I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

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1 A tel: , hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ A ɛ-δ 1. ɛ-n ɛ-δ 2. 3.

2 I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

3 3 A, B ɛ-δ ɛ-δ TA teaching assistant

4 4 ɛ-δ ɛ-δ ɛ-δ ɛ-δ ɛ-δ hara/lectures/lectures-j.html spam mail html mail

5 5 1 1

6 6 2 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 2

7 A hara/lectures/lectures-j.html x 2, x 4, sin x cos(x 2 + 2),... (1.1.1) f(x) x f(x) x f(x) D D f D x f(x) f : D R x f(x) (1.1.2) x y R 1 f x f(x) x y ρ(x, y) = x y x y x, y 4 x, y (x, y) f(x, y) x, y (x, y) xy- 3 4

8 A hara/lectures/lectures-j.html 8 n n 2 n n = 2 n = 3 5 n x x = (x 1, x 2,..., x n ) n R n R n := {x = (x 1, x 2,..., x n ) x 1, x 2,..., x n R } D n R n D R n D n f D x = (x 1, x 2,..., x n ) f(x) = f(x 1, x 2,..., x n ) f : D R (1.1.3) x f(x) x = (x 1, x 2,..., x n ) y = (y 1, y 2,..., y n ) ( n ρ(x, y) = x y = (x j y j ) 2) 1/2 j=1 (1.1.4) n = 2 n = 3 x 1, x 2, x 3 x = (x, y) z = (u, v) f(x) x f(x, y) R n domain, region ( ) x f(x) lim x a f(x) = α x a 0 f(x) α 0 (1.1.5) x a f(x) f(a) ( ) n f(x) lim x a f(x) = α x a 0 f(x) α 0 (1.1.6) x a 0 x a 0 x a a x x a x a 5 n = 2, 3 n n = 2, 3 n n n n = 1, 2, 3 n

9 A hara/lectures/lectures-j.html 9 a x a f(x) α n ( ) n f(x) x = a lim x a f(x) = f(a) x a n x, a 1.2 n f(x, y) n ( ) f(x, y) (a, b) f(a + h, b) f(a, b) f(x, b) f(a, b) lim = lim h 0 h x a x a (1.2.1) f x (a, b) f 1(a, b), f x (a, b) D 1 f(a, b) f(a, b + h) f(a, b) f(a, y) f(a, b) lim = lim h 0 h y b y b (1.2.2) f y (a, b) f 2(a, b), f y (a, b) D 2 f(a, b) (x, y) f f x (x, y) y (x, y) f x y f (a, b) (a, b) x f f x x (a, b) (C 1 - ) f(x 1, x 2,..., x n ) D f x 1, x 2,..., x n n- x = (x 1, x 2,..., x n ) f D C 1 - n- C n - x y = b x z = f(x, y) y = b

10 A hara/lectures/lectures-j.html 10 y x = a f(x,y) b a x a) x 2 + y 3, b) 2x 2 y c) sin(xy 2 ) d) (x 2 + y + z 3 ) 2 0 (x, y) = (0, 0) e) f(x, y) = 2xy (x, y) (0, 0) x 2 +y f f f f(x, y) f x (x, y) 0 f x x x x y f f(x, y) = g(y) g (1.2.3)

11 A hara/lectures/lectures-j.html 11 f(x, y) f x f x D x f D 1 f x f 1 f f x f 1 f x f f x f(x, y) f f x y x- y- x, y (a, b) f(x, y) x- y- x = y x, y ( ) f(x, y) 1 v = (v x, v y ) f v (a, b) := lim h 0 f(a + hv x, b + hv y ) f(a, b) h (1.2.4) f(x, y) (a, b) v n f(x) n v f v (a) := lim h 0 f(a + hv) f(a) h (1.2.5) f(x) a v D v f(a) f(a) v D v f(a) x 1 f 1 (x) x 1 - f(a) f, g f(0, 0) = 0, (x, y) (0, 0) f(x, y) = 2xy x 2 + y 2 (1.2.6) g(0, 0) = 0, (x, y) (0, 0) g(x, y) = (x, y) xy x2 + y 2 (1.2.7) f x (0, 0) = f y (0, 0) = 0, (x, y) (0, 0) f x (x, y) = 2y(y2 x 2 ) (x 2 + y 2 ) 2, f y(x, y) = 2x(x2 y 2 ) (x 2 + y 2 ) 2 (1.2.8) g x (0, 0) = g y (0, 0) = 0, (x, y) (0, 0) g x (x, y) = y 3 (x 2 + y 2 ), g x 3 y(x, y) = (1.2.9) 3/2 (x 2 + y 2 ) 3/2 (0, 0) ( 1 2, 1 2 ) 6

12 A hara/lectures/lectures-j.html f(x) x = a f (a) y = f(x) x a f(x) f(x) f(a) + f (a) (x a). (1.2.10) x = a f(x) (1.2.6) (1.2.7) x = y = 0 f f(x, y) x = y f(x, x) = 1 x 0 8 x y x x- x = y x- x y y- ( n x = (x 1, x 2,..., x n ) x a = (x j a j ) 2) 1/ ( ) D n f(x) D a A 1, A 2,..., A n x a f(x) = f(a) + n j=1 j=1 A j (x j a j ) + f(x), f(x) with lim x a x a = 0 (1.2.11) f x = a (1.2.6) (1.2.7) f, g (0, 0) (1.2.11) a x a x- y D n f(x) D a f(x) x = a x = a 1. f(x) 2. f(x) x j ( ) (1.2.11) A j = f x j (a) (j = 1, 2,..., n), (1.2.12) 3. n v = (v 1, v 2,..., v n ) D v f(a) = n A j v j = j=1 n j=1 f x j v j (1.2.13) 7 8

13 A hara/lectures/lectures-j.html f (1.2.11) f(x) f(a) 2. x 1 x 2, x 3,... a 2, a 3,... (1.2.11) f(x 1, a 2, a 3,..., a n ) f(a 1, a 2, a 3,..., a n ) x 1 a 1 = A 1 + f(x) x 1 a 1 (1.2.14) x 1 a 1 f x 1 x 2, x 3,... a 2, a 3,... x 1 a 1 = x a (1.2.11) f(x) f x 1 A 1 x 1 a 1 x 2 3. (1.2.11) f(a + hv) f(a) h x = a + hv = n A j hv j + f(a + hv) j=1 h = n j=1 A j v j + f(a + hv) h (1.2.15) f(a + hv) h = f(x) x a (1.2.16) (1.2.11) (1.2.15) h 0 (1.2.13) f(x, y) (1.2.11) (1.2.11) A, B f(x, y) = f(a, b) + A(x a) + B(y b) (1.2.17) z = f(x, y) z c = A(x a) + B(y b) c = f(a, b) (a, b, c) S (1.2.11) z = f(x, y) x a z = f(x, y) S S z = f(x, y) (a, b) (1.2.11) z = f(x, y) z = f(x, y) (1.2.6) (1.2.7) f, g (0, 0) (C 1 ) D n f(x) C 1 - D f(x) D D (a, b) a := (a, b), x = (x, y) f(x, y) f(a, b) = { f(x, y) f(a, y) } + { f(a, y) f(a, b) } (1.2.18) x a (1.2.11)

14 A hara/lectures/lectures-j.html 14 (1.2.18) x a y f C 1 f(a, y) f(a, b) = f y (a, ỹ) (y b) (1.2.19) ỹ y b f y = f y y x x a x (1.2.18) f(x, y) f(a, y) = f x ( x, y) (x a) (1.2.20) f(x, y) f(a, b) = f x ( x, y) (x a) + f y (a, ỹ) (y b) (1.2.21) f x ( x, y), f y (a, ỹ) f C 1 - f x, f y (u, v) f x (u, v) = f x (a, b) + g(u, v), with lim g(u, v) = 0, (1.2.22) (u,v) (a,b) f y (u, v) = f y (a, b) + h(u, v), with lim h(u, v) = 0 (1.2.23) (u,v) (a,b) (1.2.22) u = x, v = y (x, y) (a, b) ( x, y) (a, b) (1.2.21) f x ( x, y) = f x (a, b) + g( x, y), with lim g( x, y) = 0 (1.2.24) (x,y) (a,b) f y (a, ỹ) = f y (a, b) + h(a, ỹ), with lim h(a, ỹ) = 0 (1.2.25) (x,y) (a,b) f(x, y) f(a, b) = f x (a, b) (x a) + f y (a, b) (y b) + g( x, y) (x a) + h(a, ỹ) (y b) (1.2.26) g, h f(x, y) lim (x,y) (a,b) f f(x, y) = 0 x = (x, y), a = (a, b) (1.2.27) x a f(x, y) = f(a, b) + A(x a) + B(y b) + f(x, y), with A = f x (a, b), B = f y (a, b) (1.2.28) (1.2.11) n-

15 A hara/lectures/lectures-j.html chain rule f(x) g(y) h(x) = f(g(x)) (1.3.1) h f g f g sin(x 3 ) f(x) = sin x g(x) = x 3 h(x) h (x) = f ( g(x) ) g (x) (1.3.2) sin(x 3 ) x f g n A. f(z) z(x) h(x) = f ( z(x) ) x B. f(z) z(x, y) h(x, y) = f ( z(x, y) ) x, y C. f(x, y) x(t), y(t) h(t) = f ( x(t), y(t) ) t D. f(x, y), x(u, v), y(u, v) h(u, v) = f ( x(u, v), y(u, v) ) u, v A B h(x, y) x y y f(x, y) x B A h(x, y) x = f ( z(x, y) ) z(x, y), x h(x, y) y = f ( z(x, y) ) z(x, y) y (1.3.3) C D C B A 1.3.1, Case A g(x) I f(y) J g g(i) = {g(x) x I} 10 J h(x) = f ( g(x) ) I g I x g(x) g x g I g {g(x) x I} g(i) g G G(I) = {g(x) x I} g g(i)

16 A hara/lectures/lectures-j.html ( ) g(x) I x = a x f(z) b = f(a) z h(x) = f ( g(x) ) x = a h (x) = f ( g(x) ) g (x), z = g(x), w = h(z) dw dx = dw dz dz dx g(x) x = a f(z) z = f(a) h(x) x = a h(a + ɛ) h(a) ɛ = f( g(a + ɛ) ) f ( g(a) ) ɛ = f( g(a + ɛ) ) f ( g(a) ) g(a + ɛ) g(a) g(a + ɛ) g(a) ɛ (1.3.4) ɛ 0 g (a) g(a + ɛ) g(a) f g(a) ɛ 0 g(a + ɛ) g(a) = 0 g(x) 1 1 g(a + ɛ) g(a) = 0 (1.3.4) g(a + ɛ) g(a) = 0 h(g(a + ɛ)) h(g(a)) = g (a) g (a) = lim ɛ 0 g(a + ɛ) g(a) ɛ ɛ g(ɛ) g(ɛ) g(a + h) g(a) g (a)ɛ lim = 0 (1.3.5) ɛ 0 ɛ 11 (1.3.5) g(ɛ) := g(a + h) g(a) g (a)ɛ ɛ (1.3.6) g(a + ɛ) = g(a) + ɛg (a) + ɛ g(ɛ), with lim ɛ 0 g(ɛ) = 0 g(0) = 0 (1.3.7) g(a) = b f f(b + η) = f(b) + ηf (b) + η f(η), with lim η 0 f(η) = 0 f(0) = 0 (1.3.8) ( ) ( ) h(a + ɛ) h(a) f g(a + ɛ) f g(a) lim = lim ɛ 0 ɛ ɛ 0 ɛ (1.3.7) (1.3.8) (1.3.9) f ( g(a + ɛ) ) f ( g(a) ) = f ( g(a) + ɛg (a) + ɛ g(ɛ) ) f ( g(a) ) = f ( b + ɛg (a) + ɛ g(h) ) f(b) = f(b) + ηf (b) + η f(η) f(b) = ηf (b) + η f(η) with η := ɛg (a) + ɛ g(ɛ) (1.3.10) 11 g g g(ɛ) p(ɛ) a ɛ

17 A hara/lectures/lectures-j.html 17 h(a + ɛ) h(a) ηf (b) + η lim = lim f(η) [ = lim ɛ 0 ɛ ɛ 0 ɛ ɛ 0 {g (a) + g(h)}{f (b) + f(η)} ] ] [ = f (b)g (a) + lim f (b) g(ɛ) + g (a) f(η) + g(ɛ) f(η) ɛ 0 (1.3.11) (1.3.7) g(ɛ) (1.3.8) (1.3.10) η [ ] lim η = lim ɛg (a) + ɛ g(h) = 0 g (a) = 0 (1.3.12) ɛ 0 ɛ 0 12 (1.3.8) f(η) (1.3.11) h (a) = f (b)g (a) = f (g(a))g (a) 1.3.2, Case B B J f(z) D g(x, y) g f h(x, y) = f ( g(x, y) ) D g(x, y) R (a, b) x f(z) c = g(a, b) h(x, y) = f ( g(x, y) ) (a, b) x h x (a, b) = f ( g(a, b) ) g x (a, b), z = g(x, y), w = f(z) w x = dw dz z x (1.3.13) x y , Cases C & D C f(x, y) x(t), y(t) h(t) = f(x(t), y(t)) t x 0 = x(t) y 0 = y(t), x 1 = x(t + ɛ), y 1 = y(t + ɛ) 13 ( ) ( ) h(t + ɛ) h(t) f x(t + ɛ), y(t + ɛ) f x(t), y(t) f(x 1, y 1 ) f(x 0, y 0 ) lim = lim = lim ɛ 0 ɛ ɛ 0 ɛ ɛ 0 ɛ f(x 1, y 1 ) f(x 1, y 0 ) f(x 1, y 0 ) f(x 0, y 0 ) = lim + lim (1.3.14) ɛ 0 ɛ ɛ 0 ɛ ɛ x 1 = x(t + ɛ) y 0 = y(t) ɛ 0 y 0 f x (x 0, y 0 ) x (t) x 2 ɛ f(x 2, y 1 ) f(x 2, y 0 ) lim ɛ 0 ɛ with y 0 = y(t), y 1 = y(t + ɛ) (1.3.15) 12 lim f(ɛ) = lim g(ɛ) = 0 lim f(ɛ)g(ɛ) = 0 13 ɛ 0 ɛ 0 ɛ 0 x(t + ɛ)

18 A hara/lectures/lectures-j.html 18 x = x 2 y ɛ case B f y (x 2, y) y (t) x 2 x 0 f y (x 0, y 0 ) y (t) (1.3.14) h (t) = f x (x 0, y 0 ) x (t) + f y (x 0, y 0 ) y (t) (1.3.16) (1.3.16) (1.3.14) g(x, y) x(t) = y(t) = t t 0 g(0, 0) = 0, (x, y) (0, 0) g(x, y) = xy x2 + y 2. (1.3.17) (1.2.9) g x (0, 0) = g y (0, 0) = 0 t h(t) = g(x(t), y(t)) = t 2, h (t) = 1 2 (1.3.18) h (0) 0 (1.3.16) x = y = 0 h (0) = 0 g(x, y) (1.3.16) C f(x, y) C 1 - x(t), y(t) t h(t) = f ( x(t), y(t) ) t h (t) = f x x (t) + f y y (t), z = f(x, y) dz dt = z dx x dt + z dy y dt (1.3.19) f (x(t), y(t)) f n d dt f( x 1 (t), x 2 (t),..., x n (t) ) = n j=1 f x j dx j dt. (1.3.20) (1.3.17) C 1 (1.3.14) f(x, y) y f C 1 f(x 1, y 1 ) f(x 1, y 0 ) = f y (x 1, y 3 ) (y 1 y 0 ) (1.3.21) y 3 y 0 y 1 ɛ ɛ 0 [ f(x 1, y 1 ) f(x 1, y 0 ) lim = lim f y (x 1, y 3 ) y ] 1 y 0 ɛ 0 ɛ ɛ 0 ɛ (1.3.22) f C 1 f y (x, y) x, y ɛ 0 x 1 x 0 y 3 y 0 lim f y(x 1, y 3 ) = f y (x 0, y 0 ) (1.3.23) ɛ 0

19 A hara/lectures/lectures-j.html 19 (1.3.22) y 1, y 0 y 1 y 0 lim ɛ 0 ɛ y(t + ɛ) y(t) = lim = y (t) (1.3.24) ɛ 0 ɛ f(x 1, y 1 ) f(x 1, y 0 ) lim = f y (x 0, y 0 ) y (t) (1.3.25) ɛ 0 ɛ (1.3.14) n- D ( ) f(x, y) C 1 - x(u, v), y(u, v) u, v C 1 - z = h(u, v) = f ( x(u, v), y(u, v) ) C 1 - z u = z x x u + z y y u, z v = z x x v + z y y v. (1.3.26) u v xy f(x, y) = x, y r, θ x2 + y2 x = r cos θ, y = r sin θ h(r, θ) = f ( x(r, θ), y(r, θ) ) r, θ h r, θ (x, y) (u, v) α, β, γ, δ αδ βγ f x f y f u f v u = αx + βy, v = γx + δy (1.3.27) α, β, γ, δ C 1 - f(x, y) a, b 1) f y 0 f 2) x + f y 0 3) a f x + b f y 0 (1.3.28) 1) ) 3) (u, v) f u chain rule f C 1 f ( ) f(x, y) x(t), y(t) t h(t) = f ( x(t), y(t) ) t h (t) = f x x (t) + f y y (t), z = f(x, y) dz dt = z dx x dt + z dy y dt (1.3.29) n-

20 A hara/lectures/lectures-j.html ( ) f(x, y) x(u, v), y(u, v) z = h(u, v) = f ( x(u, v), y(u, v) ) (u, v) z u = z x x u + z y y u, z v = z x x v + z y y v (1.3.30) x 0 = x(t), x 1 = x(t + ɛ), y 0 = y(t), y 1 = y(t + ɛ) h(t + ɛ) h(t) = f(x 1, y 1 ) f(x 0, y 0 ) = A(x 1 x 0 ) + B(y 1 y 0 ) + o ( x 1 x y 1 y 0 2) (1.3.31) A = f x (x 0, y 0 ), B = f y (x 0, y 0 ) x(t), y(t) p = x (t), q = y (t) x 1 x 0 = x (t)ɛ + o(ɛ) = pɛ + o(ɛ), y 1 y 0 = y (t)ɛ + o(ɛ) = qɛ + o(ɛ) (1.3.32) ( ) h(t + ɛ) = h(t) + A{pɛ + o(ɛ)} + B{qɛ + o(ɛ)} + o {pɛ + o(ɛ)}2 + {qɛ + o(ɛ)} 2 ) = h(t) + (Ap + Bq)ɛ + o(ɛ) + o( (p2 + q 2 )ɛ 2 + o(ɛ 2 ) (1.3.33) ) ) o( (p2 + q 2 )ɛ 2 + o(ɛ 2 ) = o( p2 + q 2 ɛ + o(ɛ) = o(ɛ) (1.3.34) (1.3.33) h(t + ɛ) = h(t) + (Ap + Bq)ɛ + o(ɛ) (1.3.35) h(t) t Ap + Bq (1.2.10) 1.4 f(x) x f (x) f (x) x f (x) = f (2) (x) n- f (n) (x) f(x, y) f x (x, y) x, y x f x 2 f x 2 = ( ) f x x f x y 2 f y x = ( ) f 2 f, y x x y = ( ) f 2 f, x y y 2 = ( ) f (1.4.1) y y x y x y y x y

21 A hara/lectures/lectures-j.html f, g, h 2 f x y = 2 f y x f(x, y) = x 2 + y 2, g(x, y) = x 2 y 3, h(x, y) = cos(x 2 y) (1.4.2) k- f(x, y) k- f(x, y) x y k n- n 3 k- C k - 2 f x y 2 f y x g x g y 2 g x y 2 g y x f(x, y) C 2-2 f x y = 2 f y x (1.4.3) n- C 2 - n- C k - k C 2-15 D A D f x = f x, f yx = 2 f x y D f yx f xy D f xy = f yx D f x, f y, f yx A f yx f xy A f xy = f yx D f x, f y A A f xy = f yx (a, b) f (h, k) := f(a + h, b + k) f(a + h, b) f(a, b + k) + f(a, b) (1.4.4) hk h k ( lim lim h 0 k 0 (h, k) ) 1 = lim hk h 0 1 = lim h 0 h [ ( f(a + h, b + k) f(a + h, b) f(a, b + k) f(a, b) )] lim h k 0 k k [ f f ] (a + h, b) y y (a, b) = ( f ) = 2 f x y x y ( lim lim k 0 h 0 (h, k) ) = 2 f hk y x (1.4.5) (1.4.6) h, k d)

22 A hara/lectures/lectures-j.html 22 b+k b+φk b a a+θh a+h ϕ(x) := f(x, b + k) f(x, b) b, k x ϕ(a + h) ϕ(a) = ϕ (a + θh)h (1.4.7) (h, k) = ϕ(a + h) ϕ(a) = ( ) f x (a + θh, b + k) f x (a + θh, b) h (1.4.8) 0 < θ < 1 a + θh a, h, θ ψ(y) := f x (a + θh, y) y 16 ψ(b + k) ψ(b) = ψ (b + φk)k (1.4.9) f x (a + θh, b + k) f x (a + θh, b) = f xy (a + θh, b + φk) k (1.4.10) 0 < φ < 1 ψ(y) C 1 - f C 2 - x = a + θh b + φk (1.4.7) (1.4.10) (h, k) hk = f xy(a + θh, b + φk) hk hk = f xy (a + θh, b + φk) (1.4.11) 0 < θ < 1, 0 < φ < 1 θ, φ 0 1 f xy h, k f xy (a, b) (h, k) lim = f xy (a, b) (1.4.12) (h,k) (0,0) hk h 0 k 0 (1.4.5) (1.4.6) (1.4.12) f yx (a, b) f xx = 2 f x z = f(x, y) y x 2 f xx 16 (1.4.7) a, b, h, k θ θ ψ(y) b b + k (1.4.10) (1.4.7) b + k, b (1.4.7) (1.4.10)

23 A hara/lectures/lectures-j.html 23 z = f(x, y) y x- f yy x y- f xy = f yx x y ( ) ( ) ( ) u = x, v 1 1 y ( ) ( ) ( ) x = u y 1 1 v (1.4.13) (x, y) 45 (u, v) u, v f yx f x = u f x u + v f x v = 1 [ f 2 u f ] v (1.4.14) x = u x u + v x v = 1 [ 2 u ] v y 2 f y x = 1 [ 2 u + ] { [ 1 f 2 v u f ]} = 1 [ v 2 u + ] [ f v u f ] v = 1 [ 2 ] f 2 u 2 2 f u v + 2 f v u 2 f v 2 = 1 [ 2 ] f 2 u 2 2 f v 2 (1.4.15) (1.4.16) f uv = f vu u- v- f xy = f yx x f u u, v x f u x x = r cos θ, y = r sin θ 2 f x r, θ 2 f x = r f x r + θ f x θ = x f x2 + y 2 r y f x 2 + y 2 θ = x f r r y f r 2 (1.4.17) θ x = x r r y r 2 2 f ( x 2 = cos θ r sin θ r ) ( cos θ f θ r sin θ r θ = cos θ r sin θ r f ) θ θ (1.4.18) (1.4.19) = cos θ r ( cos θ f r ) sin θ r ( θ cos θ f r ) + cos θ ( sin θ r r f ) sin θ θ r ( sin θ θ r f ) θ (1.4.20)

24 A hara/lectures/lectures-j.html 24 sin θ r ( θ cos θ f r ) = sin θ r cos θ θ f r + sin θ cos θ r r 2 f x 2 = cos2 θ 2 f r 2 + sin2 θ f sin 2θ f + r r r 2 θ + sin2 θ r 2 ( f ) = sin2 θ f sin θ cos θ + θ r r r 2 f sin 2θ θ2 r 2 f r θ 2 f r θ (1.4.21) (1.4.22) f(x, y) x = r cos θ, y = r sin θ f(x, y) := 2 f x f y 2 (1.4.23) f r, θ f y x = f xy 0 f x y ( ) fx = 0 (1.4.24) y f x (x, y) = g(x) (1.4.25) g x f(x, y) g x y x y g(x) G(x) (1.4.25) { } f(x, y) G(x) = 0 (1.4.26) x f(x, y) G(x) y g(x) G(x) f(x, y) = G(x) + h(y) (1.4.27) h y f x y

25 A hara/lectures/lectures-j.html 25 2 ɛ-δ 3 ɛ-δ 2.1 ɛ-n 17 ɛ-n lim n a n = α n a n α 18 lim a n = 0 1 n a k n n k= ɛ-n 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)

26 A hara/lectures/lectures-j.html 26 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 = 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

27 A hara/lectures/lectures-j.html 27 1 n n N n N N N N = 10 4 N = N = N n a n = 1/n n n n ɛ > 0 n a n α ɛ ɛ ɛ ɛ = 10 6 ɛ = ɛ = N ɛ a n α N ɛ 19 a n α a n α n a n = 1/n ɛ = n > 100 n > 100 a n α < ɛ = 10 6 n > n > a n α < 10 6 ɛ = n > ɛ = n > ɛ > 0 lim n a n = α ɛ = 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)) (2.1.6) n 19

28 A hara/lectures/lectures-j.html 28 n a n b n c n 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 n N(ɛ) n = 1, 2, 3,... a 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 (2.1.7) (2.1.8) (2.1.6) 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) (2.1.9) 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 } a n = n

29 A hara/lectures/lectures-j.html a n n lim a n = α lim a n = β n n α = β ɛ-n a n b n = 1 n n k=1 a k lim n a n = α lim n b n = α ɛ-n lim a a 1 + a a n n = α = lim 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 ρ 1 = ρ 2 = ρ 3 =... = 1 j=1 = α 2.2 ɛ-δ 20 n a n x x a f(x) 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

30 A hara/lectures/lectures-j.html 30 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) 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 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) f(x) lim f(x) x 0 ɛ-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 a x a x a x a ɛ-δ

31 A hara/lectures/lectures-j.html 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) = x lim x f(x) = lim n a n = α lim x a f(x) = b lim x a a 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 f(x) a, b lim f(x) = b x a lim f(x) = lim f(x) = b x a+0 x a 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

32 A hara/lectures/lectures-j.html 32 lim x xα =, lim x xα = 0, lim x +0 xα = 0, α > 0 (2.2.10) lim x +0 xα =, α < 0 (2.2.11) x α = e α log x e x log x x lim x a f(x) = b lim n a n = a n a n a {a n } lim n f(a n) = b < x a < δ n a n a 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

33 A hara/lectures/lectures-j.html ( ) A N A a a N A bounded from above N A upper bound M A a 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 A A A inf A sup A A A web page

34 A hara/lectures/lectures-j.html ( ) S S [ ] [ ] S S [ ] ( 6) A α = sup A (i) A x x α (ii) α < α α α < x x A β = inf A A A α x α I. α = sup A (i), (ii) α A A x α x A (i) O.K. (ii) (ii) α α < x x A α α x x A α A α A (ii) II. (i), (ii) α = sup A α A (i) α A α α α α x x A (ii) α α A lim n a n = α a n α ɛ > 0, N(ɛ), n > N(ɛ) = a n α < ɛ (3.2.1) α

35 A hara/lectures/lectures-j.html 35 e ( e = lim 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 28 lim N N n=0 x n n n! lim N N n=0 x n n n! (3.2.4) ( ) a 1 a 2 a 3... a n... a n (monotone) increasing (monotone) decreasing (monotone) non-decreasing (monotone) non-increasing strictly increasing n n 28 e x

36 A hara/lectures/lectures-j.html ( ) {a n } M n a n < M {a n n 1} M {a n } n ( 7) {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 a n 2 n a n 2 n a n {a 1, a 2, a 3,...} α α = lim n a n a n α a n α ɛ ɛ > 0 n n ɛ > 0 α = α ɛ α {a n } α < a m α m m N(ɛ) := m n N(ɛ) α < a m a n α a n α < ɛ (3.2.5) a n ɛ-n lim n a n = α

37 A hara/lectures/lectures-j.html ( ) {a n } ( ) lim inf a n := lim inf a n n N n N, lim sup n ( ) a n := lim sup a n N n N (3.3.1) lim inf a n = lim a n lim sup a n = lim a n n n n n (3.3.1) b N = inf n N a n = inf{a n n N}, c N = sup a n = sup{a n n N} (3.3.2) n N lim inf n a n := lim N b N, lim sup a n := lim c N (3.3.3) n N lim sup lim inf sup inf lim n a n lim sup lim inf ( ) (i) +, a n lim inf a n lim sup a n (3.3.4) n n (ii) {a n } M, n, a n M (iii) lim a n = α lim sup a n = lim inf a n = α (3.3.5) n n (iii) β γ (i) (ii). {a n } (3.3.2) b N, c N n b n b n+1 b n+2 c n+2 c n+1 c n (3.3.6) b n c n b m c n b n, c n m n b n a m c n (3.3.7)

38 A hara/lectures/lectures-j.html 38 {a n } {b N } why? lim N b N (3.3.3) lim inf a n lim sup a n (iii). lim n a n = α ɛ N n N = α ɛ < a n < α + ɛ (3.3.8) N n N inf sup ( ) ɛ N α ɛ b N α + ɛ α ɛ c N α + ɛ (3.3.9) {b N } α ɛ b N n N α ɛ b n c N c N α + ɛ n N c n α + ɛ ( ) ɛ N n > N α ɛ b n c n α + ɛ (3.3.10) (3.3.7) m, n b n c m ( ) ɛ N n > N α ɛ b n c n α + ɛ (3.3.11) ɛ N n > N ( ) b n α ɛ c n α ɛ (3.3.12) lim b n = α lim c n = α ɛ-n n n lim b n = α lim c n = α ɛ-n n n ɛ > 0 N 1, N 2, { ( n > N1 = b n α < ɛ ) ( m > N 2 = c m α < ɛ )} (3.3.13) N 1 N 2 N ( ) ɛ > 0 N, n > N = b n α < ɛ c n α < ɛ (3.3.14) b n c n ɛ > 0 N, n > N = α ɛ < b n c n < α + ɛ (3.3.15) (3.3.7) b n a n c n ɛ > 0 N, n > N = α ɛ < a n < α + ɛ (3.3.16) lim n a n = α ɛ-n ( 9) a 1, a 2, a 3,... b 1, b 2,... a < b n a a n b [a, b] [a, a+b 2 ] [ a+b a+b 2, b] 2 {a n } [a, b] a n I 1 a n I 1 I 1

39 A hara/lectures/lectures-j.html 39 a n I 1 a n I 2 [a, b] I 1 I 2 I 3... I l b a 2 l {a n } I l a n b l {b l } lim sup b l lim inf b l m l b m I l l l {b l } ( ) a n Cauchy sequence ɛ > 0 N(ɛ) m, n N(ɛ) a m a n < ɛ (3.4.1) ε 1 ε 2 N(ε 1 ) N(ε 2 ) n a n a n a m m, n ( 11 ) a n a n lim sup lim inf ɛ-δ

40 A hara/lectures/lectures-j.html N(ɛ) a n := 1 n b n := 1 n 2 c n := ( 1)n n d n := ( 1)n n {a n }, {b n }, {c n } α c n a 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 a n, b n (3.2.4) e x e x = n=0 x n n! 30 x x > 0 x sin x = x x3 3! + x5 5! x7 7! + x 0 < r < 1 {a n } a n+2 a n+1 r a n+1 a n n = 1, 2, 3,... a n α ɛ > 0 N(ɛ) n > N(ɛ) a n α < ɛ/2 m, n > N(ɛ/2) a m α < ɛ 2, a n α < ɛ 2 (3.4.2) m, n a m a n = (a m α) + (α a n ) a m α + α a n < ɛ 2 + ɛ 2 = ɛ (3.4.3) (3.4.1) 30 e x e x

41 A hara/lectures/lectures-j.html β := lim inf n a n, γ := lim sup a n (3.4.4) n β γ (3.3.2) b N := inf a m, c N := sup a n m N n N β = γ {a n } ɛ > 0, N ( ) l, m N = a l a m < ɛ (3.4.5) ɛ > 0, N ( ) l, m N = a l a m < ɛ (3.4.6) N, m l N sup sup a l = c N l N ( ) ɛ > 0, N m N = c N a m ɛ (3.4.7) m N inf inf m N a l = b N ɛ > 0, N c N b N ɛ (3.4.8) c n b n {b n } {c n } {c n b n } N c N b N ɛ n N c n b n ɛ (3.4.8) ɛ > 0, N ( n N = ) c n b n ɛ lim n (c n b n ) = 0 ɛ-n β = γ (3.4.9) n x a ( 11 ) lim x a f(x) f(x) (C) ɛ > 0 δ(ɛ) > 0 0 < x a < δ(ɛ) 0 < y a < δ(ɛ) x, y f(x) f(y) < ɛ lim f(x) x a (C) lim x a f(x) lim x a f(x) = b

42 A hara/lectures/lectures-j.html 42 a n a lim n a n = a {a n } lim n f(a n) = b (C) a n a lim a n = a {a n } {f(a n )} n lim f(a n) n {a n } lim f(a n) n {a n } {a n } (C) (C)

43 A hara/lectures/lectures-j.html a f(x) a lim x a f(x) = f(a) ɛ δ(ɛ) 0 < x a < δ(ɛ) x f(x) f(a) < ɛ (4.1.1) ɛ > 0, δ(ɛ) > 0, 0 < x a < δ(ɛ) = f(x) f(a) < ɛ (4.1.2) 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) (4.1.3) x b 0 f(a) lim x a f(x) f(x) = x x f(x) x = a x = a 4.2, ( p.93 2) a f(x) x = a f(a) > 0 a f(x) > 0 x f(x) > 1 2 f(a) f(a) < 0

44 A hara/lectures/lectures-j.html 44 f(x) x = a ɛ-δ ( 19) [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 = f(a) < F < f(b) f(a) > f(b) f(a) = f(b) f(a) = F c = a g(x) := f(x) F g(c) = 0 c x = a x g(a) < 0 g(x) x = a g(x) < 0 a y a x < y x g(x) < 0 y Y g(b) > 0 y b Y Y α g(α) = 0 g(α) < 0 g(α) > 0 g(α) < α x g(x) < 0 Y α Y α g(α) > α x g(x) > 0 δ > 0 α δ < x α x g(x) < 0 Y α Y α g(α) = 0 c = α ( 20) f(x) = 1/x (0, 1) g(x) = x (0, 1) g(x) = 2 x 2 x [a, b] x [a, b] n (n + 1) n

45 A hara/lectures/lectures-j.html 45 (n + 1) f(x) x n n = 1, 2, 3,... x 1, x 2, x 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 ) (4.2.1) l y l α α l z l x f f(x) f(α) f(x) f(α) (4.2.2) x f(α) 4.3 x α α x > 0 x α α a n x α = lim n xa n (4.3.1) x α α lim n an = α {an} p.116

46 A hara/lectures/lectures-j.html ( ) x = a f(x) f(x) f(a) lim x a x a (5.1.1) 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 (5.1.1) x ( ) f (a) := f(x) f(a) lim x a 0 x a (5.1.2) f(x) a left derivative f + (a) := f(x) f(a) lim x a+0 x a (5.1.3) f(x) a right derivative f a f (a) f (a) = f + (a) f a f (a) = f (a) = f + (a)

47 A hara/lectures/lectures-j.html ( 25) f(x) x = a f a p.129 Weierstrass 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... (5.1.4) 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) (5.1.5) ( ) n f (k) (x) g (n k) (x), k ( ) n n C k = k n! k! (n k)! (5.1.6) 32 ( ) ( ) ( ) n n 1 n 1 = + (5.1.7) k k k 1 I f(x) n f (n) (x) I C n - m < n C n - C m ( Rolle 28) f(x) [a, b] (a, b) f(a) = f(b) = 0 f (ξ) = 0 (a < ξ < b) (5.2.1) ξ ) ( a k b n k n ( 32 (a + b) n n = n k=0 k k)

48 A hara/lectures/lectures-j.html 48 ξ a, b f(x) f (x) = 0 f(x) f(x) (a, b) 33 f(x) ξ f ξ (a, b) ξ ξ f(ξ) f(x) ξ f (ξ) = lim h 0 f(ξ + h) f(ξ) h (5.2.2) h h > 0 h < 0 h 0 h a ξ b x a ξ b x Lagrange ( 27) f(x) [a, b] (a, b) ξ f(b) f(a) b a = f (ξ) (a < ξ < b) (5.2.3) 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 ( 30) 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) (5.2.4) ξ 33

49 A hara/lectures/lectures-j.html 49 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) = ( ) 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 ( p.131, 26) 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 pp

50 A hara/lectures/lectures-j.html ( p ) f(x) [a, b] (a, b) (a, b) f (x) > 0 = [a, b] f(x) (a, b) f (x) < 0 = [a, b] f(x) (a, b) f (x) = 0 [a, b] f(x) f f f (x) > 0 f (x) < 0 f(x) = x ( 29) f(x) I = (a, b) I f (x) 0 I f(x) I f (x) 0 I f(x) x = a f(x) local maximum r > 0, 0 < x a < r = f(x) < f(a) (5.3.1) f x = a x = a f(x) local minimum r > 0, 0 < x a < r = f(x) > f(a) (5.3.2) r > 0, x a < r = f(x) f(a) (5.3.3) f a f(x) x = a maximum f f(a) f x f(x) < f(a) (5.3.4) x minimum local global

51 A hara/lectures/lectures-j.html 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 f (x) x f(x) y = f(x) f (x) f (x) y = f(x) f (x) > 0 x f (x) < 0 x f f pp convex function concave function

52 A hara/lectures/lectures-j.html f(x) f(a) f(x) = f(a) + n=1 f (n) (a) (x a) n (5.4.1) n! a = 0 e x = 1 + x + x2 2 + x3 3! + x4 4! + = 1 n! xn (5.4.2) n=0 sin x = x x3 3! + x5 5! x7 7! + = ( 1) n (2n + 1)! x2n+1 (5.4.3) n=0 cos x = 1 x2 2! + x4 4! x6 6! + = n=0 ( 1) n (2n)! x2n (5.4.4) sin x cos x sin x cos x 2π sin π = 0 sin x cos x (5.4.2) e x, sin x 34 (5.4.1) x a f(x) f(a) e x, sin x (5.4.2)

53 A hara/lectures/lectures-j.html ( ) f(x) I n a I x I a x ξ n 1 f(x) = f(a) + (5.4.5) n 1 S n (x) := f(a) + k=1 k=1 f (k) (a) k! (x a) k + f (n) (ξ) (x a) n (5.4.5) n! f(x) = S n (x) + R n (x), (5.4.6) f (k) (a) k! (x a) k, R n (x) := f (n) (ξ) (x a) n (5.4.7) n! S n (x) n R n (x) n 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 f (n) (x) ξ ξ [ n 1 F (x) := f(x) f(a) + F (x) (5.4.6) R n (x) k=1 f (k) (a) (x a) ], k G(x) := (x a) n (5.4.8) k! 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) (5.4.9) G(a) = G (a) = G (a) =... = G (n 1) (a) = 0, G (n) (a) = n! (5.4.10) F (x) F (a) G(x) G(a) = F (ξ 1 ) G (ξ 1 ) ξ 1 ξ 1 a x F (a) = G (a) = (5.4.11) F (ξ 1 ) G (ξ 1 ) = F (ξ 1 ) F (a) G (ξ 1 ) G (a) = F (ξ 2 ) G (ξ 2 ) (5.4.12) 35

54 A hara/lectures/lectures-j.html 54 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 ) 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! (5.4.13) (5.4.14) (5.4.15) f(x) (x a) k S n (x) (5.4.6) S n (x) f(x) R n (x) (n ) x = 0 f(x), g(x) x 0 f(x) g(x) lim x 0 x n = 0 n (5.4.16) 0 g(x) f(x) n n f(x) g(x) x n ( ) lim x a f(x) = lim x a h(x) = 0 lim x a f(x) h(x) = 0 f(x) h(x) f(x) = o( h(x) ) o f(x) h(x) x a K > 0 δ > 0 0 < x a < δ = f(x) < K (5.4.17) h(x) f(x) h(x) f(x) = O ( h(x) ) O 36

55 A hara/lectures/lectures-j.html 55 x f(x) g(x) f(x) g(x) f(x) g(x) f(x) g(x) f(x) = Ω ( g(x) ) (5.4.16) f(x) g(x) = o(x n ) o (5.4.18) ( ) 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) (5.4.19) n! S n (x) f(x) (n 1) f(x) = n 1 k=0 f (k) (0) x k + o(x n 1 ) (5.4.20) k! lim x 0 R n (x) = 0 (5.4.21) xn 1 (5.4.21) ( ) 1) 0 f (n) δ > 0 M > 0 x < δ f (n) (x) < M (5.4.22) f(x) = n 1 k=0 f (k) (0) x k + O(x n ) (5.4.23) k! 2) 0 f (n) f(x) C n - 1) C f(x) = c n (x a) n + c n 1 (x a) n c 1 (x a) + c 0

56 A hara/lectures/lectures-j.html 56 f(x) = e x e x a = 0 e x = n 1 k=0 x k k! + R n(x), R n (x) := 1 (n 1)! x 0 e y (x y) n 1 dy (5.4.24) x lim n R n(x) = 0 x e x = k=0 x k k! (5.4.25) sin, cos n 1 sin x = ( 1) k x 2k+1 (2k + 1)! + ( 1)n (2n 1)! k=0 x 0 (sin y) (x y) 2n 1 dy (5.4.26) 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)! (5.4.27) 37 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=

57 A hara/lectures/lectures-j.html 57 x 0 f(x) = 1 + x + x 2 + o(x 2 ) g(x) = 3 + x + o(x) f(x)g(x) f(x)/g(x) x = 0 g(x) 0 tan x x = 0 a a) log(1 + 3x) b) e ax c) log(1 + x 2 ) d) tan x e) cosh x a), b), e) c), d) n n log(1 + x) = n k=0 ( 1) k x k+1 k + 1 x + 0 ( y) n+1 dy (5.4.28) 1 + y n [ n ( 1) k x k+1 ] ( 1) k x k+1 log(1 + x) = lim = n k + 1 k + 1 k=0 1. n k=0 1 n 1 + y = ( y) k + ( y)n y k=0 (5.4.29) (5.4.30) 1 + y y y x n (5.4.28) x 1 1+y x > 1 3. S n+1 (x) := n ( 1) k x k+1, R n+1 (x) := k + 1 k=0 (5.4.28) x 0 ( y) n+1 dy (5.4.31) 1 + y log(1 + x) = S n+1 (x) + R n+1 (x) (n = 0, 1, 2,...) (5.4.32)

58 A hara/lectures/lectures-j.html 58 k k = 1 k = 0 S n+1, R n+1 n + 1 S n+1 n + 1 R n+1 S n+1 p.1 R n 1.3 p (5.4.32) n x n [ ] a n, b n a n, b n n lim an + b n = lim a n + lim b n n n n a n = S n+1 (x), b n = R n+1 (x) 1 < x 1 lim n R n+1(x) = 0 p.1 lim n S n+1(x) C := log(1+x) n C = a n + b n, a n = C b n (5.4.33) n 1 a n b n C = lim n a n + lim n b n (5.4.34) 1 < x 1 b n = R n+1 (x) [ log(1 + x) = lim S n+1(x) + lim R n+1(x) = lim S n n+1(x) + 0 = lim n n n n k=0 ( 1) k x k+1 ] k + 1 (5.4.35) p.1 (1) n n 1 < x 1 lim R n+1(x) = 0 n n (5.4.32) x > 1 x = 2 S n+1 (x), R n+1 (x) n = 1, 2, 3, (x a) n 2. x x

59 A hara/lectures/lectures-j.html 59 Euler e iθ = cos θ + i sin θ, θ R (5.4.36) x = iθ e iθ (ix) k = = ( 1) l x 2l k! (2l)! + i ( 1) l x 2l+1 (5.4.37) (2l + 1)! k=0 l=0 k i k cos θ+i sin θ sin, cos 2π e a+b = e a e b sin x sin x l= ( Taylor ) 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 (5.4.38) R n (x) n lim R n(x) = 0 n f(x) = lim n S n(x) = k=0 f (k) (a) (x a) k (5.4.39) k! n f I 1 < x 1 n x > 1 f(x) C N - (5.4.38) 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 f(x) = f(a) + x a f (y)dy (5.4.40)

60 A hara/lectures/lectures-j.html 60 x a f (y)dy = x a { d dy (x y)} f (y)dy = = (x a)f (a) + x a [ ] x (x y)f (y) + a x a (x y)f (y)dy (x y)f (y)dy (5.4.41) II. n n + 1 n N 1 n (5.4.38) (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. (5.4.42) (5.4.38) (n 1)! (5.4.38) n < x 1 [ n ( 1) k 1 x k ] ( 1) k 1 x k log(1 + x) = lim = n k k k=1 k=1 (5.5.1) log log log log(1 + x) (5.5.1) n n 1 + x = 1 + y 1 y, y = x 2 + x (5.5.2) ( 1 + y ) log(1 + x) = log = 2 ( y + y3 1 y 3 + y ) (5.5.3) - sin, cos sin x cos x sin, cos

61 A hara/lectures/lectures-j.html x x : log(1 + x) n log(1 + x) log(1 + x) n = 1, 3, 7 n = 2, 4 log(1 + x) n (5.5.3) n n = 1, 2, 3, 4 log(1 + x) n 3 log(1 + x) sin x, cos x x (sin x) = cos x, (cos x) = sin x x sin, cos x sin 2 x + cos 2 x = 1 sin α = 0, cos β = 0 α, β sin, cos α, β π π 1 2π sin, cos π

62 A hara/lectures/lectures-j.html ( ) f(x, y) C 1 - f(a + h, b + k) = f(a, b) + f f (a + θh, b + θk)h + (a + θh, b + θk)k (5.6.1) x y θ 0 < θ < 1 θ a, b, h, k C 1 n a = (a 1, a 2,..., a n ) h = (h 1, h 2,..., h n ) 0 < θ < 1 f(a + h) = f(a) + j=1 f x j (a + θh) h j (5.6.2) g(t) = f(a + th, b + tk) t f(a + h, b + k) f(a, b) = g(1) g(0) = g (θ) (5.6.3) g x(t) = a + th, y(t) = b + yk n g (t) = f x x (t) + f y y (t) = f x h + f y k (5.6.4) C ( ) f(x, y) f(a + h, b + k) = f(a, b) + f f (a + θh, b + θk)h + (a + θh, b + θk)k (5.6.5) x y θ 0 < θ < 1 θ a, b, h, k

63 A hara/lectures/lectures-j.html /3 8/ /3 8/3 web page URL A 8/7

64 A hara/lectures/lectures-j.html 2 - sin, cos sin x cos x sin, cos S(x) C(x) S(x) := n=0 ( 1) n x 2n+1, C(x) := (2n + 1)! S(x) a N := N n=0 n=0 ( 1) n x 2n+1 (2n + 1)! ( 1) n x 2n (2n)! N S(x) S(x) sin x C(x) cos x x S (x) = C(x), C (x) = S(x) x sin, cos x {S(x)} 2 + {C(x)} 2 = 1 sin, cos S(x + y) = S(x)C(y) + C(x)S(y) S(α) = 0, C(β) = 0 α, β S(x), C(x) S(x), C(x) α S(x + α) = S(x), C(x + α) = C(x) x α, β π π 1 2π S(x), C(x) π S(x), C(x) sin x, cos x sin x, cos x S(x), C(x) sin x, cos x

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