III ϵ-n ϵ-n lim n a n = α n a n α 1 lim a n = 0 1 n a k n n k= ϵ-n 1.1
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1 III ϵ-n ϵ-n lim n = α n n α 1 lim n = 0 1 n k n k= ϵ-n n α n n α lim n = α ϵ Nϵ n > Nϵ n α < ϵ ϵ n > Nϵ n α < ϵ Nϵ ϵ > 0 Nϵ n > Nϵ = n α < ϵ
2 III 2 Nε 1 Nε 2 ε 1 ε 2 α ε ε 2 1 n Nϵ N ϵ ϵ n > Nϵ n α < ϵ n Nϵ n α ϵ Nϵ Nϵ Nϵ ϵ N ϵ N ϵ lim n = n n n lim n = + M NM n > NM n > M lim n = + lim n = { n } n n N n N
3 III 3 N N N = 10 4 N = N = N n n = 1/n n n n ϵ > 0 n n α ϵ ϵ ϵ ϵ = 10 6 ϵ = ϵ = N ϵ n α N ϵ 2 n α n α n n = 1/n ϵ = n > 100 n > 100 n α < ϵ = 10 6 n > n > n α < 10 6 ϵ = n > ϵ = n > ϵ > 0 lim n = α ϵ = N lim n = α N ϵ ϵ-n N ϵ n n α n n α ϵ n α N n ϵ n α ϵ-n ϵ Nϵ ϵ N n = 1, 2, 3,... n = 1 n, b n = 1 log2 + log2 + log n, c 1 n = log2 + log2 + log n n n n b n c n
4 III 4 n b n c n b n c n n n n 1/n b n log n c n 10 8 n n n N ϵ n α ϵ ϵ n n n α ϵ ϵ-n n Nϵ n = 1, 2, 3,... n = 3, b n = 1 n, c n = 1, d n = 1 n n n 10, 10 2, 10 3, 10 4, 10 5, 10 6,... e n = n f n = n + 3 n, g n = sin n n, h n = n + 1 n, p n = 2n + 1 n + 1, q 1 n = logn ϵ-n ϵ-n lim n = α, lim b n = β lim n + b n = α + β. lim n = α, lim b n = β lim nb n = αβ. lim n = α, lim b n n = β β 0 lim = α b n β. b n m b m = 0 {b n } n = n n n lim n = α lim n = β α = β
5 III 5 ϵ-n n b n = 1 n n k=1 k lim n = α lim b n = α ϵ-n lim n n = α = lim = α n 1 n ρ 1, ρ 2, ρ 3,... n / n b n := ρ j j ρ j j=1 lim n = α lim b n = α ρ 1, ρ 2, ρ 3, ρ 1 = ρ 2 = ρ 3 =... = 1 j=1 1.2 ϵ-δ n n x x fx fx, b fx x b lim x fx = b ϵ δϵ 0 < x < δϵ x fx b < ϵ ϵ > 0 δϵ > 0 0 < x < δϵ = fx b < ϵ x > 0 x = fx f b f = b x
6 III 6 δε 2 b ε 2 ε 2 ε 1 ε 1 x δε 1 ϵ-n 0 < x < δϵ fx b < ϵ 0 < x δϵ fx b ϵ 0 < x ϵ-n ϵ, δ ϵ, δ x fx b ϵ-n ϵ δ ϵ-n α fx b < ϵ δϵ δϵ 1 lim x 0 x, > 0 2 lim x 2 2x + 3 x 0, 3 lim x 2 2x x 1 1 x lim, 5 lim x x x 1 x 1, 6 lim sin 1 x 0 x, x lim x x 1 + x 1 x 8 lim x 0 x 9 lim x 0 x fx lim fx x 0 ϵ-δ x = 10 1, 10 2, 10 3, 10 4,... fx := x { } { } lim fx = α lim gx = β lim fx + gx = α + β lim fxgx = αβ x x x x ϵ-δ
7 III , 2, 3,... { n } { n } { n } 1, 2, 3, 4, 5, 6,... 1, 3, 5, 7, 9,... 1, 4, 9, 16, 25,... 1, 2, 5, 10, 100, 10032, , { n } L n n < L K n n > K K, L { n } n n L n K { n } {b n } {b n } 1, 2, 3,... K L ccumultion point K L
8 III 8 n 2 n = 1.4, 2 = 1.41, 3 = 1.414,... 2 II Bolzno-Weiertrss 1.4 lim n = α n α ϵ > 0 Nϵ n > Nϵ n α < ϵ α e e = lim n n e x = 1 + x + x2 2! + x3 3! + = lim N N n=0 x n n! x e x e x 3 lim N N n=0 x n n n! lim N N n=0 x n n n! e x
9 III n... n monotone incresing monotone decresing monotone non-decresing monotone non-incresing. strictly incresing n n { n } lim n { n } lim n { n } lim n = + { n } lim n = + ± lim n n 2 n n 2
10 III 10 n n ϵ-δ α { n } {b k } α { n } α α {b k } α k b k α {b k } { n} {b k } k 1 b k1 > α n 1 k b k b k1 > α b k α {b k } { n } k n b k = n n = b k n n α n n n α n m m n α {b k } k n = b k n n n α { n }, {b k } {b n } α ϵ > 0 Kϵ > 0 k > Kϵ = b k α < ϵ k > Kϵ α ϵ < b k n = b k n α ϵ < n { n } n 1 α ϵ < n1 n > n 1 α ϵ < n1 n ϵ > Kϵ Kϵ k 1 n1 = b k1 n 1 n > n 1 α ϵ < n 1.4.9
11 III ϵ > 0 n 1 > 0 n > n 1 α ϵ < n < α lim n = α ϵ-δ { n } α α n Cuchy sequence ϵ > 0 Nϵ m, n Nϵ m n < ϵ ε 1 ε 2 Nε 1 Nε 2 n n n m m, n n n lim sup lim inf
12 III Nϵ n := 1 n b n := 1 n 2 c n := 1n n d n := 1n n { n }, {b n }, {c n } α c n n := log n + n k=1 1 k b n := c 1 := 1, n 1 c n+1 := 1 2 n 1 k 1 k=1 k c n + α c n c n n, b n e x e x = n=0 x n n! x x > 0 x sin x = x x3 3! + x5 5! x7 7! + x 0 < r < 1 { n } n+2 n+1 r n+1 n n = 1, 2, 3,... n x lim x fx fx C ϵ > 0 δϵ > 0 0 < x < δϵ 0 < y < δϵ x, y fx fy < ϵ 1.6
13 III A N A N A bounded from bove N A upper bound M A M A bounded from below M A lower bound A A bounded A [0, 1] A A A A A A A A supremum sup A A A A infimum inf A A A A inf A sup A A A S S [ ] [ ] S S [ ] n { n } lim sup k k n { n } { n } lim sup n lim n lim inf k k n lim inf n lim n sup k n k n +
14 III 14 α α α { n } n α ϵ > 0 Nϵ/2 n > Nϵ/2 n α < ϵ/2 m, n > Nϵ/2 m α < ϵ 2, n α < ϵ m, n m n = m α + α n m α + α n < ϵ 2 + ϵ 2 = ϵ β := lim inf n, γ := lim sup n β γ b N := inf m N m, c N := sup n n N β = γ { n } ϵ > 0, N l, m N = l m < ϵ ϵ > 0, N l, m N = l m < ϵ N, m l N sup sup l = c N l N ϵ > 0, N m N = c N m ϵ
15 III 15 m N inf inf m N l = b N ϵ > 0, N c N b N ϵ c n b n {b n } {c n } {c n b n } N c N b N ϵ n N c n b n ϵ ϵ > 0, N n N = c n b n ϵ lim c n b n = 0 ϵ-n β = γ
16 III [, b] f n x n = 1, 2, 3,... i {f n x} [, b] fx x lim f nx = fx ϵ > 0 x [, b] Nϵ, x n > Nϵ, x = f n x fx < ϵ ii {f n x} [, b] fx ϵ > 0 Nϵ x [, b] n > Nϵ = f n x fx < ϵ lim f nx = fx N x x N N x I = [, b] f n x n = 1, 2, 3, f n x n 0 < x < 1/n f n x = [ 1 ] lim f n xdx = lim f nx dx lim < x < 1/n g n x =
17 III 17 [ 1 ] lim g n xdx = 0 fx = lim f nx [ 1 ] lim f n xdx = 0 lim [ lim g nx dx fxdx { } ] f n x fx dx = g n x = f n x fx [, b] g n x x lim g nx = 0 lim g n x = 0 lim g nx = 0 ϵ > 0 Nϵ, x n > Nϵ, x = g n x < ϵ x g n x < ϵ g n xdx g n x dx b ϵ lim g nx = N x x g n x < ϵ n f n x g n x = f n x 0 f n x n n x g n x < ϵ [, b] f n x n = 1, 2, 3,... fx lim f n xdx = { } lim f nx dx = fxdx α f n x fx f fx
18 III 18 ϵ > 0 Nϵ x [, b] n > Nϵ = f n x fx < ϵ ϵ f n xdx f n xdx fxdx = fxdx f n x fx dx {f n x fx}dx ϵ dx = ϵb ϵ n n n f n x i I = [, b] n=0 f nx F x f n x [ n=0 ] f n xdx = [ n=0 ] f n x dx = F xdx. iii n=0 nx n R fx n=0 nx n R, R [, b] [ ] fxdx = n x n dx = 0, b = x x < R x 0 ftdt = n=0 n=0 n n + 1 xn Arzelà 5.10, 5.11 I =, b {f n x} n n, x M n 0 x I fx M fx := lim f nx I dx n lim fxdx = lim f nx dx = fxdx
19 III 19 I =, {f n x} gx g f n n 0 x I f n x gx gxdx < fx := lim f nx I dx n lim fxdx = lim f nx dx = fxdx I = [, b] f n x n = 1, 2, 3,... x I lim f nx = fx n f n x x fx x [ 1, 1] 0 1 x 0 f n x = nx 0 < x 1/n 1 1/n < x n fx = lim f 0 1 x 0 nx = 1 0 < x x = I {f n } fx f n x fx
20 III 20 fx I = [, b] c lim fx = fc, ϵ > 0, δ > 0 x c < δ = fx fc < ϵ x c f n x fx fc f n x f n c fx fc = fx f n x + f n x f n c + f n c fc fx fc fx f n x + f n x f n c + f n c fc f n x fx f n c fc f n f n f n x f n c f n x c x c n c n x n x n x c n n x c n c = n x > = 1 x > 0 n 1 0 = 1 x c n ϵ > ϵ Nϵ x n Nϵ ϵ n Nϵ ϵ x f Nϵ x δϵ, Nϵ > 0 x c < δϵ, Nϵ ϵ ϵ, Nϵ, δϵ, Nϵ ϵ ϵ ϵ x c fx fc < 3ϵ fx x = c 2.4 f n x fx i f n x I = [, b] C 1 - { d dx f nx} I {f n x} x 0 I {f n x} I C 1 - ii f n x I C 1 - n [ ] d lim dx f nx = d [ ] lim dx f nx = d dx fx. d dx f nx I n f nx
21 III 21 x 0 I n f nx I C 1 - [ ] d dx f nx n=0 = d [ ] f n x. dx n=0 iii n nx n R fx n nx n R, R d dx fx = n n x n 1 n=1 2.5 R {x, y x [, b], y [c, d]} fx, y fx, y x y Iy y Iy fx, ydx fx, y R y [c, d] Iy R fx, y y [c, d] Iy y y d dy Iy = d [ ] fx, ydx = dy [ ] fx, y dx y 2.6 fx = n x n f x = n nx n 1?? n=0 n= Newton, Leibnitz x
22 III 22
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sup inf (ε-δ 4) 2018 1 9 ε-δ,,,, sup inf,,,,,, 1 1 2 3 3 4 4 6 5 7 6 10 6.1............................................. 11 6.2............................... 13 1 R R 5 4 3 2 1 0 1 2 3 4 5 π( R) 2 1 0
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() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
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Chap11.dvi
. () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x
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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.
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)
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f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
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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IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (
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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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1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
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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