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1 A S hr/lectures/lectures-j.html ϵ-n 1 ϵ-n lim n n = α n n α 2 lim n = 0 1 n k n n k= ϵ-n n α n n α lim n n = α ϵ N(ϵ) n > N(ϵ) n α < ϵ (1.1.1) ϵ n > N(ϵ) n α < ϵ (1.1.2) N(ϵ) ( ϵ > 0 N(ϵ) n > N(ϵ) = n α ) < ϵ (1.1.3)

2 A S hr/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ϵ) N ϵ ϵ- (1.1.3) n > N(ϵ) n α < ϵ n N(ϵ) n α ϵ N(ϵ) N(ϵ) N(ϵ) ϵ N ϵ N ϵ lim n n = n n n lim n n = + M N(M) n > N(M) n > M (1.1.4) lim n = + lim n = { n } n n n n N n N N N N = 10 4 N = N = N n

3 A S hr/lectures/lectures-j.html 3 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 3 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 n = α ϵ = N lim n 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 N(ϵ) ϵ N n = 1, 2, 3,... n = 1 n, b n = 1 log(2 + log(2 + log n)), c 1 n = log(2 + log(2 + log n)) (1.1.5) n n n b n c n n b n c n b n c n n n n 1/n b n log n 3

4 A S hr/lectures/lectures-j.html 4 c n 10 8 n n n N ϵ n α ϵ ϵ n n n α ϵ ϵ-n n N(ϵ) 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 = 0 (1.1.6) (1.1.7) (1.1.5) 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 = log(n + 1) (1.1.8) ϵ-n ϵ-n lim n = α, lim b n = β lim ( n + b n ) = α + β. n n n lim n = α, lim b n = β lim nb n = αβ. n n n lim n = α, lim b n n = β β 0 lim = α n n n b n β. b n m b m = 0 {b n } n = n n n lim n = α lim n = β n n α = β ϵ-n n b n = 1 n n k=1 k lim n n = α lim n b n = α

5 A S hr/lectures/lectures-j.html 5 ϵ-n lim n n = α = lim = α n n 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,... n n ρ 1 = ρ 2 = ρ 3 =... = 1 j=1 1.2 ϵ-δ 4 n n x x f(x) f(x), b f(x) x b lim x f(x) = b ϵ δ(ϵ) 0 < x < δ(ϵ) x f(x) b < ϵ (1.2.1) ( ϵ > 0 δ(ϵ) > 0 0 < x < δ(ϵ) = f(x) b ) < ϵ (1.2.2) x > 0 x = f(x) f() b f() = b x δ(ε 2 ) b ε 2 ε 2 ε 1 ε 1 x δ(ε 1 ) 4 2.1

6 A S hr/lectures/lectures-j.html 6 ϵ-n 0 < x < δ(ϵ) f(x) b < ϵ 0 < x δ(ϵ) f(x) b ϵ 0 < x ϵ-n ϵ, δ ϵ, δ x f(x) b ϵ-n ϵ δ ϵ-n α f(x) b < ϵ δ(ϵ) δ(ϵ) 1) lim x 0 x, > 0 ( 2) lim x 2 2x + 3 x 0 1 x 2 1 4) lim, 5) lim x x x 1 x 1, x 3 3 7) lim x x 8) lim x 0 ) ( ), 3) lim x 2 2x + 3. (1.2.3) x x 1 x x 6) lim sin 1 x 0 x, (1.2.4) 9) lim x 0 x (1.2.5) f(x) lim f(x) x 0 ϵ-δ x = 10 1, 10 2, 10 3, 10 4,... f(x) := x { } { } lim f(x) = α lim g(x) = β lim f(x) + g(x) = α + β lim f(x)g(x) = αβ x x x x ϵ-δ ( ) 1, 2, 3,... { n } 5 2.2

7 A S hr/lectures/lectures-j.html 7 { 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 n 2 n (1.3.1) 1 = 1.4, 2 = 1.41, 3 = 1.414,... 2 II

8 A S hr/lectures/lectures-j.html lim n = α n n α ( ) ϵ > 0 N(ϵ) n > N(ϵ) n α < ϵ (1.4.1) α e ( e = lim n (1.4.2) n n) ( ) n... n (monotone) incresing (monotone) decresing (monotone) non-decresing (monotone) non-incresing. strictly incresing n n 6 p.55

9 A S hr/lectures/lectures-j.html ( 2.2.4) { n } lim n n { n } lim n n { n } lim n = + { n } n lim n = n + ± lim n n n 2 n n 2 n p ϵ-δ f(x) lim x f(x) = f() ϵ δ(ϵ) x < δ(ϵ) x f(x) f() < ϵ (1.5.1) ϵ > 0, δ(ϵ) > 0, ( x < δ(ϵ) = ) f(x) f() < ϵ (1.5.2) 0 < x < δ(ϵ) x < δ(ϵ) 0 < lim x f(x) f() x 0 <

10 A S hr/lectures/lectures-j.html f(x) f(x) lim f(x) = x +0 f() f(x) lim f(x) = f() x 0 right continuous, left continuous continuous to the right, continuous to the left. f(x) [, b] c (, b) lim f(x) = f(c) lim f(x) = f(), lim x c x +0 f(x) = f(b) (1.5.3) x b 0 f() lim x f(x) f(x) = x x f(x) x = x = ϵ-δ ( p.49) f(x) x = f δ > 0 x < δ x f(x) < f() + 1 (1.5.4) f() > 0 f(x) > 0 δ > 0 x < δ x f(x) > f() 2 f() < 0 (1.5.5) ( p.50) f g b = f() h(x) = g(f(x)) ( p.50) f, g (1) f(x) + g(x) f(x) g(x) (2) f(x)g(x) (3) g() 0 f(x)/g(x)

11 A S hr/lectures/lectures-j.html ( 2.2.6) [, b] f(x) f() f(b) F f(c) = F c [, b] x b f(x) f() f(b) f(x) = x 2 2 f(x) = 0 x x = ± 2 x x x = ( 2.2.8) f(x) = 1/x (0, 1) g(x) = x (0, 1) g(x) = sin x x sin x 1.6 x α α x > 0 x α α n x α = lim n x n (1.6.1) x α α lim n n = α {n}

12 A S hr/lectures/lectures-j.html ( ) x = f(x) f(x) f() lim x x (2.1.1) f(x) x = derivtive f () df () dx f(x) differentible f I f I f () f () f derived function derivtive x x 0 (2.1.1) x ( ) f () := f(x) f() lim x 0 x (2.1.2) f(x) left derivtive f +() := f(x) f() lim x +0 x (2.1.3) f(x) right derivtive f f () f () = f + () f f () = f () = f + () 8 2.3

13 A S hr/lectures/lectures-j.html f(x) x = f p.129 Weierstrss ( Rolle 2.3.9) f(x) [, b] (, b) f() = f(b) ξ f (ξ) = 0 ( < ξ < b) (2.2.1) ξ, b f(x) f (x) = 0 f(x) f(x) (, b) 10 f(x) ξ f ξ (, b) ξ ξ f(ξ) f(x) ξ f (ξ) = lim h 0 f(ξ + h) f(ξ) h (2.2.2) h h > 0 h < 0 h 0 h ξ b x ξ b x

14 A S hr/lectures/lectures-j.html 14 Lgrnge ( ) f(x) [, b] (, b) ξ f(b) f() b = f (ξ) ( < ξ < b) (2.2.3) ξ, b g(x) = f(x) f() x b {f(b) f()} 0 = g (ξ) = f (ξ) 1 b {f(b) f()} < ξ < b ( p.64 3) f(x) g(x) [, b] (, b) (, b) g (x) 0 f(b) f() g(b) g() = f (ξ) g (ξ) ( < ξ < b) (2.2.4) ξ g (x) 0 g() g(b) f(b) f() k := F (x) := f(x) f() k{g(x) g()} F () = F (b) = 0 g(b) g() F f, g F (ξ) = 0 ξ f (ξ) kg (ξ) = ( ) I f x, y I x < y f(x) < f(y) f I x, y I x < y f(x) f(y) f I x, y I x < y f(x) > f(y) f I x, y I x < y f(x) f(y) f I

15 A S hr/lectures/lectures-j.html ( ) f(x) I = (, b) I f (x) 0 = I f(x) I f (x) > 0 = I f(x) I f (x) = 0 I f(x) I f (x) > 0 f () > 0 x = p.135 f (x) > 0 f (x) < 0 f(x) = x f(x) n- n- n th derivtive f (n) (x) f (x), f (x), f (x) f (2) (x) = d2 dx 2 f(x) = d { d } dx dx f(x), f (3) (x) = d3 dx 3 f(x) = d [ d { d }] f(x), dx dx dx... (2.3.1) f (0) (x) f(x) Leibniz d { f(x)g(x)} = f (x)g(x) + f(x)g (x), dx n d n dx n { f(x)g(x)} = n k=0 d 2 dx 2 { f(x)g(x)} = f (x)g(x) + 2f (x)g (x) + f(x)g (x) (2.3.2) ( ) n f (k) (x) g (n k) (x), k ( ) n n! := n C k = k k! (n k)! (2.3.3) 12 ( ) ( ) ( ) n n 1 n 1 = + (2.3.4) k k k 1 I f(x) n f (n) (x) I C n - m < n C n - C m ( 12 ( + b) n n = n ) ( k=0 k k b n k n k)

16 A S hr/lectures/lectures-j.html x = f(x) locl mximum r > 0, 0 < x < r = f(x) < f() (2.3.5) f x = x = f(x) locl minimum r > 0, 0 < x < r = f(x) > f() (2.3.6) r > 0, x < r = f(x) f() (2.3.7) f f(x) x = mximum f f() f x f(x) < f() (2.3.8) x minimum locl globl p x = f(x) (i) f(x) x = x = f(x) f () = 0 (ii) f(x) x = f () = 0. f () > 0 f(x) x = b. f () < 0 f(x) x = c. f () = 0 f(x) x = (ii)-c f (x) x f(x) y = f(x) f (x) f (x) y = f(x) f (x) > 0 x f (x) < 0 x

17 A S hr/lectures/lectures-j.html 17 f f convex function concve function f(x) f() f(x) = f() + n=1 f (n) () (x ) n (2.4.1) n! = 0 e x = 1 + x + x2 2 + x3 3! + x4 4! + = 1 n! xn (2.4.2) n=0 sin x = x x3 3! + x5 5! x7 7! + = ( 1) n (2n + 1)! x2n+1 (2.4.3) n=0 cos x = 1 x2 2! + x4 4! x6 6! + = n=0 ( 1) n (2n)! x2n (2.4.4) sin x cos x sin x cos x 2π sin π = 0 sin x cos x (2.4.2) e x, sin x 14 (2.4.1) x f(x) f() ( ) f(x) I n I e x, sin x (2.4.2)

18 A S hr/lectures/lectures-j.html 18 x I x ξ n 1 f(x) = f() + (2.4.5) n 1 S n (x) := f() + k=1 k=1 f (k) () k! (x ) k + f (n) (ξ) (x ) n (2.4.5) n! f(x) = S n (x) + R n (x), (2.4.6) f (k) () k! (x ) k, R n (x) := f (n) (ξ) (x ) n (2.4.7) n! S n (x) n R n (x) n f(x) x = = 3 f(x) x = 3 x =... x =... x =... x = 2 x = 2 = 0 Mclurin y = x x x = y y = 0 y x x = ξ x b R n (x) x, R n (x) ξ x, f (n) (x) ξ ξ [ n 1 F (x) := f(x) f() + F (x) (2.4.6) R n (x) k=1 f (k) () (x ) ], k G(x) := (x ) n (2.4.8) k! F, G F (x) f(x) (x ) k G(x) n F () = F () = F () =... = F (n 1) () = 0, F (n) () = f (n) () (2.4.9) G() = G () = G () =... = G (n 1) () = 0, G (n) () = n! (2.4.10) F (x) F () G(x) G() = F (ξ 1 ) G (ξ 1 ) ξ 1 ξ 1 x (2.4.11) 15

19 A S hr/lectures/lectures-j.html 19 F () = G () = F (ξ 1 ) G (ξ 1 ) = F (ξ 1 ) F () G (ξ 1 ) G () = F (ξ 2 ) G (ξ 2 ) (2.4.12) ξ 2 ξ 2 ξ 1 F (k) () = G (k) () = 0 k n 1 F (k) (ξ k ) G (k) (ξ k ) = F (k) (ξ k ) F (k) () G (k) (ξ k ) G (k) () = F (k+1) (ξ k+1 ) G (k+1) (ξ k+1 ) ξ k+1 ξ k+1 ξ k k n 1 F (x) F () G(x) G() = F (n) (ξ n ) G (n) (ξ n ) ξ n ξ n x F (x) (x ) n = f (n) (ξ n ) n! (2.4.13) (2.4.14) (2.4.15) (2.4.6) n 1 R n (x) n lim n R n(x) = 0 f(x) = lim n S n(x) = k=0 f (k) () (x ) k (2.4.16) k! lim n S n R n (2.4.6) n R n S n n f(x) R n (x) n f I f(x) = c n (x ) n + c n 1 (x ) n c 1 (x ) + c 0 f(x) = e x e x = 0 e x = n 1 k=0 x k k! + R n(x), R n (x) := eξ n! xn (2.4.17)

20 A S hr/lectures/lectures-j.html 20 ξ 0 x x lim n R n(x) = 0 x e x = k=0 x k k! (2.4.18) sin, cos sin x = S n (x) + R n (x), S n (x), R n (x) (2.4.19) x lim n R n(x) = 0 x sin x = ( 1) k x 2k+1 (2k + 1)! cos x = k=0 k=0 ( 1) k x2k (2k)! (2.4.20) 16 sin x n = 1, 2,..., 8 y = S n (x) y = sin x n n n = 11, 21, 31, 41 n = 10, 20, 30, 40 y = sin x n x 2 n=1 n= n= sin x 1 sin x x x n= n=

21 A S hr/lectures/lectures-j.html f(x) (x ) k S n (x) (2.4.6) S n (x) f(x) R n (x) f(x) = 1/(1 x) (n ) x = 0 f(x), g(x) x 0 f(x) g(x) lim x 0 x n = 0 n (2.4.21) 0 g(x) f(x) n n f(x) g(x) x n ( 2.3.5) lim f(x) = lim h(x) = 0 x x lim x f(x) h(x) = 0 f(x) h(x) f(x) = o( h(x) ) o f(x) x h(x) K > 0 δ > 0 ( 0 < x < δ = f(x) ) < K h(x) f(x) h(x) f(x) = O ( h(x) ) O (2.4.22) x f(x) g(x) f(x) g(x) f(x) g(x) f(x) g(x) f(x) = Ω ( g(x) ) (2.4.21) f(x) g(x) = o(x n ) n o (2.4.23) 17

22 A S hr/lectures/lectures-j.html ( ) f(x) x = 0 n f(x) = S n (x) + R n (x), S n (x) := n 1 k=0 f (k) (0) x k, k! R n (x) := f (n) (θx) x n (0 < θ < 1) (2.4.24) n! S n (x) f(x) (n 1) f(x) = n 1 k=0 f (k) (0) x k + o(x n 1 ) (2.4.25) k! lim x 0 R n (x) = 0 (2.4.26) xn 1 (2.4.26) ( ) 1) 0 f (n) δ > 0 M > 0 n x x < δ f (n) (x) < M (2.4.27) f(x) = n 1 k=0 f (k) (0) x k + O(x n ) (2.4.28) k! 2) 0 f (n) f(x) C n - 1) C S n ( )) f(x) g(x) = n j=0 jx j g(x) f(x) n g(x) 0, 1,..., n j f n g n g 1 (x) = j x j, g 2 (x) = j=0 j = b j 0 j n x 0 x 0 g 1 (x) g 2 (x) x n g 1(x) f(x) x n n b j x j (2.4.29) j=0 + f(x) g 2(x) x n (2.4.30) g 1 (x) g 2 (x) lim x 0 x k = 0 (0 k n) (2.4.31)

23 A S hr/lectures/lectures-j.html 23 g 1 (x) g 2 (x) = n ( j b j )x j (2.4.32) k = 0, 1, 2,... (2.4.31) k b k = 0 k = 0, 1, 2,... j=0 f(x) S n f(x) (n 1) 1/(1 3x) tn x x = 0 tn x = sin x cos x p (x ) n 2. x x Euler e iθ = cos θ + i sin θ, θ R (2.4.33) x = iθ e iθ (ix) k = k! k=0 = l=0 ( 1) l x2l (2l)! + i ( 1) l x 2l+1 (2l + 1)! l=0 (2.4.34) k i k cos θ + i sin θ sin, cos 2π e +b = e e b

24 A S hr/lectures/lectures-j.html ( 2.5.8) f(x) I C n - I I x I f(x) = S n (x) + R n (x), S n (x) := n 1 k=0 f (k) () x (x ) k, R n (x) := k! f (n) (y) (n 1)! (x y)n 1 dy (2.4.35) f(x) C N - (2.4.35) n N n I. n = 1 x f (y)dy = f(x) f() f() f (0) (x) := f(x) I. n = 2 n = 1 x f (y)dy = x f(x) = f() + { d dy (x y)} f (y)dy = = (x )f () + x x f (y)dy (2.4.36) [ ] x (x y)f (y) + x (x y)f (y)dy (x y)f (y)dy (2.4.37) II. n n + 1 n N 1 n (2.4.35) (n 1)! x x { f (n) (y)(x y) n 1 dy = f (n) (y) 1 d n dy (x y)n} dy = 1 [ ] x f (n) (y) (x y) n + 1 x f (n+1) (y) (x y) n dy n n = 1 n f (n) () (x ) n + 1 n x f (n+1) (y) (x y) n dy. (2.4.38) (2.4.35) (n 1)! (2.4.35) n + 1

25 A S hr/lectures/lectures-j.html 25 3 f f = e x2 F F 3.1 f(x) 18 f(x) f(x) > 0 < b f(x)dx [, b] y = f(x) x ( ) < b [, b] f(x) f(x)dx [, b] n n = x 0 < x 1 < x 2 <... < x n 1 < x n = b [, b] P [x i 1, x i ] i = 1, 2,..., n P P = mx 1 i n (x i x i 1 ) [x i 1, x i ] ζ i i = 1, 2,..., n ζ 1, ζ 2,..., ζ n ζ P ζ f R(f; P, ζ) = n f(ζ i ) (x i x i 1 ) (3.1.1) i=1 P 0 P P ζ P 0 R(f; P, ζ) P, ζ f(x) [, b] f(x)dx f(x)dx lim P 0 R(f; P, ζ) (3.1.2) 18 f(x) F (x) 19

26 A S hr/lectures/lectures-j.html 26 = b > b b f(x)dx = 0 f(x)dx = b f(x)dx > b b f(x)dx f(x) > 0 n = 5 R(f; P, ζ) y=f(x) x ζ 0 1 x ζ 1 2 x ζ 2 3 x ζ 3 4 x ζ 4 5 x 5 x f(x) x = lim f(x) = f() f(x) [.b] x [, b] c lim f(x) = f(c) ϵ δ x c c [, b] ϵ > 0 δ(ϵ, c) > 0 x c < δ(ϵ, c) = f(x) f(c) < ϵ (3.2.1) 20

27 A S hr/lectures/lectures-j.html 27 δ ϵ c c c 0 δ(ϵ, c) f(x) [, b] δ(ϵ, c) c [, b] c δ(ϵ) f(x) [, b] I ( 3.1.2) f(x) I c [, b] δ(ϵ) > 0 ( ) c I x c < δ(ϵ) = f(x) f(c) < ϵ (3.2.2) δ(ϵ) c ( 3.1.4) < b [, b] ϵ > 0 δ(ϵ) > 0 ( ) x, y [, b] x y < δ(ϵ) = f(x) f(y) < ϵ (3.2.3) δ(ϵ) x, y [, b] ( 3.1.8) A N A N A bounded from bove N A upper bound M A M A bounded from below M A lower bound A bounded A [0, 1] A A A A ( ) A A A A supremum sup A A A A infimum inf A

28 A S hr/lectures/lectures-j.html 28 A A A inf A sup A A A ( ) S S [ ] [ ] S S [ ] (3.1.2) Dirichlet 0 x 1 f(x) = f(x)dx (3.3.1) 1 x Lebesgue Drboux f [, b] f(x) f(x) [, b]?? P [x i 1, x i ] f(x) m i (f; P ), M i (f; P ) n s(f; P ) m i (f; P ) (x i x i 1 ), i=1 n S(f; P ) M i (f; P ) (x i x i 1 ) (3.3.2) i=1 s(f; P ) S(f; P ) P s(f) = sup{s(f; P ) P [, b] }, S(f) = inf{s(f; P ) P [, b] } (3.3.3) s(f) S(f)

29 A S hr/lectures/lectures-j.html 29 n = 5 x 0 x 1 x 2 x 3 x 4 x 5 x ζ s(f; P ) R(f; P, ζ) S(f; P ) (3.3.4) s(f; P ) S(f; P ) (Drboux ) P P 0 (3.3.3) s S s(f) = S(f) lim s(f; P ) = s(f), lim S(f; P ) = S(f) (3.3.5) P 0 P 0 P s(f; P ) S(f; P ) s(f) S(f) (3.3.6) f [, b] ( ) f [, b] s(f) = S(f) f f(x)dx = s(f) = S(f) (3.3.7) ( 3.2.3) f(x) [, b] f [, b]

30 A S hr/lectures/lectures-j.html 30 [ 1, 1] f(x) = x 2, x S(f; P ) [, b] P 1, P 2 P 12 = P 1 P 2 S(f; P 12 ) S(f; P 1 ) S(f; P 12 ) S(f; P 2 ) s(f; P ) S(f; P ) x 0 x 1 x 2 x 3 x 4 x 5 x x 0 x 1 x 2 y 1 x 3 x y 4 2 x 5 P 1 P 1 P 2 x P 1 = (x 0, x 1, x 2, x 3, x 4, x 5 ) S(f; P 1 ) P 2 = (y 0, y 1, y 2, y 3 ) y 0 =, y 3 = b P 1 P 2 S(f; P 1 P 2 ) S(f; P 1 P 2 ) S(f; P ) s(f; P ) S s S(f; P ) P S(f; P ) P inf S(f) P S(f) S(f; P ) P S(f) S(f; P ). (3.3.8) S(f) S(f; P ) inf S(f) S(f; P ) P ϵ > 0, P S(f; P ) S(f) + ϵ. (3.3.9) (3.3.9) P 0 P ( ) (??) ϵ > 0 δ > 0, P < δ = S(f; P ) S(f) + ϵ (3.3.10)

31 A S hr/lectures/lectures-j.html 31 (3.3.9) P P P P P P P P P, P S S(f; P P ) S(f; P ). (3.3.11) n P M, m [, b] f S(f; P ) S(f; P P ) n(m m) P (3.3.12) P P n (M m) P (3.3.11) (3.3.12) S(f; P ) S(f; P P ) + n(m m) P S(f; P ) + n(m m) P (3.3.13) (3.3.9) S(f; P ) S(f; P ) + n(m m) P S(f) + ϵ + n(m m) P (3.3.14) P n(m m) P < ϵ n P P ϵ > 0 δ > 0 P < δ = S(f; P ) S(f) + 2ϵ (3.3.15) ϵ 2ϵ ϵ (3.3.10) Drboux ϵ > 0 δ > 0 P < δ S(f; P ) < S(f) + ϵ s(f; P ) > s(f) ϵ (3.3.16) (3.3.15) s(f; P ) R(f; P, ζ) S(f; P ) (3.3.17) P < δ ζ s(f) ϵ s(f; P ) R(f; P, ζ) S(f; P ) S(f) + ϵ (3.3.18) s(f) = S(f) δ 0 ϵ 0 lim R(f; P, ζ) = s(f) = S(f) (3.3.19) P 0 S(f) s(f) = c > 0 sup, inf s(f; P ) s(f) = S(f) c S(f; P ) c (3.3.20) P, P S(f; P ) s(f; P )

32 A S hr/lectures/lectures-j.html s(f) = S(f) ϵ > 0 0 S(f; P ) s(f; P ) < ϵ P (3.3.21) ϵ > 0 δ > 0 ( ) P < δ = 0 S(f; P ) s(f; P ) < ϵ (3.3.22) b ϵ = ϵ f δ > 0 b x y < δ = f(x) f(y) < ϵ (3.3.23) P P < δ I i = [x i 1, x i ] x, y x y < δ f(x) f(y) < ϵ I i f M i m i 0 M i m i ϵ i 0 S(f; P ) s(f; P ) = i (M i m i )(x i x i 1 ) i ϵ (x i x i 1 ) = ϵ b = ϵ (3.3.24) (3.3.22) f(x)dx = 0, b = 1, f(x) = x (3.3.25) f(x) f(x) = sin 3.3.3

33 A S hr/lectures/lectures-j.html b f(x) = 0 < b f(x)dx f(x)dx = b ( ) (i) < c < b c f(x)dx + f(x)dx = c f(x)dx f(x)dx (3.4.1) (ii) (3.4.1), b, c (i) f(x) 0 [, b] c [, c] [c, b] (ii) (i) ( ) (i) f(x)dx, g(x)dx { } b f(x) ± g(x) dx = f(x)dx ± (ii) f(x)dx α g(x)dx (3.4.2) { } b αf(x) dx = α f(x)dx (3.4.3) f f(x)dx (3.4.4) (i), (ii) (i), (ii)

34 A S hr/lectures/lectures-j.html < b (i) (ii) [, b] f(x) 0 = [, b] f(x) g(x) = f(x)dx 0 (3.4.5) f(x)dx g(x)dx (3.4.6) (iii) [, b] f(x) f(x) f(x)dx > 0 (i) f(x) 0 (ii) h(x) = g(x) f(x) (i) (iii) c b c f(c) > 0 f(x) f(x) x = c δ > 0 x c < δ x b f(x) f(c) 2 (3.4.7) c δ c + δ b c ± δ f(x)dx = c δ f(x)dx + c+δ c δ f(x)dx + f(x)dx (3.4.8) c+δ f(x) 0 (3.4.7) c+δ c δ f(x)dx (3.4.8) c+δ c δ f(c) f(c) dx = 2δ = f(c) δ > 0 (3.4.9) < b f(x)dx f(x) dx. (3.4.10) f(x) f(x) f(x) (3.4.11) b (ii) (3.4.10) f(x) dx f(x)dx f(x) dx (3.4.12)

35 A S hr/lectures/lectures-j.html ( 3.3.6) < b [, b] f(x) g(x) 0 [, b] ξ f(x)g(x)dx = f(ξ) g(x)dx (3.4.13) g(x) 1 ξ f(x)dx = f(ξ) (b ) (3.4.14) [, b] M, m g(x) 0 [, b] mg(x) f(x)g(x) Mg(x) m g(x)dx mg(x)dx f(x)g(x) dx Mg(x)dx = M g(x)dx (3.4.15) (3.4.13) f(x)dx = 0 (3.4.13) (3.4.13) g(x)dx 0 g(x) 0 g(x)dx > 0 (3.4.15) g(x)dx m f(x)g(x)dx g(x)dx M (3.4.16) m, M [, b] f(x) f(x) x b f(x) m M f(x)g(x)dx g(x)dx = f(ξ) (3.4.17) ξ [, b] (3.4.13) f(ξ) = 1 b f(x)dx, f(ξ) = f(x)g(x)dx g(x)dx (3.4.18) f(x) [, b] f(x) g(x) ( 3.3.5) I f(x) I F (y) := y F I y F (y) f(x)dx (3.4.19) d F (y) = f(y). (3.4.20) dy F (y) d F (y + h) F (y) 1 F (y) = lim = lim dy h 0 h h 0 h y+h y f(x)dx (3.4.21)

36 A S hr/lectures/lectures-j.html 36 h +0 h 0?? [y, y + h] ξ hf(ξ) d F (y + h) F (y) F (y) = lim = lim f(ξ) (3.4.22) dy h 0 h h 0 ξ y y + h h 0 ξ y f f(ξ) f(y) f(y) f(x) f(x) F (x) f(x) < b [, b] f(x) f F (x) (i) F 1 (x) f(x) C F 2 (x) := F 1 (x) + C (3.4.23) F 2 (x) f(x) (ii) F 1 (x) F 2 (x) f(x) x C F 2 (x) F 1 (x) = C ( x b) (3.4.24) (i) F 1 = f F 2 = f (ii) F 1, F 2 f d dx {F 2(x) F 1 (x)} = f(x) f(x) = 0 (3.4.19) c ( ) [α, β] C 1 - ϕ(t) ϕ(α) =, ϕ(β) = b b α < t < β ϕ(t), b [, b] [b, ] f(x) f(x)dx = β α f ( ϕ(t) ) ϕ (t) dt (3.4.25) ( ) [, b] C 1 - f(x), g(x) f(x) g (x) dx = [ ] b f(x)g(x) f (x) g(x) dx (3.4.26)

37 A S hr/lectures/lectures-j.html 37 x > 0 log x log x := du (3.5.1) u 1/u log log 1 = 0 log x (0, ) (log x) = 1/x. x, y > 0 log(xy) = log x + log y log x x lim log x = lim log x =. x x +0 y = log x y = exp x x log x y y = log x x y x y x = exp y exp 1 e exp 0 = 1 exp x (, ) (exp x) = exp x. x, y exp(x + y) = (exp x) (exp y) exp x x lim exp x = 0 lim exp x =. x x + x := exp(x log ) (3.5.2) x e x = exp x p.107 α x > 0 d dx xα = α x α 1 (3.5.3) x α+1 (α 1) α + 1 x α dx = (3.5.4) log x (α = 1) e = exp 1 ( lim x = e (3.5.5) x x)

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

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 III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 1 1 1.1 ϵ-n ϵ-n lim n = α n n α 1 lim n = 0 1 n k n k=1 0 1.1.7 ϵ-n 1.1.1 n α n n α lim n = α ϵ Nϵ n > Nϵ n α < ϵ 1.1.1 ϵ n > Nϵ n α < ϵ 1.1.2