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1 14 S hara/lectures/lectures-j.html r 1 S phone: , Office hours: 1 4/11 web download. I. 1. ϵ-δ , , 3.7 II Lagrange III , 4., IV

2 14 S hara/lectures/lectures-j.html r web download I, II 1, 1 A = B = A 4 : 6 = max{ A, B } A B A, B 1 1 1

3 14 S hara/lectures/lectures-j.html r 3 hara/lectures/lectures-j.html 14 spam mail html mail

4 14 S hara/lectures/lectures-j.html r a < b n Z 1 N Q R C x R x A Z Q a < b a = b a b 1 a > b a = b a b a < x < b (a, b) a x b [a, b] n! = n (n 1) (n ) 1 n! = 1 a = b a b a := b a b f(x) := x sin x f(x) x sin x unique, uniquely

5 14 S hara/lectures/lectures-j.html r f, g, h x = 3 a (a) f(x) = e x = exp(x ) (b) g(x) = 1 + x + x (c) h(x) = cos(ax) (1) : A4 B5

6 14 S hara/lectures/lectures-j.html r 6 5 5/9 5/16 3 f, g, h x = a (a) f(x) = e x3 +x = exp(x + x 3 ) x 9 (b) g(x) = 1 + x + x 4 x 6 (c) h(x) = cos(ax 3 ) x 9 3* sin x sin(1) sin(1) a < b a x b f(x) g(x) b f(x)dx b a a g(x)dx : A4 B5 Golden Week web page

7 14 S hara/lectures/lectures-j.html r 7 1 R N f f(x) f() + f () x + f () x + f () x 3 6 f(), f (), f (), f () f(x) = e x f (x) = x e x, f (x) = ( + 4x ) e x, f (x) = (1x + 8x 3 ) e x f() = 1, f () =, f () =, f () = f(x) x + = 1 + x. g, h g(x) = (1 + x + x ) 1/, g (x) = 1 (1 + x) (1 + x + x ) 1/, g (x) = 3 4 (1 + x + x ) 3/ g (x) = 9 8 (1 + x) (1 + x + x ) 5/ g() = 1, g () = 1, g () = 3 4, g () = 9 8 g(x) x x 3 16 x3. h h(x) = cos(ax), h (x) = a sin(ax), h (x) = a cos(ax) h (x) = +a 3 sin(ax) h(x) 1 a x 1 f(x) = 1 + x f(x) 1 + x f(x) 1 + x f(x) = 1 + x + x a f(x) x + 1 f(x) 1 + x

8 14 S hara/lectures/lectures-j.html r 8 sin, cos sin, cos x sin(x) := n= ( 1) n (n + 1)! xn+1, cos(x) := e z := n= z n n! n= ( 1) n (n)! xn z n n π (sin x) = cos x, (cos x) = sin x x sin, cos sin, cos ( d ) a n x n = dx n= n= d ( an x n) dx sin x + cos x = 1 sin, cos sin x, cos x 1 sin α =, cos β = α, β sin, cos α, β π π 1 π sin, cos π

9 14 S hara/lectures/lectures-j.html r web 4 f, g sin(xy) f(x, y) = sin ( xy ) x + y (x + y ), g(x, y) = (x = y = ) (1) f, g (, ) () f, g (, ) x, y (3) f, g (, ) (cos θ, sin θ) θ π (), (3) : A4 B5 (a) t = x + x 3 e t t t = x + x 3 = x (1 + x) t 5 = x 1 (1 + x) 5 x 1 t 4 x 8 e t = 1 + t + t + t3 6 + t4 4 + t = x (1 + x) x 9 f(x) = 1 + x (1 + x) + 1 x4 (1 + x) x6 (1 + x) x8 (1 + x) 4 + = 1 + x (1 + x) + 1 x4 (1 + x + x ) x6 (1 + 3x + 3x + x 3 ) x8 (1 + 4x + ) + = 1 + x + x 3 + x4 + x5 + 3 x6 + x x8 + x9 3 +

10 14 S hara/lectures/lectures-j.html r 1 (b) t = x + x 4 t 3 x 6 t 4 x 8 t t = 1 + t t 8 + t g(x) = x (1 + x ) 1 8 x4 (1 + x ) x6 (1 + x ) 3 + = x (1 + x ) 1 8 x4 (1 + x + ) x6 (1 + ) = 1 + x x x6 + (c) t = ax 3 t 3 h(x) = 1 t + = 1 a x6 + 3 sin x x = x, x 3, x 5, x 7,... x 5 sin x = x x x sin x = x x3 6 + x (x y) 3 sin y dy x (x y) 5 sin y dy x = 1 sin 1 x x 1 6 (x y) 3 sin y dy 1 6 (x y) 3 dy = x x (x y) 5 sin y dy 1 1 x sin(1) = (x y) 5 dy = 1 7 sin(1) =.8478 sin(1).84 1 x 5 sin 6 sin x 1 sin x x x 1 1 x (x y) 5 sin y dy 1 1 x (x y) 5 y dy = 1 54

11 14 S hara/lectures/lectures-j.html r 11 1/7 x 5 sin 6 x 6 sin 7 x 6 sin x = x x3 6 + x cos y x (x y) 6 cos y dy 1 7 x x (x y) 6 cos y dy 1/7 3 x 3 sin(1) = (x y) 6 dy = 1 7! = 1 54 sin(1) = sin x = x x x (x y) 4 cos y dy 1 4 x (x y) 4 cos y dy 1 4 x sin(1) = (x y) 4 dy = 1 1 sin(1) = /1 sin x x sin x x x3 3! sin x x x3 3! + x5 5! sin x x x3 3! + x5 5! x7 7! x π

12 14 S hara/lectures/lectures-j.html r f(a + rt) f(a) D t f(a) := lim r + r r a t - r D t f(a) := lim r f(a + rt) f(a) r 5 g(x, y) = y sin(xy ) (a) g x (b) g y (c) g x (d) ( ) g x y : A4 B5

13 14 S hara/lectures/lectures-j.html r 13 sin() lim = lim h h h h /h = 4 f, g 3 (1) f lim f(x, y) = f(, ) (x,y) (,) (x, y) (, ) (x, y) (, ) x + y (3)

14 14 S hara/lectures/lectures-j.html r 14 f x sin x x sin x x sin ( xy ) xy x + y f(x, y) f(, ) f g g(, ) = g(x, y) (x, y) = (t, t) sin(t ) lim g(t, t) = lim t t t = 1 lim sin(t ) t t = 1. g g x y x y g(x, ) = g(, y) = 45 () f f(h, ) f(, ) (, ) = lim = lim = x h h h h f f(, h) f(, ) (, ) = lim = lim = y h h h h f g g g(h, ) g(, ) (, ) = lim = lim = x h h h h g g(, h) g(, ) (, ) = lim = lim = y h h h h (1) g (3) lim lim r + r lim r r t = (cos θ, sin θ) D t f(r cos θ, r sin θ) f(, ) sin( r cos θ sin θ ) D t f(, ) = lim = lim r r r r lim r + cos θ sin θ lim r sin θ cos θ = sin θ cos θ = θ =, π/, π, 3π/ g g(r cos θ, r sin θ) g(, ) D t g(, ) = lim = lim r r r sin(r cos θ sin θ) r r = lim r sin(r cos θ sin θ) r 3

15 14 S hara/lectures/lectures-j.html r 15 cos θ sin θ cos θ sin θ = θ =, π/, π, 3π/ x y () f x = a x a x a a f(a) f a f x x- t- x a t t x- x

16 14 S hara/lectures/lectures-j.html r x, y r θ x = r cos θ, y = r sin θ x, y f(x, y) x, y g(r, θ) = f(r cos θ, r sin θ) r, θ g f x + f y 7 f(x, y) = (1 + x + y) 1/3 x = y = x, y, x, y, x, xy, y (1) () 1 () (1) : A4 B5

17 14 S hara/lectures/lectures-j.html r 17 5 g x = y cos(xy ) y = y 4 cos(xy ) g y = y sin(xy ) + y cos(xy ) xy = y sin(xy ) + xy 3 cos(xy ) x g x = ( ) y 4 cos(xy ) = y 4 ( sin(xy ) y ) = y 6 sin(xy ) x ( ) g = ( ) y sin(xy ) + xy 3 cos(xy ) = y cos(xy )y + y 3 cos(xy ) xy 3 sin(xy )y x y x = 4y 3 cos(xy ) xy 5 sin(xy ) (d) xy 3 cos(xy ) x

18 14 S hara/lectures/lectures-j.html r f, g x, y f(x, y) = xy(x + y 1), g(x, y) = x y e x y : A4 B5 6 f x = g r r x + g θ θ x f y = g r r y + g θ θ y ( ) x, y r, θ x r = cos θ, y x = sin θ, r θ = r sin θ, y θ = r cos θ

19 14 S hara/lectures/lectures-j.html r 19 r r x x x r r r 1 x x 1 x θ x y θ = r y y r θ y θ (*) = cos θ sin θ f g = cos θ x r sin θ g r θ, r sin θ r cos θ f y = 1 r r cos θ sin θ r sin θ g = sin θ r + cos θ g r θ, cos θ = cos θ sin θ r ( ) sin θ cos θ r x, y (*) x r x x, y [ f ( g ) x = r r r x + ( g ) θ ] r θ r x x + g [ ( r ) r r r x x + ( r θ x [ ( g ) + r r θ x + ( g ) θ ] θ θ θ x x + g [ ( θ ) r θ r x x + ( θ θ x ) θ x ) θ x ] ] ( ) y x y (***) (**) (***) [ f x = g r cos θ + g ( sin θ ) ] cos θ + g [ cos θ cos θ + cos θ ( r θ r r r θ [ g + r θ cos θ + g ( θ sin θ ) ] ( sin θ ) + g [ ( sin θ r r θ r r = cos θ g sin θ g r r r θ + sin θ g r θ + sin θ g sin θ g + r r r θ sin θ r ) cos θ + θ ) ] ( sin θ r ) ( sin θ r y f y = sin θ g sin θ g + r r r θ + cos θ g r θ + cos θ g sin θ g r r r θ f x + f y = g r + 1 g r θ + 1 g r r r g x cos θ r x g g r x.y g r = f x x r + f y f f = cos θ + sin θ y r x y ) ]

20 14 S hara/lectures/lectures-j.html r r θ r f x [ g r = cos θ f x x r + f x y ] [ y f y + sin θ r y r + f x y g f f = r cos θ r sin θ θ x y r sin θ x r g θ = f x x θ + f y f f = r sin θ + r cos θ y θ x y ] [ f x x θ + f x y = cos θ f x + sin θ f y + sin θ f x y ] [ y f y + r cos θ θ y θ + f x y = r cos θ f f r sin θ x y + r sin θ f x r sin θ f x y + r cos θ f y r g r + 1 g r θ = f x + f y 1 [ cos θ f ] f + sin θ r x y g r r ] f x + f y = g r + 1 g r θ + 1 [ cos θ f f + sin θ r x y = g r + 1 g r θ + 1 g r r [ r r x x (1) ] x θ g f f = cos θ + sin θ r x y, g f f = r sin θ + r cos θ θ x y f f f f x y x y () r x r x, y r = x + y x r x = x/ x + y = x/r = cos θ θ θ = arctan(y/x) (1) g f g 1 () g f f x + f y = g (3) f x + f g y (4) g f x + f y = A g r + B g θ + C g r + D g r θ + E g θ A E r, θ (5) x, y B D

21 14 S hara/lectures/lectures-j.html r 1 (6) A, C, E f = x +y = r, f = r 4, f = x = r cos θ 3 7 (1) f x = 1 3 (1 + x + y) /3 = 3 (1 + x + y) /3, f y = 1 (1 + x + y) /3 3 f x = 4 9 (1 + x + y) 5/3 = 8 9 (1 + x + y) 5/3, f x y = 4 9 (1 + x + y) 5/3, f y = (1 + x + y) 5/3 9 f x = 3, f y = 1 3, f x = 8 9, f x y = 4 9, f y = 9 f(x, y) x y 4 9 x 4 9 xy 1 9 y = 1 + ( 3 x + 1 ) ( 3 y 3 x + 1 ) 3 y () t = x + y (1 + t) 1/3 (1) f t 1 9 t = 1 + x + y 3 (x + y)

22 14 S hara/lectures/lectures-j.html r f = f x = y(3x + y 1) = f y = x(x + 3y 1) A B A = B = 4 Case 1. y = x = (x, y) = (, ) Case. y = x + 3y 1 = (x, y) = ±(1, ) Case 3. x = 3x + y 1 = (x, y) = ±(, 1) Case 4. x + 3y 1 = 3x + y 1 = x y = x = ±y (x, y) = ±(1/, 1/) (x, y) = ±(1/, 1/) f x = 6xy, f y = 6xy, f x y = 3(x + y ) 1 H(x, y) = 36x y {3(x + y ) 1} 4 Case 1. H(, ) = 1 < Case. H(±1, ) = 4 < Case 3. H(, ±1) = 4 < Case 4. H(±1/, ±1/) = 9/4 1/4 = > Case 4 Case 4 f x f (±1/, ±1/) = 3/ >, x f (±1/, 1/) = 3/ < x (±1/, ±1/) f 1 8

23 14 S hara/lectures/lectures-j.html r 3 (±1/, 1/) f 1 8 g = g x = (1 x ) y e x y = g y = (1 y ) x e x y Case 1. y = x = (x, y) = (, ) Case. y = 1 y = Case 3. x = 1 x = Case 4. 1 x = 1 y = (x, y) = (±1/, ±1/ ). g x = xy( 3 + x ) e x y, g y = xy( 3 + x ) e x y, g x y = ( 1 + x )( 1 + y ) e x y Case 1. (, ) g x = g y =, H(, ) = 1 < Case 4. (x, y) = ±(1/, 1/ ) g x y = 1 g x = g y = e 1, g x y = H(, ) = 4e > (x, y) = ±(1/, 1/ ) g x = g y = e 1, g x y = H(, ) = 4e > (x, y) = ±(1/, 1/ ) g 1/(e) (x, y) = ±(1/, 1/ ) g 1/(e)

24 14 S hara/lectures/lectures-j.html r / D f(x, y)dxdy f(x, y) D D (1) f(x, y) = x + y D = [, 4] [1, ] () f(x, y) = x y D = [1, ] [, 1] (3) f(x, y) = y D = [, ] [, 1] 1 + xy 1 D D f(x, y)dxdy f(x, y) D (1) f(x, y) = x y. D x + y x y () f(x, y) = x y D x y x + 3 (3) f(x, y) = x + y D x + y : A4 B5

25 14 S hara/lectures/lectures-j.html r f(x, y)dxdy f(x, y) D D D () f y (1) f(x, y) = x y D y x x web 7/18 () f(x, y) = y D y + y x y 3 1 x, y x y y x f (1) () (3) 3 dx 1 dy 6 x y 3 3 y +y 1 4y 3 dy 3 y dy f(x, y) dx f(x, y) dx f(x, y) 13 p.19, 5 D x + y π x y 1 D u = x + y, w = x y e x y sin(x + y) dxdy D : A4 B5

26 14 S hara/lectures/lectures-j.html r 6 9 (1) () D (x + y )dxdy = 1 4 dx dx 1 (3) x 1 dy dx y 1 + xy = 1 1 dy (x + y ) = dy xy = dy{log(1 + y) log(1)} = 1 4 dx ( x dx 1 3 x = ) = = 1 dy log(1 + y) = = 7 3 ds log(s) = 3 log 3 y dx 1 y dy 1 + xy = ( 1 dx x log x ) x = [ ( ) ] log(1 + x) x ( = ) [ ( log(1 + ) lim ) ] log(1 + x) = 3 log 3 lim x + x x + [ ( x ) ] log(1 + x) x log(1 + x) = x + O(x ) lim x + [ ( x ) ] log(1 + x) = lim x + [ ( x ) ] [ ] {x + O(x )} = lim 1 + O(x) = 1. x + 3 log 3 1

27 14 S hara/lectures/lectures-j.html r 7 1 (1) x x y y 1 x dx 1 x/ dy (xy (1 ) x/)3 ) = (x = = 1 ( 16 3 ) = 1 ( dx x( x) 3 = 1 ds ( s)s 3 4 ) = = y y 1 x x y 1 dy y dx (xy ) = = 1 (y 1 ( y) ) = 1 dy y (1 y) ( 1 dy(y y 3 + y 4 ) = = 5) = () x 1 x 3 y x y x x+3 3 dx dy x y = dx x (x + 3) x 4 = 1 3 dx(9x + 1x 3 + 4x 4 x 6 ) = 3616 x y = 1 1 y dy x y + y 9 1 y dy x y = (y 3)/ 1 dy y5/ dy ( 1 ) 4 y(y 3)3 + y3/ 3 = =

28 14 S hara/lectures/lectures-j.html r 8 (3) D x, y x, y dx 3 x 54. dy(x + y ) = 3 dx (x (3 x) + = ) (3 x)3 = = 7 3 dx (9 9x + 6x 4x 3 /3) x, y x 8 3 dx 3 x dy x = 3 dx x (3 x) = 8 = 54 3 dx (3x x 3 ) = = 81 1 =

29 14 S hara/lectures/lectures-j.html r web page web web page A A A4 A4

30 14 S hara/lectures/lectures-j.html r 3 14 D f(x, y) () a D f(x, y)dxdy (1) f(x, y) = x + y D x + y 5 (x, y) () f(x, y) = y D x, y, x + y a (x, y) x = u, y = w (3) () x = r(cos θ) 4, y = r(sin θ) : A4 B5 11 (1), () () (1) x x dx x dy x y = 1 = () 1 3 dy = y 3 y +y dx x ( x) = 1 ( = 16 5) dx y = 1 3 dx (4x 4x 3 + x 4 ) = = 16 3 dy y{( y 3) (y + y)} = ( 1 + 7) 3 (1 9) = ( (1 16) ( ) dy ( y 3 4y 3y) ) = 8 3

31 14 S hara/lectures/lectures-j.html r 31 1 b a f(x)dx a < b f(x) a < b a f(x)dx = b b a f(x)dx b a f(x)dx a < b b a f(x)dx 1 (1) (1) () (3) 3 6 dy dx 3 y/ (x+3)/ x 3 dx 1 x (x +3)/4 dx f(x, y) dy f(x, y) dy f(x, y) 13 x, y

32 14 S hara/lectures/lectures-j.html r 3 u, w u π, w 1 x, y u, w x = u + w, y = u w ( ) (x, y) (u, w) = det 1/ 1/ = 1 1/ 1/ 1 dw π/ du 1 ew sin(u) = 1 1 dw e w π/ du sin(u) = 1 (e 1) 1 = e 1.

33 14 S hara/lectures/lectures-j.html r web page web web page A A A4 A4

34 14 S hara/lectures/lectures-j.html r (1) x = r cos θ, y = r sin θ r 5, θ < π r x + y = r 5 dr π dθ r r = π 5 [ r dr r 3 4 ] 5 = π = π 5 4 = π () a = 1 u, w u, w x, y u, w x, y sweep x = u u = x u = x x u y = w x = u, y = w u = x, w = y u = x, w = y u, w, u + w a

35 14 S hara/lectures/lectures-j.html r 35 [ ] u det = 4uw w a y dxdy = du D = a a u du u (a u) 4 = dw 4uw w = 4 a a du u a u du u 4 (a u) = a u a dw w 3 = 4 du u 4 a a (a u)4 du u 4 du u 5 = a a5 5 a6 6 = a6 3. (3) x = r cos 4 θ, y = r sin 4 θ x, y r, θ cos sin 4 π/ θ [, π/] θ [π/, π] θ θ [, π/] r r x, y x + y a r (cos θ + sin θ ) a r a (x, y) r, θ = det D y dxdy = [ cos 4 θ sin 4 θ a dr r a, θ π ] 4r cos 3 θ sin θ 4r sin 3 = 4r cos 3 θ sin 3 θ(cos θ + sin θ) = 4r cos 3 θ sin 3 θ θ cos θ π/ a dθ 4r cos 3 θ sin 3 θ r sin 4 θ = 4 π/ dr r dθ (cos θ) 3 (sin θ) 7 r a 6 /3 θ t = sin θ π/ dθ (cos θ) 3 (sin θ) 7 = π/ dθ cos θ (1 sin θ) (sin θ) 7 = 1 dt (1 t ) t 7 = = 1 4 () D y dxdy = 4 a = a n 1/n 1 1 x = x(t) x (t)

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

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10% 1 2006.4.17. 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 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n

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