2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p

Size: px

Start display at page:

Download "2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p"

はすななぐも
5 years ago
Views:

1 2012 IA 8 I [0, 1] n x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 2. 1 x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 1 0 f(x)dx 3. < b < c [, c] b [, c] 4. [, b] f(x) 1 f(x) 1 f(x) [, b] 5. F (x) = x f(x)dx F (x) x = c c f(x) 6. F (x) = x f(x) F (c) f(c) c f(x) 7.

2 2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p

3 8 2 4 p p p.10 x y f(x) f(y) 7 p p S s

4 8 3 9 p p R ξ 1, ξ 2,... R,ξ (f) > R : 0 = x 0 < x 1 <... < x n = M < x 1 Mx 1 > R

5 8 4 M [x 0, x 1 ] ξ 1 1/2 M ξ m R,ξ (f) = f(ξ k )(x k x k 1 ) f(ξ 1 )(x 1 x 0 ) = Mx 1 > R 1. [, b] f M f(x) > M x [, b] f. R ξ 1, ξ 2,... R,ξ (f) > R f(x) M f(x) > M x N f(x) > N x [x i 1, x i ] x i x i 1 d d = min{x 1 x 0, x 2 x 1,..., x n x n 1 } n f f(x 0 ) > R + N(b ) d x 0 x 0 [x k0 1, x k0 ] ξ k0 = x 0 R,ξ (f) > k k 0 ( N)(x k x k 1 ) + > N(b ) + R + N(b ) d = R d R + N(b ) (x k0 x k0 1) d f(x) f(x) M f(x) > M x f(x) < M x f(x) [x h1 1, x h1 ],..., [x hl 1, x hl ] f(x) 0 x [x k1 1, x k1 ],..., [x km 1, x km ] : = x 0 < x 1 < < x n 1 < x n = b {x h1, x h2,..., x hl } {x k1, x k2,..., x km } = {x 1, x 2,..., x n } {x h1, x h2,..., x hl } {x k1, x k2,..., x km } =

6 8 5 f(x) < 0 x [x hi 1, x hi ] (1 i l) f(x) 0 x [x kj 1, x kj ] (1 j m) [x h1 1, x h1 ],..., [x hl 1, x hl ] ξ h1,..., ξ hl f(ξ h1 ),..., f(ξ hl ) N f(x) R + N(b ) f(x 0 ) > d x 0 d x 0 [x kj0 1, x kj0 ] [x kj0 1, x kj0 ] ξ kj0 f(ξ kj ) 0 1 x 0 [x kj 1, x kj ] ξ kj = x 0 ξ 1 x 1 ξ 2 x 2 ξ 3 x 3 ξ 4 x 4 ξ 5 x 5 = b ξ 1 ξ 2 N ξ 5 1: ξ R,ξ (f) = l m f(ξ hi )(x hi x hi 1) + f(ξ kj )(x kj x kj 1) i=1 j=1 l ( N)(x hi x hi 1) + 0 (x kj x kj 1) + f(x 0 )(x kj0 x kj0 1) j j 0 i=1 > N(b ) + R + N(b ) d = R d

7 8 6 f(x) = { tn x π/2 < x < π/2 0 x = ±π/2 0 n M { M x = 1/2 n f M (x) = gm (x)dx = [0, 1] 1 f 0 (x)dx = 0 1 f 1 0 (x)dx = lim f ε +0 ε (x)dx 14 [x k 1, x k ] f(x) f(x) f(x) f(x) [x k 1, x k ] 1. [, b] f [, b] S (f) = s (f) = sup [x k 1,x k ] f(x)(x k x k 1 ), inf f(x)(x k x k 1 ) [x k 1,x k ] f. f 2 f(x) [x k 1, x k ] s (f) R,ξ (f) S (f) (1)

8 [, b] f lim s (f) = lim S (f) 0 0 f f(x)dx f f 3. [, b] f. lim s (f) = lim S (f) = 0 0 lim S (f) = 0 f(x)dx f(x)dx ε δ < δ S (f) f(x)dx < ε ε sup f(x) f(ξ k ) < [x k 1,x k ] 2(b ) ξ k [x k 1, x k ] 0 S (f) R,ξ (f) < ε 2(b ) (x k x k 1 ) = ε 2 f(x) δ < δ ξ R,ξ(f) f(x)dx < ε 2 < δ S (f) f(x)dx S (f) R,ξ (f) + R,ξ(f) f(x)dx < ε

9 8 8 2, [, b] f f lim s (f) = lim S (f) 0 0 f(x)dx 5 4 (1) 4 s (f) f(x)dx S (f) (2) (1) (2) 1, 2 s 1 (f) S 2 (f) (3) 2. [, b] 2 5. f s (f) s (f) S (f) S (f). (1) x x [x k0 1, x k0 ] 2

10 8 9 f(x) O xk0 1 x x k0 2: sup f(x) sup f(x), [x k0 1,x ] [x k0 1,x k0 ] sup f(x) sup f(x) [x,x k0 ] [x k0 1,x k0 ] S (f) S (f) = sup f(x)(x k0 x k0 1) [x k0 1,x k0 ] ( sup f(x)(x x k0 1) + [x k0 1,x ] sup [x,x k0 ] f(x)(x k0 x ) ) 0 (3) s 1 (f) s 3 (f) S 3 (f) S 2 (f) (2) 0 s (f) f(x)dx s (f) f(x)dx S (f) < f(x)dx S (f) < s (f) (2) s (f) S (f)

11 [, b] f f lim (S (f) s (f)) = 0 (4) 0. (4) s (f) M (3) s (f) M S (f) S (f) s (f) M s (f) 0 (4) 0 s (f) M (4) S (f) M n = ( 1) n b n = ( 1) n + 1/n n n b n n b n ε-δ ε δ < δ = S (f) s (f) < ε S (f) s (f) S (f) s (f) = sup x,y [x k 1,x k ] (f(x) f(y))(x k x k 1 ) 2 x y [x k 1, x k ] f(x) f(y) f(x) f(y) < ε b S (f) s (f) ε 3 < δ [x k 1, x k ] x k x k 1 < δ +α f 2 inf I f(x) = sup I ( f(x)) sup I f(x) inf I f(x) = sup x,y I (f(x) f(y)) 3 (5)

12 I f ε δ x y < δ x, y f(x) f(y) < ε 7.. f [, b] f ε δ [, b] 2 x, y x y < δ = f(x) f(y) < ε b < δ S (f) s (f) = = mx f(x)(x k x k 1 ) [x k 1,x k ] min f(x)(x k x k 1 ) [x k 1,x k ] mx (f(x) f(y))(x k x k 1 ) < x,y [x k 1,x k ] ε b (x k x k 1 ) = ε f 6. (5) ε δ < δ ε(b ) S (f) s (f) < ε(b ) f(x) = e x2 f 2e 1/2 (< 0.86) ε = 0.01 δ ε [0, 1] e x2 dx ε x δ y ( x y < δ = f(x) f(y) < ε ) ε δ x y ( x y < δ = f(x) f(y) < ε ) x δ δ x δ x 1 4 (6) 5 ε > 0 x I

13 8 12 1/x (x > 0) 0 < x < y 1 x 1 y < y x < δ x, y x < 100 y x < x2 100 x x = 1 y x < 1/99 x = 0.1 y x < 1/9990 x = 0.01 y x < 1/ x 0 y x 0 δ sin(1/x) f f f < M f(x) f(y) < M x y (6) δ = ε/m O.K. f f f C 1 - f f [, b] C - 1/x sin(1/x) (0, ) [, b] C f [, b] ε ε ε < mx [,b] f(x) min [,b] f(x) [, b] x r [, b] I r (x) [, b] x r(x) I r (x) = [, b] [x r, x + r] y, z I r (x) = f(y) f(z) < ε r ε r(x) r(x) > 0 f r I r (x) y f(x) f(y) < ε 2

14 8 13 I r (x) 2 y, z f(y) f(z) f(y) f(x) + f(x) f(z) < ε (7) r(x) y x y < r(x) r(x) x y r I r (y) I r(x) (x) I r (y) 2 z, z f(z) f(z ) < ε r(y) r(x) x y r(x) + x y r I r (y) I r(x) (x) I r (y) 2 z, z f(z) f(z) ε r(y) r(x) + x y r(x) r(y) x y r(x) r(x) [, b] r 0 r 0 > 0 x y < r 0 x, y [, b] r 0 r(x) f(x) f(y) < ε. 9. [, b]. f [, b] S (f) = f(x k )(x k x k 1 ), s (f) = S (f) s (f) = f(x k 1 )(x k x k 1 ) (f(x k ) f(x k 1 )) (x k x k 1 ) = (f(b) f()) n 0 S (f) s (f) 0 (f(x k ) f(x k 1 ))

15 8 14. c [, b] f f x c + 0 x c 0 lim x c 0 f(x) = sup f(x) [,c) lim f(x) = inf f(x) x c+0 (c,b] n f(b) f() n + 1 lim f(x) x c+0 lim x c 0 f(x) < f(b) f() n c n + 1 f n f f g lim s (f) = lim S (f) s (f) 0 S (f) S (f) s (f) 0 S (f) s (f) S (f) s (f) 0 S (f) s (f) S (f) s (f)

16 f(x) [, b] f(x)dx f(x)dx f(x)dx = inf S (f), f(x)dx = sup s (f) 0 inf f(x)(b ) s (f) S (f) sup f(x)(b ) [,b] [,b] l f S (f) S (f) (M m)l s (f) s (f) (M m)l M m f(x). l = 1 x x [x k0 1, x k0 ] 3 S (f) S (f) = sup f(x)(x k0 x k0 1) [x k0 1,x k0 ] ( sup f(x)(x x k0 1) + [x k0 1,x ] sup [x,x k0 ] M(x k0 x k0 1) (m(x x k0 1) + m(x k0 x )) (M m) f(x)(x k0 x ) l > 1 l l 1, 2,..., l 0 = )

17 8 16 M f(x) (M m) m x k0 1 x x k0 3: 1 l i S i 1 (f) S i (f) (M m) i 1 i = 1 i = l S (f) S (f) (M m) l i 1 i i [, b] f i=1 lim s (f) = 0 f(x)dx, lim S (f) = 0 f(x)dx. S (f) f(x)dx 0 ε δ < δ S (f) f(x)dx < ε f(x)dx S (f) S 0 (f) f(x)dx < ε 2

18 n 0 M f(x) m f(x) δ δ < ε 4n 0 (M m) < δ 0 10 δ 0 S (f) S (f) (M m)n 0 δ < ε 2 0 f(x)dx S (f) S 0 (f) S (f) f(x)dx = S (f) S (f) + S (f) f(x)dx < ε 2 + S 0 (f) f(x)dx < ε n [, b] f n 0 1, 2,... lim (S n (f) s n (f)) = 0 (8) n

19 8 18. (4) lim (S (f) s (f)) = 0 0 (8) 11 lim (S (f) s (f)) 0 (8) 0 n 0 S n (f) s n (f) 0 ε-δ 13. ε-δ [, b] f ε S (f) s (f) < ε. ε-δ { n } n 0 lim (S n (f) s n (f)) = 0 n ε-δ ε N n > N = S n (f) s n (f) < ε ε-δ S (f) s (f) < 1 n n n S n (f) s n (f) < 1 n ε-δ δ

20 /2 n f(x) 0 ε-δ f(x) ε ε ε N < 2 N 1 ε N [0, 1] 2 N x k = k 2 N (k = 0, 1,..., 2 N ) f(x) 0 x = 1/2 n (n = 1, 2, ) [x k 1, x k ] 1/2 n n > N 1/2 n [x 0, x 1 ] n N x k = 1/2 n x k 1 = 1/2 n (n = 1, 2,..., N) 2N [x 0, x 1 ] S (f) = 2 N sup [x k 1,x k ] f(x)(x k x k 1 ) 2N 1 2 N < ε 3 3. f [, c] [, c] x m < f(x) < M m M ε { } ε δ = min 6(M m), b 2, c b 2 [, b δ] [b + δ, c] f [, b δ] 1 [b + δ, c] 2 S 1 (f) s 1 (f) < ε 3 S 2 (f) s 2 (f) < ε [, c] S (f) = S 1 (f) + sup x [b δ,b+δ] f(x)2δ + S 2 (f) < S 1 (f) + 2Mδ + S 2 (f) s (f) = s 1 (f) + inf f(x)2δ + s 2 (f) > s 1 (f) + 2mδ + s 2 (f) x [b δ,b+δ] S (f) s (f) < S 1 (f) s 1 (f)+2(m m)δ+s 2 (f) s 2 (f) < ε 3 +2(M m) ε 6(M m) + ε 3 = ε ε f [, c]

21 f [, b] [ + c, b + c] g g(x) = f(x c) g [ + c, b + c] f(x)dx = +c +c g(x)dx.. [ + c, b + c] [, b] c f(ξ k )(x k x k 1 ) = g(ξ k + c) ((x k + c) (x k 1 + c)) 15. [, b] f, g C (f + g)(x)dx = (Cf)(x)dx = C f(x)dx + f(x)dx g(x)dx.. R,ξ (f + g) = R,ξ (f) + R,ξ (g), R,ξ (Cf) = CR,ξ (f)

22 [, b] f, g fg. 13 ε S (fg) s (fg) < ε f g M { f M = mx 1 2 sup [,b] S 1 (f) s 1 (f) < S 2 (g) s 2 (g) < f(x), sup g(x) [,b] 1 2 [x k 1, x k ] fg x, y [x k 1, x k ] f(x)g(x) f(y)g(y) ε 2M ε 2M f(x)g(x) f(y)g(y) f(x)g(x) f(y)g(x) + f(y)g(x) f(y)g(y) sup (f(x)g(x) f(y)g(y)) M [x k 1,x k ] S (fg) s (fg) = sup [x k 1,x k ] ( M g(x) f(x) f(y) + f(y) g(x) g(y) } M f(x) f(y) + M g(x) g(y) sup (f(x) f(y)) + M [x k 1,x k ] (f(x)g(x) f(y)g(y))(x k x k 1 ) sup (f(x) f(y)) + M [x k 1,x k ] = M (S (f) s (f)) + M (S (g) s (g)) 1 2 sup (g(x) g(y)) [x k 1,x k ] S (f) s (f) S 1 (f) s 1 (f) < ε 2M S (g) s (g) S 2 (g) s 2 (g) < ε 2M sup (g(x) g(y)) [x k 1,x k ] ) (x k x k 1 )

23 8 22 S (fg) s (fg) < ε 17. [, b] f 1/f 1/f [, b] [, b] 2 x, y 1 f(x) 1 f(y) 1 1 f(x) f(y) f(x) f(y) L 1/f [, b] I ( ) 1 sup I f inf 1 I f L2 sup f inf f I I ( ) ( ) 1 1 S s L 2 (S (f) s (f)) f f f /f 18. [, b] f, g x [, b] f(x) g(x) f(x)dx g(x)dx.. ξ R,ξ (f) R,ξ (g)

24 f [, b] f f(x)dx f(x) dx. f 6 S ( f ) s ( f ) = sup x,y [x k 1,x k ] ( f(x) f(y) ) (x k x k 1 ) t, s t s t s 6 S ( f ) s ( f ) sup x,y [x k 1,x k ] f f x [, b] (f(x) f(y)) (x k x k 1 ) = S (f) s (f) f(x) f(x), f(x) f(x) f(x)dx f(x) dx f(x)dx f(x) dx 20. f M m f(x)dx = µ(b ) µ [m, M] f µ = f(x) x [, b]. µ. R,ξ (f) m(b ) R,ξ (f) M(b ) m(b ) 6 t = t s + s t s + s t s t s f(x)dx M(b )

25 8 24 m µ = f(x)dx M b f f(x) = µ x 21. < c < b f [, b] f [, b] [, c] [c, b] f [, c] [c, b] [, b] f(x)dx = c f(x)dx + c f(x)dx. f [, b] S (f) s (f) < ε c c c [, c] c b [c, b] 1 2 i = 1, 2 S i (f) s i (f) S 1 (f) s 1 (f) + S 2 (f) s 2 (f) = S (f) s (f) < ε f [, c] [b, c] f [, c] [b, c] [, c] 1 [b, c] 2 S 1 (f) s 1 (f) < ε 2 S 2 (f) s 2 (f) < ε 2 [, b] 1 2 S (f) s (f) = S 1 (f) + S 2 (f) s 1 (f) s 2 (f) < ε f [, b] { n} { n} [, c] [c, b] 0 [, b] { n } n n n S n (f) c f(x)dx S (f) f(x)dx n c S n (f) f(x)dx S n (f) = S n (f) + S n (f)

26 8 25 f(x)dx = c f(x)dx + c f(x)dx < b > b f(x)dx f(x)dx = b f(x)dx 21, b, c 10 F (x) f(x) F (x) = f(x) F (x) f(x) F (x) f(x) F (x) f(x) F (x) f(x) F (x) f(x) 0 F (x) G(x) f(x) F (x) G(x) (F (x) G(x)) = F (x) G (x) = f(x) f(x) = 0 2 x, y (F (x) G(x)) (F (y) G(y)) = 0 (x y) = 0 f f [, b] c x c f(t)dt (x [, b]) 7 f 21 x f(t)dt x f(t)dt = d c d c f(t)dt 7 x c f(t)dt + C (C R)

27 f M x+h x f(t)dt f(t)dt = c c x+h x f(t)dt x+h x f(t) dt M h f f 23. [, b] f F (x) = x c f(t)dt F = f. 20 ( 1 x+h ) x f(t)dt f(t)dt = 1 h h c c x+h x f(t)dt = f(x + θh) 0 1 θ h 0 x + θh x f ( 1 x+h ) x lim f(t)dt f(t)dt = f(x) h 0 h c c 8 cos x R cos x x cos tdt ( R) sin x + c ( 1 c 1) cos x sin x + C (C R) C > 1 sin x + C cos x 8

28 F f f(x)dx = F (b) F () 25. f x f(t)dt f F f f(x)dx = F (b) F () 5 x < 0 f(x) = 0 x 0 f(x) = 1 1 x x = 0 0 { x (x 0) f(x)dx = 0 (x < 0) x = p x f(x) = q q 0 x f(x)

29 8 28 g(x) = { 0 x 0 1 x = F F (x) = 3 x4 sin 1 (x 0) x 0 (x = 0) F F (x) = x sin 3 x 3 1 cos 1 (x 0) x 2 x 0 (x = 0) ( )

() 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.

() 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. () 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >