I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10

1 2007.4.13. 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 0. 1. 1. 2. 3. 2. ɛ-δ 1. ɛ-n ɛ-δ 2.1 2. 3.1 3. 4.1 2,3

2 3. 2.2 1. 2. 4. 1. 2. 2.3 3. 2.4 4. 2.5 5. I, II 1, 2 ɛ-δ 100 A = 0.50 + 0.50 A 4 : 6 = max{ A, } A A 10

3 10% A, B ɛ-δ ɛ-δ

4 ɛ-δ ɛ-δ ɛ-δ ɛ-δ http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html E-mail hara@math.kyushu-u.ac.jp spam mail html mail

5 1 2 1 2

6 3 a < b n Z 1 N Q R a < b a = b a b a > b a = b a b a < x < b (a, b) a x b [a, b] n! = n (n 1) (n 2) 2 1 n 0! = 1 unique, uniquely 3

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 7 1 1.1 f x f(x) f(x) = x 2 x x 2 x f(x) x f(x) x 1 x 2 f(x 1 ) = f(x 2 ) x x f(x) y = f(x) (1.1.1) x y x = g(y) y = f(x) (1.1.2) y x g f x y = f(x) y x f x y y = f(x) y = g(x) (1.1.2) x, y y = g(x) x, y x, y (1.1.2) y x y = x 2 x = ± y y > 0 x f(x) x y y = x 2 x 0 y x x = y x 0 f(x) = x 2 g g(y) = y y = e x x = log y y = sin x y x x sin x x π/2 y = sin x x y y x sin x = arcsin y (1.1.3) y = sin x (1.1.4)

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 8 cos 0 x π y = cos x x = arccos y (1.1.5) arccos tan x < π/2 y = tan x x = arctan y (1.1.6) arctan f g y = f(x) g(f(x)) = g(y) = x (1.1.7) x g (f(x)) f (x) = 1 g (f(x)) = 1 f (x) (1.1.8) y = f(x) x = g(y) y g (y) = 1 f (x) = 1 f ( g(y) ) (1.1.9) g(y) f g(y) g(y) = arcsin y f f(x) = sin x f (x) = cos x d dy arcsin y = 1 cos ( arcsin y ) = 1 cos x x = arcsin y (1.1.10) y = sin x cos x = ± 1 x 2 x π/2 cos cos x = 1 x 2 d dy arcsin y = 1 1 y 2 (1.1.11) arcsin y d dy arccos y = 1 sin ( arccos y ) = 1 1 y 2 (1.1.12) d dy arctan y = 1 { sec ( arctan y )} 2 = { cos ( arctan y )} 2 = 1 1 + y 2 (1.1.13)

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 9 1.2 arctan x arctan 0 = 0 x 0 (arctan t) dt = arctan x arctan 0 = arctan x (1.2.1) arctan x = x 0 1 dt (1.2.2) 1 + t2 1 1 + t 2 = ( 1) n t 2n t < 1 (1.2.3) n=0 (1.2.2) arctan x = x 0 ( n=0 ( 1) n t 2n ) dt?? = x ( 1) n t 2n dt = n=0 0 n=0 2n + 1 = x x3 3 + x5 5 x7 +... (1.2.4) 7 ( 1) n x2n+1 x < 1 arctan x (1.2.4) x 1 x p.27 (1.2.4) arctan x x x = 1 π 4 = 1 1 3 + 1 5 1 7 + 1... (1.2.5) 9 π 1/4 π P.28 arctan x arctan x tan sin tan x tan x tan x = a 1 x + a 3 x 3 + a 5 x 5 + a 7 x 7 + (1.2.6) a 1, a 3, a 5,... x tan x a n tan arctan x arctan tan x tan(arctan x) = x a 1 arctan x + a 3 (arctan x) 3 + a 5 (arctan x) 5 + = x (1.2.7)

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 10 arctan x (1.2.4) a 1 ( x x3 3 + x5 5 x7 7 +... ) + a 3 ( x x3 3 + x5 5 x7 7 +... ) 3 + a5 ( x x3 3 + x5 5 x7 7 +... ) 5 + = x (1.2.8) x 1 x x, x 3, x 5, x 7,... 4 a 1, a 3, a 5,... x ( a 1 x + a ) ( 1 3 + a 3 x 3 a1 ) ( + 5 a 3 + a 5 x 5 + a 1 7 + 14 15 a 3 5 ) 3 a 5 + a 7 x 7 + = x (1.2.9) a 1 = 1, a 1 3 + a 3 = 0, a 1 5 a 3 + a 5 = 0, a 1 7 + 14 15 a 3 5 3 a 5 + a 7 = 0 (1.2.10) a 1 = 1, a 3 = 1 3, a 5 = 2 15, a 7 = 17 315 tan x (1.2.11) tan x = x + x3 3 + 2 15 x5 + 17 315 x7 + (1.2.12) sin x sin x = tan x 1 + (tan x) 2 (1.2.13) 1/ 1 + x 1 1 + x = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + (1.2.14) a 0, a 1, a 2,... 1 = (1+x)(a 0 +a 1 x+a 2 x 2 + ) 2 = a 0 +(1+2a 0 a 1 )x+(a 2 1+2a 0 a 2 +2a 0 a 1 )x 2 +(a 2 1+2a 0 a 2 +2a 0 a 3 +2a 1 a 2 )x 3 +... (1.2.15) a 0 = 1 a 1 = 1 2 a 2 = 3 8 a 3 = 5 16 (1.2.16) cos x = 1 1 + (tan x) 2 = 1 1 2 (tan x)2 + 3 8 (tan x)4 5 16 (tan x)6 + (1.2.17) tan x (1.2.12) sin x = tan x cos x = cos x = 1 1 2 x2 + 1 24 x4 1 720 x6 + = 1 x2 2! + x4 4! x6 6! + (1.2.18) (x + x3 3 + 2 15 x5 + 17 )( 315 x7 + 1 1 2 x2 + 1 24 x4 1 ) 720 x6 + = x x3 6 + x5 120 x7 5040 + = x x3 3! + x5 5! x7 7! + (1.2.19) 4

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 11 2 ɛ-δ 3 ɛ-δ 2.1 ɛ-n 5 ɛ-n lim n a n = α n a n α 6 lim a n = 0 1 n a k n n k=1 0 2.1.7 ɛ-n 2.1.1 a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ (2.1.1) ɛ > 0 N(ɛ) ( n > N(ɛ) = a n α ) < ɛ (2.1.2) 5 2.1 6

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 12 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (2.1.2) n > N(ɛ) a n α < ɛ n N(ɛ) a n α ɛ N(ɛ) N(ɛ) N(ɛ) ɛ N ɛ N ɛ lim n a n = + 2.1.2 a n a n n lim n a n = + M N(M) n > N(M) a n > M (2.1.3) M > 0 N(M) ( ) n > N(M) = a n > M (2.1.4) lim n a n = M < 0 N(M) ( ) n > N(M) = a n < M (2.1.5) lim a n = + lim a n = {a n } n n

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 13 2.1.1 2.1.1 1 n n N n N N N N = 10 4 N = 10 10 N = 10 100 N n a n = 1/n n n n ɛ > 0 n a n α ɛ ɛ ɛ ɛ = 10 6 ɛ = 10 14 ɛ = 10 200 N ɛ a n α N ɛ 7 a n α a n α n a n = 1/n ɛ = 0.0001 n > 100 n > 100 a n α < 0.0001 ɛ = 10 6 n > 20000 n > 20000 a n α < 10 6 ɛ = 10 12 n > 10 20 ɛ = 10 100 n > 10 300 ɛ > 0 lim n a n = α ɛ = 10 300 N lim n a n = α N ɛ ɛ-n N ɛ N a n α N a n α ɛ a n α N ɛ a n α ɛ-n ɛ N(ɛ) ɛ N n = 1, 2, 3,... a n = 1 n, b n = 1 log(2 + log(2 + log n)), c 1 n = log(2 + log(2 + log n)) + 10 8 (2.1.6) 7

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 14 n n 1 10 100 10 3 10 4 10 5 10 6 10 8 10 16 a n 1 10 1 10 2 10 3 10 4 10 5 10 6 10 8 10 16 b n 1.00938 0.80577 0.73645 0.69834 0.67321 0.65494 0.64084 0.62006 0.57692 c n 1.00938 0.80577 0.73645 0.69834 0.67321 0.65494 0.64084 0.62006 0.57692 a n b n c n b n c n n n a n 1/n b n log n c n 10 8 n n n N ɛ a n α ɛ ɛ n n a n α ɛ ɛ-n 2.1.2 2.1.7 2.1.3 n N(ɛ) n = 1, 2, 3,... a n = 3, b n = 1 n, c n = 1, d n = 1 n n 2 + 1 1 n 10, 10 2, 10 3, 10 4, 10 5, 10 6,... e n = 0 (2.1.6) n (2.1.7) (2.1.8) 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) (2.1.9) ɛ-n 2.1.4 ɛ-n lim a n = α, lim b n = β lim (a n + b n ) = α + β. n n n lim a n = α, lim b n = β lim a nb n = αβ. n n n lim a n = α, lim b a n n = β β 0 lim = α n n n b n β. b n m b m = 0 {b n } 2.1.5 a n = 1 + 1 n 2.1.6 1

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 15 2.1.6 a n n lim a n = α lim a n = β n n α = β ɛ-n 2.1.7 a n b n = 1 n n k=1 a k lim n a n = α lim n b n = α ɛ-n 2.1.8 2.1.7 lim a a 1 + a 2 + + a n n = α = lim = α n n n a 1 a n ρ 1, ρ 2, ρ 3,... ( n ) / ( n ) b n := ρ j a j ρ j j=1 lim a n = α lim b n = α ρ 1, ρ 2, ρ 3,... n n 2.1.7 ρ 1 = ρ 2 = ρ 3 =... = 1 j=1 2.2 ɛ-δ 8 n a n x x a f(x) 2.2.1 f(x) a, b f(x) x a b lim x a f(x) = b ɛ δ(ɛ) 0 < x a < δ(ɛ) x f(x) b < ɛ (2.2.1) ɛ > 0 δ(ɛ) > 0 ( 0 < x a < δ(ɛ) = f(x) b ) < ɛ (2.2.2) x a > 0 x = a f(x) a f(a) b f(a) = b x a 8 2.1 p.49

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 16 δ(ε 2 ) b ε 2 ε 2 ε 1 ε 1 x δ(ε 1 ) a ɛ-n 0 < x a < δ(ɛ) f(x) b < ɛ 0 < x a δ(ɛ) f(x) b ɛ 0 < x a ɛ-n ɛ, δ ɛ, δ x a f(x) b ɛ-n ɛ δ ɛ-n α f(x) b < ɛ δ(ɛ) 2.2.2 δ(ɛ) 1) lim x 0 x, a > 0 ( 2) lim x 2 2x + 3 x 0 ) ( ), 3) lim x 2 2x + 3. (2.2.3) x 1 1 x 2 1 4) lim, 5) lim x 0 1 + x x 1 x 1, 6) lim sin 1 x 0 x, (2.2.4) x 3 a 3 7) lim x a x a 1 + x 1 x 8) lim x 0 x 9) lim x 0 x (2.2.5) 2.2.3 f(x) lim f(x) x 0 ɛ-δ 0.001 x = 10 1, 10 2, 10 3, 10 4,... f(x) := x { } { } 2.2.4 lim f(x) = α lim g(x) = β lim f(x) + g(x) = α + β lim f(x)g(x) = αβ x a x a x a x a ɛ-δ

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 17 2.2.1 lim x a f(x) = b lim x f(x) = b ɛ > 0 L(ɛ) x > L(ɛ) x f(x) b < ɛ lim f(x) = b ɛ > 0 L(ɛ) x x < L(ɛ) x f(x) b < ɛ lim x a f(x) = + M > 0 δ(m) x a < δ(m) x f(x) > M lim x a f(x) = M > 0 δ(m) x a < δ(m) x f(x) < M lim f(x) = lim x x f(x) = lim n a n = α lim x a f(x) = b lim x a a 2.2.5 f(x) a, b lim f(x) = b a x > a x a x a+0 ɛ > 0 δ(ɛ) > 0 0 < x a < δ(ɛ) x f(x) b < ɛ ( ) ɛ > 0 δ(ɛ) > 0 0 < x a < δ(ɛ) = f(x) b < ɛ (2.2.6) lim f(x) = b a x < a x a x a 0 ( ) ɛ > 0 δ(ɛ) > 0 δ(ɛ) < x a < 0 = f(x) b < ɛ (2.2.7) lim x a+0 lim lim lim ± a x a + x a 0 x a 2.2.6 f(x) a, b lim f(x) = b x a lim f(x) = lim f(x) = b x a+0 x a+0 2.2.7 lim x ex =, lim x ex = 0, lim log x =, (2.2.8) x lim log x =, lim x +0 e x 1 sin x = 1, lim = 1, (2.2.9) x 0 x x 0 x

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 18 lim x xα =, lim x xα = 0, x α = e α log x lim x +0 xα = 0, α > 0 (2.2.10) lim x +0 xα =, α < 0 (2.2.11) e x log x x 2.3 lim x a f(x) = b lim n a n = a n a n a {a n } lim n f(a n) = b 9 2.3.1 0 < x a < δ n a n a 2.3.1 10 ɛ > 0 δ > 0 x(0 < x a < δ f(x) b ɛ) lim n a n = a n a n a {a n } lim n f(a n) = b {a n } ɛ > 0 δ = 1/n n δ f(x) b ɛ x 0 < x a < δ = 1/n δ = 1/n x a n n = 1, 2, 3,... {a n } 0 < a n a < 1/n lim n a n = α a n a a n f(a n ) b ɛ lim n f(a n) = b 9 10

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 19 2.4 ɛ-δ 2.4.1 a f(x) a lim x a f(x) = f(a) ɛ δ(ɛ) x a < δ(ɛ) x f(x) f(a) < ɛ (2.4.1) ɛ > 0, δ(ɛ) > 0, ( x a < δ(ɛ) = f(x) f(a) ) < ɛ (2.4.2) 0 < x a < δ(ɛ) x a < δ(ɛ) 0 < lim x a f(x) f(a) x a 0 < 2.4.2 f(x) a f(x) a lim f(x) = x a+0 f(a) a f(x) a lim f(x) = f(a) x a 0 right continuous, left continuous continuous to the right, continuous to the left. f(x) [a, b] c (a, b) lim f(x) = f(c) lim f(x) = f(a), lim x c x a+0 f(x) = f(b) (2.4.3) x b 0 f(a) lim x a f(x) 2.4.3 f(x) = x x 2.4.4 f(x) x = a x = a

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 20 ɛ-δ 2.4.5 ( p.49) a f(x) x = a a f δ > 0 x a < δ x f(x) < f(a) + 1 (2.4.4) f(a) > 0 a f(x) > 0 δ > 0 x a < δ x f(x) > f(a) 2 f(a) < 0 (2.4.5) f(x) x = a ɛ-δ p.50 2.4.6 ( p.50) f a g b = f(a) h(x) = g(f(x)) a δ 2.4.7 ( p.50) f, g a (1) f(x) + g(x) f(x) g(x) a (2) f(x)g(x) a (3) g(a) 0 f(x)/g(x) a

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 21 3 2.1.7 2 11 12 3.1 3.1.1 ( ) a 1, a 2, a 3,... {a n } {a n } {a n } 3.1.2 ( ) {a n } N n a n < N n a n > M M, N {a n } n 3.1.3 ( ) {a n } {b n } {b n } a n 2 n (3.1.1) a 1 = 1.4, a 2 = 1.41, a 3 = 1.414,... 2 11 12 II II

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 22 3.2 13 lim a n = α a n n α ( ) ɛ > 0, N(ɛ), n > N(ɛ) = a n α < ɛ (3.2.1) α e ( e = lim 1 + 1 n (3.2.2) n n) e x = 1 + x + x2 2! + x3 3! + = lim N N n=0 x n n! (3.2.3) x e x e x 14 lim N N n=0 x n n n! lim N N n=0 x n n n! (3.2.4) 3.2.1 ( ) a 1 a 2 a 3... a n... a n (monotone) increasing (monotone) decreasing (monotone) non-decreasing (monotone) non-increasing. 4.2.4 13 2.2 14 e x

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 23 strictly increasing n n 3.2.2 ( 2.2.4) {a n } lim n a n {a n } lim n a n {a n } lim a n = + {a n } n lim a n = n + ± lim n a n 3.2.2 a n 2 n a n 2 n 3.2.2 a n ɛ-δ α 3.1.3 {a n } {b k } α {a n } α α

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 24 {b k } α k b k α (3.2.5) {b k } {a n } {b k } k 1 b k1 > α n 1 k b k b k1 > α b k α {b k } {a n } k n b k = a n (3.2.5) a n = b k a n a n α a n n a n α n m a m a n α {b k } k a n = b k n n a n α (3.2.6) {a n }, {b k } {b n } α ɛ > 0 K(ɛ) > 0 ( ) k > K(ɛ) = b k α < ɛ (3.2.7) k > K(ɛ) α ɛ < b k (3.2.8) a n = b k n α ɛ < a n {a n } n 1 α ɛ < a n1 n > n 1 α ɛ < a n1 a n ɛ > 0 (3.2.7) K(ɛ) K(ɛ) k 1 a n1 = b k1 n 1 n > n 1 α ɛ < a n (3.2.9) (3.2.6) ɛ > 0 n 1 > 0 n > n 1 α ɛ < a n < α (3.2.10) lim n a n = α ɛ-δ {a n } α α

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 25 3.3 3.3.1 ( 2.2.6) [a, b] f(x) f(a) f(b) F f(c) = F c [a, b] x a b f(x) f(a) f(b) f(x) = x 2 2 f(x) = 0 x x = ± 2 x x x = 2 3.3.1 f(a) < F < f(b) f(a) > f(b) f(a) = f(b) f(a) = F c = a 2.4.5 g(x) := f(x) F g(c) = 0 c x = a x g(a) < 0 g(x) 2.4.5 x = a g(x) < 0 a y a x < y x g(x) < 0 y Y g(b) > 0 y b Y Y [a, b] 2 n Y a n n a n n Y y b a a n b 3.2.2 15 a n α Y a a n b a α b g(α) = 0 g(α) < 0 g(α) > 0 g(α) < 0 2.4.5 α x g(x) < 0 Y α n a n > α a n α g(α) > 0 2.4.5 α x g(x) > 0 δ > 0 α δ < x α x g(x) < 0 n a n α δ a n α g(α) = 0 c = α 15

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 26 3.3.2 ( 2.2.8) f(x) = 1/x (0, 1) g(x) = x (0, 1) g(x) = sin x x sin x 3.3.2 [a, b] x [a, b] n (n + 1) n (n + 1) f(x) x n n = 1, 2, 3,... x 1, x 2, x 3,... 3.1.3 y l = x il y l x 1, x 2,... i l y l i l α := lim l y l α f(x) x a x b x f(α) f(x) y l i l x z l i l f(x) y l f(z l ) f(y l ) (3.3.1) l y l α α l z l x f f(x) f(α) f(x) f(α) (3.3.2) x f(α) 3.4 x α α x > 0 x α α a n x α = lim n xa n (3.4.1) x α α 16 16 lim n an = α {an}

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 27 4 4.1 4.1.1 ( ) x = a f(x) f(x) f(a) lim x a x a (4.1.1) f(x) x = a derivative f (a) df (a) dx f(x) a differentiable f I f I a f (a) a f (a) f derived function derivative x a x a 0 (4.1.1) x 4.1.2 ( ) 4.1.1 f (a) := f(x) f(a) lim x a 0 x a (4.1.2) f(x) a left derivative f +(a) := f(x) f(a) lim x a+0 x a (4.1.3) f(x) a right derivative f a f (a) f (a) = f + (a) f a f (a) = f (a) = f + (a)

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 28 2.3.4 2.3.7 4.1.3 f(x) x = a f a p.129 Weierstrass 4.2 4.2.1 ( Rolle 2.3.9) f(x) [a, b] (a, b) f(a) = f(b) f (ξ) = 0 (a < ξ < b) (4.2.1) ξ ξ a, b f(x) f (x) = 0 f(x) f(x) (a, b) 17 f(x) 3.3.2 ξ f ξ (a, b) ξ ξ f(ξ) f(x) ξ f (ξ) = lim h 0 f(ξ + h) f(ξ) h (4.2.2) h h > 0 h < 0 h 0 h a ξ b x a ξ b x 17

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 29 Lagrange 4.2.2 ( 2.3.10) f(x) [a, b] (a, b) ξ f(b) f(a) b a = f (ξ) (a < ξ < b) (4.2.3) ξ a, b g(x) = f(x) f(a) x a b a {f(b) f(a)} 0 = g (ξ) = f (ξ) 1 b a {f(b) f(a)} a < ξ < b 4.2.3 ( p.64 3) f(x) g(x) [a, b] (a, b) (a, b) g (x) 0 f(b) f(a) g(b) g(a) = f (ξ) g (ξ) (a < ξ < b) (4.2.4) ξ g (x) 0 g(a) g(b) f(b) f(a) k := F (x) := f(x) f(a) k{g(x) g(a)} F (a) = F (b) = 0 g(b) g(a) F f, g F (ξ) = 0 ξ f (ξ) kg (ξ) = 0 4.2.1 4.2.4 ( ) 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 3.2.1

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 30 4.2.5 ( 2.3.12 2.3.14) f(x) I = (a, 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 a f (a) > 0 x = a p.135 f (x) > 0 f (x) < 0 f(x) = x 3 4.3 f(x) n- n- n th derivative 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... (4.3.1) 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) (4.3.2) ( ) n f (k) (x) g (n k) (x), k ( ) n n! := n C k = k k! (n k)! (4.3.3) 18 ( ) ( ) ( ) n n 1 n 1 = + (4.3.4) k k k 1 I f(x) n f (n) (x) I C n - m < n C n - C m - ( 18 (a + b) n n = n ) ( k=0 k a k b n k n k)

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 31 4.3.1 4.3.1 x = a f(x) local maximum r > 0, 0 < x a < r = f(x) < f(a) (4.3.5) f x = a x = a f(x) local minimum r > 0, 0 < x a < r = f(x) > f(a) (4.3.6) r > 0, x a < r = f(x) f(a) (4.3.7) f a f(x) x = a maximum f f(a) f x f(x) < f(a) (4.3.8) x minimum local global p.69 70 4.3.2 x = a f(x) (i) f(x) x = a x = a f(x) f (a) = 0 (ii) f(x) x = a f (a) = 0 a. f (a) > 0 f(x) x = a b. f (a) < 0 f(x) x = a c. f (a) = 0 f(x) x = a (ii)-c 4.3.2 f (x) x f(x) y = f(x) f (x) f (x) y = f(x) f (x) > 0 x f (x) < 0 x

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 32 f f convex function concave function

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 33 4.4 f(x) f(a) f(x) = f(a) + n=1 f (n) (a) (x a) n (4.4.1) n! a = 0 e x = 1 + x + x2 2 + x3 3! + x4 4! + = 1 n! xn (4.4.2) n=0 sin x = x x3 3! + x5 5! x7 7! + = ( 1) n (2n + 1)! x2n+1 (4.4.3) n=0 cos x = 1 x2 2! + x4 4! x6 6! + = n=0 ( 1) n (2n)! x2n (4.4.4) sin x cos x sin x cos x 2π sin π = 0 sin x cos x (4.4.2) e x, sin x 19 (4.4.1) x a f(x) f(a) 4.4.1 4.4.1 ( ) f(x) I n a I x I a x ξ n 1 f(x) = f(a) + (4.4.5) k=1 f (k) (a) k! (x a) k + f (n) (ξ) (x a) n (4.4.5) n! f(x) = S n (x) + R n (x), (4.4.6) 19 e x, sin x (4.4.2)

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 34 n 1 S n (x) := f(a) + k=1 f (k) (a) k! (x a) k, R n (x) := f (n) (ξ) (x a) n (4.4.7) n! S n (x) n R n (x) n f(x) x = a a = 3 f(x) x = 3 x =... x =... x =... x = 2 x = 2 a = 0 Maclaurin y = x a x x = a y y = 0 y x x = a ξ a x b R n (x) x, a R n (x) ξ x, a 4.4.1 f (n) (x) 4.4.7 ξ ξ 4.4.1 20 [ n 1 F (x) := f(x) f(a) + F (x) (4.4.6) R n (x) k=1 f (k) (a) (x a) ], k G(x) := (x a) n (4.4.8) k! 4.2.3 F, G F (x) f(x) (x a) k G(x) n F (a) = F (a) = F (a) =... = F (n 1) (a) = 0, F (n) (a) = f (n) (a) (4.4.9) G(a) = G (a) = G (a) =... = G (n 1) (a) = 0, G (n) (a) = n! (4.4.10) 4.2.3 F (x) F (a) G(x) G(a) = F (ξ 1 ) G (ξ 1 ) ξ 1 ξ 1 a x F (a) = G (a) = 0 4.2.3 (4.4.11) F (ξ 1 ) G (ξ 1 ) = F (ξ 1 ) F (a) G (ξ 1 ) G (a) = F (ξ 2 ) G (ξ 2 ) (4.4.12) ξ 2 ξ 2 a ξ 1 F (k) (a) = G (k) (a) = 0 k n 1 F (k) (ξ k ) G (k) (ξ k ) = F (k) (ξ k ) F (k) (a) G (k) (ξ k ) G (k) (a) = F (k+1) (ξ k+1 ) G (k+1) (ξ k+1 ) (4.4.13) 20

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 35 ξ k+1 ξ k+1 a ξ k k n 1 F (x) F (a) G(x) G(a) = F (n) (ξ n ) G (n) (ξ n ) ξ n ξ n a x F (x) (x a) n = f (n) (ξ n ) n! (4.4.14) (4.4.15) 4.4.2 4.4.1 (4.4.6) n 1 R n (x) n lim n R n(x) = 0 f(x) = lim n S n(x) = k=0 f (k) (a) (x a) k (4.4.16) k! lim n S n R n (4.4.6) n R n S n n f(x) R n (x) n f I 4.4.3 f(x) = c n (x a) n + c n 1 (x a) n 1 +... + c 1 (x a) + c 0 f(x) = e x e x a = 0 e x = n 1 k=0 x k k! + R n(x), R n (x) := eξ n! xn (4.4.17) ξ 0 x x lim n R n(x) = 0 x e x = k=0 x k k! (4.4.18)

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 36 sin, cos sin x = S n (x) + R n (x), S n (x), R n (x) (4.4.19) 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)! (4.4.20) 21 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=3 5 7 2 n=11 21 31 41 1 sin x 1 sin x 0 0 2 4 6 8 10 0 0 10 20 30 40 x x -1-1 -2 n=2 4 6 8-2 n=10 20 30 40 21

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 37 4.4.4 f(x) (x a) k S n (x) (4.4.6) S n (x) f(x) R n (x) f(x) = 1/(1 x) 4.4.2 (n ) x = 0 f(x), g(x) x 0 f(x) g(x) lim x 0 x n = 0 n (4.4.21) 0 g(x) f(x) n n f(x) g(x) x n 22 4.4.3 ( 2.3.5) lim f(x) = lim h(x) = 0 x a x a lim x a f(x) h(x) = 0 f(x) h(x) f(x) = o( h(x) ) o f(x) x a h(x) K > 0 δ > 0 ( 0 < x a < δ = f(x) ) < K h(x) f(x) h(x) f(x) = O ( h(x) ) O (4.4.22) x f(x) g(x) f(x) g(x) f(x) g(x) f(x) g(x) f(x) = Ω ( g(x) ) (4.4.21) f(x) g(x) = o(x n ) n o (4.4.23) 22

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 38 4.4.4 ( ) 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) (4.4.24) n! S n (x) f(x) (n 1) f(x) = n 1 k=0 f (k) (0) x k + o(x n 1 ) (4.4.25) k! lim x 0 R n (x) = 0 (4.4.26) xn 1 (4.4.26) 4.4.5 ( ) 1) 0 f (n) δ > 0 M > 0 n x x < δ f (n) (x) < M (4.4.27) f(x) = n 1 k=0 f (k) (0) x k + O(x n ) (4.4.28) k! 2) 0 f (n) f(x) C n - 1) C - 4.4.4 S n 4.4.6 ( 2.5.6 1)) f(x) g(x) = n j=0 a jx j g(x) f(x) n g(x) a 0, a 1,..., a n a j f n g n g 1 (x) = a j x j, g 2 (x) = j=0 a 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 (4.4.29) j=0 + f(x) g 2(x) x n (4.4.30) g 1 (x) g 2 (x) lim x 0 x k = 0 (0 k n) (4.4.31)

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 39 g 1 (x) g 2 (x) = n (a j b j )x j (4.4.32) k = 0, 1, 2,... (4.4.31) a k b k = 0 k = 0, 1, 2,... j=0 f(x) S n f(x) (n 1) 1/(1 3x) tan x x = 0 tan x = sin x cos x p.82 4.4.5 1. (x a) n 2. x x 4.4.6 Euler e iθ = cos θ + i sin θ, θ R (4.4.33) 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 (4.4.34) k i k cos θ + i sin θ sin, cos 2π e a+b = e a e b

A http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 40 4.4.7 4.4.7 ( 2.5.8) f(x) I C n - I a I x I f(x) = S n (x) + R n (x), S n (x) := n 1 k=0 f (k) (a) x (x a) k, R n (x) := k! a f (n) (y) (n 1)! (x y)n 1 dy (4.4.35) f(x) C N - (4.4.35) n N n I. n = 1 x a f (y)dy = f(x) f(a) f(a) f (0) (x) := f(x) I. n = 2 n = 1 x a f (y)dy = x a f(x) = f(a) + { d dy (x y)} f (y)dy = = (x a)f (a) + x a x a f (y)dy (4.4.36) [ ] x (x y)f (y) + a x a (x y)f (y)dy (x y)f (y)dy (4.4.37) II. n n + 1 n N 1 n (4.4.35) (n 1)! x a x { f (n) (y)(x y) n 1 dy = f (n) (y) 1 d a 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 a n a = 1 n f (n) (a) (x a) n + 1 n x a f (n+1) (y) (x y) n dy. (4.4.38) (4.4.35) (n 1)! (4.4.35) n + 1