dt T )2. = 4π 2 m T 2 (1 + dm m )(1 2dT T ). = 4π 2 m T 2 (1 + dm m 2dT T ) dm m = ±0.1% dt T = ±0.2% k = 0.5% 2 k = 4π 2 m T 2 1 log k = log(4π 2 ) + log m 2 log T dk k = dm m 2 log dt T = 0.5% 2
3 x No.4 5 B 1 K dk = 0 W Q(K K 0 ) = 0 K = K 0 + W Q = = 0.173% 2 dk = W V Q(K K 0) = Q V V ( K K 0 W ) Q K K 0 W Q = F K 0 W Q dk = df df = Q V F F (t) = A exp ( QV ) t K(t) = A exp ( QV ) t + K 0 + W Q 0 0 No.5 2 C t = 0 T dx dx x = ax bxy = (a by) T 0 dx x = log x(t ) log x(0) = T 0 T (a by) = at b y(t) 0 3
4 x(t ) = x(0) T 0 y(t) = T Y at bt Y = 0 Y = a b 2 dy y = ( c + dx) log y(t ) log y(0) = 0 = ct + dxt = 0 X = c d x y dx = ax bxy rx = (a r)x bxy dy = cy + dxy ry = (c + r)y + dxy X = c + r d > X Y = a r b No.1 x y < Y No.6 3 C y T dy = ky y(0) = y y(t) = A exp( kt) y(t) = y exp( kt) y(t ) = y exp( kt ) y y exp( kt ) + y 0 = y y = y 0 1 exp( kt ) 4
5 No.7 5 A 1 2 ẍ = rω 2 sin(ωt) ÿ = rω 2 cos(ωt) x sin ωt = ±1, cos ωt = 0 y = r B y cos ωt = ±1, sin ωt = 0 y = 0, 2r A C ẍ 2 + ÿ 2 = r 2 ω rω 2 x x B y y A C 2 2 No.8 1 B Taylor y(x + h) = y(x) + hy (x) + h2 2 y (ξ) h 2 Y i+1 = Y i + h(x i + Y i ) = 1.1Y i i = 1.1Y i i 5
6 x i = 0.1iY 0 = 1 Y 1 = = 1.1 Y 2 = = 1.22 Y 3 = = Y 4 = = y = 2e e = 2e Taylor No.9 2 B dx = f(x) > 0 x f(x) f(x) = 0 0 < x < x 0 x = x 0 f(x) No.10 1 B 1 dp 0 (t) = ap 0 (t) < 0 (P 0 (t) P 0 (t) dp 1 (t) = a(p 0 (t) P 1 (t)) P 0 > P 1 P 0 P 1 dp 1(t) = 0 2 dp 0 (t) = ap 0 P 0 (0) = 1 P 0 (t) = e at 6
7 dp 1 (t) = ap 1 (t) + ae at dp 1 (t) = ap 1 (t) P 1 (t) = Ce at C C(t) P 1 (t) = C(t)e at dp 1 (t) dp 1 (t) = ap 1 (t) + ae at = C (t)e at ac(t)e at = ac(t)e at + ae at C (t) = a C(t) = at + C 1 P 1 (0) = 0 C 1 = 0 P 1 (t) = ate at No.11 2 A x x = 0 d 2 y dx 2 = e ax dy dx = 1 a e ax + C 1 2 = 1 a + C 1 C 1 = a x dy dx = 1 a e ax a y = 1 ( a 2 e ax ) x + C 2 a 7
8 x a = 0 a = 1 2 y(x) 2 e ax x dy = 0 a = 2 dx No.12 4 B 1 F df dx df = f = f 0 + av 3 df dx = df d dv 1 dx = f 0 + av 3 V = f 0 V + av 2 ( ) df = f 0 dx V 2 + 2aV = 0 V = 3 f0 2a 2 t F F = (f 0 + av 3 )t L L = V t Lagrange g g(t, V ) = V t + k((f 0 + av 3 )t F ) g V = t + 3kaV 2 t = 0 g t = V + k(f 0 + av 3 ) = 0 2 t V = 3kaV 2 t k(f 0 + av 3 ) f 0 + av 3 = 3aV 3 8
9 V = 3 f0 2a Lagrange H.7 No.13 3 A x y Lagrange f(x, y) = x 3 + y 3 + k(x 2 + y 2 3) f x = 3x2 + 2kx = 0 f y = 3y2 + 2ky = 0 x = 0 y = 0 x = y 2 x = 0 x = y x = 0 y = 3 x = y = x 3 + y 3 = x 3 + y 3 = No.14 1 A 1 xy P (r cos θ, r sin θ) = (aωt cos ωt, aωt sin ωt) v x = dx = aω cos ωt aω2 t sin ωt v y = dy = aω sin ωt + aω2 t cos ωt 9
10 v = v 2 x + v 2 y = a 2 ω 2 + a 2 ω 4 t 2 = aω 1 + ω 2 t 2 2 v 2 = v r = dr = aω v θ = r dθ = aω2 t v 2 r + v 2 θ = aω 1 + ω 2 t 2 IV No.15 2 C g(x) = sin ωx d 2 g(x) dx 2 + g(x) = 0 ω = 0 ω = 1 g(x) = A sin x + B cos x f(x) = A sin x + B cos x 1 sin 2x 3 f(0) = 0 B = 0 f (x) = A cos x 2 cos 2x 3 f (0) = A 2 3 = 1 3 A = 1 f(x) = sin x 1 3 sin 2x = sin x 2 3 sin x cos x = sin x ( cos x ) f(x) = 0 x x = 0, ±π, ±2π 2 10
11 2 2 1 f(x) = 1 sin 2x 3 f(x) = C sin 2x g(x) g(x) = e kx k g(x) = sin ωx 2 No.16 5 B 1 2 x 3 (4m + 1)x 2 + 2(m + 3)x + 2(m 3) = x 3 x 2 + 6x 6 2m(2x 2 x 1) = (x 1)(x m(2x + 1)) x = 1 2 x 2 4mx + 6 2m = 0 x = 1 2 f(x) = x 2 4mx + 6 2m 2 D/4 = (2m) 2 (6 2m) = 4m 2 + 2m 6 = (2m + 3)(m 1) > 0 m < 3 2, m > 1 4m > 0, 6 2m > 0 0 < m < 3 x = 1 f(1) = 7 6m 0 1 < m < 7 6, 7 6 < m <
12 x = 1 m = x3 x 2 + 6x 6 4x 2 2x + 6 = (x2 + 6)(x 1) 2(2x + 1)(x 1) = x (2x + 1) y = g(x) = x (2x + 1) y = m x = 1 x > 0 2 y = g(x) x = 1 2 x = No.17 3 A ϕ x = x 2 + y 2 (2x) = x x 2 + y 2 1 x y 2 ϕ x 2 = x2 + y 2 x 2x (x 2 + y 2 ) 2 = x2 + y 2 (x 2 + y 2 ) 2 2 ϕ y 2 = y2 + x 2 (x 2 + y 2 ) 2 2 ϕ x ϕ y 2 = 0 i z = x + iy z = x iy f(z) f(z) 2 ϕ x ϕ y 2 = 0 ϕ(x, y) = 1 2 ln zz = 1 2 ln z ln z No.18 5 A 1 r V V = 4 3 πr3 12
13 r = 4 dr = 3 dv = 4πr2 dr dv = 4π 42 3 = 192π 2 r V r = 1 + 3t V = 4 3 π(1 + 3t)3 = 4π 3 (27t3 + 27t 2 + 9t + 1) dv = 4π 3 (81t2 + 54t + 9) = 4π(27t t + 3) t = 1 dv = 192π 1 2 II I No.19 5 A 1 (x a) 2 + (y a) 2 = r 2 x 2(x a) + 2(y a) dy dx = 0 x = 1 y = 3 dy dx = 3 2(1 a) + 2(3 a) 3 = 0 a = (x, y) = (1, 3) (1 a) 2 + (3 a) 2 = r 2 y y > 0 ) y a = r 2 (x a) 2 x dy dx = (x a) r2 (x a) 2 13
14 (x, y) = (1, 3) dy dx = 3 3 = (1 a) (1 a) (1 a) = = r2 (1 a) 2 3 a 3 a a < 3 1 a = x 2 + y 2 = r 2 3 y = x ±3r r (x, y) = (, ) x y a (1, 3) (y > x ) 3r r + a = 1, + a = r 4a = 10 a = No.20 1 A t > 0 r t + r2 t 2 t r2 t = 2r e x + 1 e x 2 e x 1 e x = 2 e y + 9 e y 2 e y 9 e y = 6 ( e x + 1 ) (e y e x + 9e ) y 12 2 x y No.21 3 A 14
15 h V = πr 2 h h = V πr 2 S = 2πrh + 2πr 2 = 2V r + 2πr2 ds dr = 2V r 2 + 4πr = 0 r = ( ) 1 V 3 2π II 1 No.22 3 A 1 f(a) = g(a) = 0 g(x) lim x a f(x) = lim g (x) x a f (x) d dx ( x 3) = x d dx ( 1 9 x 3) = 2 9 x 9 + x 3 9 x lim = lim = 1 x 0 9 x 3 x x 2 x (1 + x) n. = 1 + nx 9 + x 3 = x 9 3 =. ( x ) 3 = x x 3 = 3 1 x 9 3 =. ( 3 1 x ) 3 = x x 3 lim = 1 x 0 9 x 3 15
16 2 1 No.23 4 A 1 y = 3 4 x ( x 2 + y 2 = x x + 5 ) 2 2 = x x = ( x 6 5) f(x, y) = x 2 + y 2 + k(3x + 4y 10) f = 2x + 3k = 0 x f = 2y + 4k = 0 y x : y = 3 : 4 x = 3t, y = 4t 9t + 16t = 25t = 10 t = 2 5 x 2 + y 2 = 25t 2 = 4 3 x 2 + y 2 2 3x + 4y 10 = = 2 x 2 + y = 4 I
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5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx
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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
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 ),
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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r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
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
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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
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211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
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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
数学演習：微分方程式
( ) 1 / 21 1 2 3 4 ( ) 2 / 21 x(t)? ẋ + 5x = 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 x(t) = sin 5t? ẋ
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.
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. ttp://www.morikita.co.jp/books/mid/00571 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. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (
(. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2
18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,
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
t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
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
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
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ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
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
No.2 1 2 2 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 δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
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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f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a
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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
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2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t) = x 0 + v 0 t 1 2 gt2 v(t) = v 0 gt t x = x 0 + v2 0 2g v2 2g 1.1 (x, v) θ
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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 +
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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
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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)
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chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
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1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
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4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
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Chap11.dvi
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