O f(x) x = A = lim h f( + h) f() h A (differentil coefficient) f f () y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) * t (v
|
|
|
- ようた つねざき
- 5 years ago
- Views:
Transcription
1 I A 62 B 66 big
2 O f(x) x = A = lim h f( + h) f() h A (differentil coefficient) f f () y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) * t (velocity) p(t) = (x(t), y(t), z(t)) ( dp dx dt = dt, dy dt, dz ) dt. ( ) = x x 2 * geometry geo 2
3 2. x =, x = *2 o(h) = f( + h) f() Ah o(h) h h o(h) lim h h ( ) f( + h) f() = lim A = h h o(h) f(x) x = A = f () f( + h) = f() + f ()h + o (h), g( + h) = g() + g ()h + o 2 (h) f( + h)g( + h) = (f() + f ()h + o (h))(g() + g ()h + o 2 (h)) = f()g() + (f ()g() + f()g ())h + f ()g ()h 2 + (f() + f ()h)o 2 (h) + (g() + g ()h)o (h) + o (h)o 2 (h) h o 3 (h) f( + h)g( + h) = f()g() + (f ()g() + f()g ())h + o 3 (h), o 3 (h) lim h h = (f(x)g(x)) = f (x)g(x) + f(x)g (x). ( ). (i) (f(x)g(x)) = f (x)g(x) + f(x)g (x). *2 Weierstrss 3
4 (ii) *3 {f(g(x))} = f (g(x))g (x). 3. (ii) o(h) =, x =, 4. (x ) = x 2 x x+h x lim h h ( ) = x x 2. = x lim h h h A x A y = x x = A A + A = > e (e x ) = e x e *4 *5 log x e x (log x) = /x (x > ) Remrk. e.2. (i) > y = x x = e x log = e log d dx x = e x log log = x log. (ii) > x x (x > ) x = e log x d dx x = e log x ( log x) = x. *3 *4 bse *5 (nturl logrithm) ln x 4
5 5. > y = x y = x 6. x x (x > ) 7. x < (log( x)) = x (e x ) = e x, (log x ) = x (x ), (xα ) = αx α (x > )..3. ( log x + ) x 2 + = x x ( ) ( ).4. f(x) = (f(x)) 9. ( ) = (f(x)) 2 f (x) = f (x) f(x) f(x) 2. ( ) f(x) = f (x)g(x) f(x)g (x) g(x) g(x) 2 Remrk. sin(x + h) = sin x cos h + cos x sin h sin(x + h) sin x h lim h = cos h h sin x + sin h h cos h sin h, lim h h h cos x 5
6 y h x cos x x = y = cos x x = y = *6 (sin h h tn h (sin x) = cos x (cos x) = sin x (sin x) = cos x, (cos x) = sin x, (tn x) =. tn x (cos x) 2 = + (tn x)2.. rdin 2. 2 ( > ) log x, x, e x x *6 (rdin).7 rd rd 6
7 x e x x e bx (b > ) f(x) = x e bx, x > f(x) f (x) = x e bx ( bx) x /b f (x) + f(x) x = /b y C /b x lim x + f(x) = C > f(x) C C > f(x) x e x x e x x e x/2 e x/2 b = /2 x e x/2 C C > x e x Ce x/2 *7 lim x + e x/2 = lim x + x e x = log x x t = log x log x lim x + x = lim t + t e t = lim t + *7 ( ) t / = e t 7
8 x + t + log x << x << e x (x + ) 3. >, b > x << e bx (x + ) < < b x x b, e x e bx Remrk. x << e bx x x < e bx x = 4, b = x = 2 2 = 6 > 3 2 > e lim x x/x log x << x log x lim x log(x/x ) = lim x x =. lim x x/x =. 4. lim x + xx 5. > x < lim n n x n = 6. (i) y = x 2 e x. (ii) y = x log x (x > ). 3 (inverse function) y = f(x) g x = g(y) g(f(x)) = x, f(g(y)) = y 8
9 f(x) sin x ( π/2 x π/2), cos x ( x π), tn x ( π/2 < x < π/2) rcsin x, rccos x, rctn x. *8 (inverse trigonometric function) x rcsin x rctn x x y + k y x x + h 7. rccos x + rcsin x = π 2 ( x π/2) 8. < < π/2 sin x = sin x *8 sin x sin x sin x (sin x) 9
10 9. 5π/4 sin x (rcsin x) =, (rctn x 2 x) = + x 2. *9 y = f(x) x = g(y) g (y) = f (x) = f (g(y)). y + k = f(x + h), x + h = g(y + k) g(y + k) g(y) lim k k h = lim h f(x + h) f(x) = f (x). y = sin x x = rcsin y rcsin (y) = 2. rctn x 2. f(x) = ex e x 2 (sin x) = cos x = y 2 (i) (ii) g(y) (iii) g(y) y 4 b (f(x) + g(x)) dx = b f(x) dx + b g(x) dx *9
11 b f(x) dx = lim n n k= f( + k(b )/n) b n b f(x) dx = c f(x) dx + b c f(x) dx [, b] f(x) = x < x < x 2 < < x n = b ξ j [x j, x j ] lim j= n f(ξ j )(x j x j ), = mx{x x,, x n x n } ξ j f(x) [, b] * (integrl) b f(x) dx f (integrnd) W j (f, ) f(ξ j ) b x x j ξ j x j * B. Riemnn ( ) A.-L. Cuchy ( )
12 Remrk. lim x j x j dx [, b] V = b I(t) t L = b Q = b (dx ) 2 + dt S(x) dx. I(t) dt. ( ) 2 dy + dt ( ) 2 dz dt. dt (r, θ) r = f(θ) θ = α, θ = β S = 2 β α f(θ) 2 dθ θ j r j α = θ < θ < < θ n = β θ j ξ j θ j r j = f(ξ j ) θ j = θ j θ j θ j 2π πr2 j = 2 r2 j θ j 2
13 r = f(θ) (θ j θ θ j ) S n f(ξ j ) 2 (θ j θ j ) 2 j= S = 2 β α f(θ) 2 dθ * (crdioid): r = ( + cos θ) ( π θ π) 4. ( ). (i) b f(x) dx + c b f(x) dx = (ii) f(x) g(x) ( x b) c f(x) dx. b b f(x) dx f(x) dx b b g(x) dx. f(x) dx. Remrk. x 24. b f(x) dx = log x = b f(y) dy =. x log(xy) = log x + log y t dt * f(θ) (θ j θ θ j ) θ ξ j, ξ j 2 f(ξ j ) 2 (θ j θ j ) S f(ξ 2 j ) 2 (θ j θ j ) j 3 j
14 25. b > > b b b t dt > b > b x > * 2 x lim dt = +, x t lim t dt > 2 x dt t x x + dt =, t 4.2 (Dirichlet). f(x) f(x) = { x x [, b] f(x) x = lim f(x), x lim f(x) x (i) (x < ) f(x) = (x = ) 2 (x > ) x = *2 e x 4
15 (ii) g(x) = { sin(/x) (x ) (x = ) x = 4.4. [, b] f Proof. x y = f(x) x ( ) * 3 S = x < < x n = b [x j, x j ] f W j (f) f W (f, ) = mx{w j (f)} j S f(ξ j )(x j x j ) W (f, )(b ) j= f W (f, ) * 4 S = lim j= f(ξ j )(x j x j ) *3 *4 5
16 x = c ( < c < b) c [x k, x k ],,, lim f(ξ j )(x j x j ) = j c f(x) dx, lim j f(ξ j )(x j x j ) = b c f(x) dx c f(x) x = c f(ξ k )(x k x k ) M(x k x k ) M b f(t)dt, b b x x f(t)dt x (indefinite integrl) (definite integrl) F (x) = f(x) F (x) f(x) (primitive function) * 5 f(x) dx f(x) dx x b f(x) dx 26 ( ). *5 6
17 4.5 (). f(x) [, b] d x f(t) dt = f(x). dx Proof. S(x) f(t) x t x + h M h, m h m h S(x + h) S(x) h M h f(t) t = x h S(x + h) S(x) lim h h = lim h M h = lim h m h = f(x). M h m h S(x + h) S(x) x x + h 4.6 ( * 6 ). f(x) f(x) F (x) b f(x)dx = F (b) F () F (b) F () = [ ] b F (x) *6 7
18 Proof. ( d x ) F (t) dt F (x) = F (x) F (x) = dx x C x F (t) dt F (x) = C F (t) dt = F (x) + C x x = = F () + C C = F () x = b b F (t) dt = F (b) + C = F (b) F (). b > b b f(x) dx = f(x) dx b 27. f(x) f() = x f (t) dt f(x) ( < x < b) (i) f (x) ( < x < b) f (, b) (incresing) (ii) f (x) > ( < x < b) f (, b) (strictly incresing) 8
19 * 7 2 x 2 dx = rcsin x, x dx = rctn x. x2 + A dx = log(x + x 2 + A) x 2 + dx = π ( ). (integrtion by prts) f(x)g(x) = f (x)g(x)dx + f(x)g (x)dx. (integrtion by substitution) b f(g(x))g (x)dx = g(b) g() f(y)dy. f(g(x))g (x) f(x) F (x) g(x) F (g(x)) Remrk. (i) (ii) y = g(x), g (x) = dy/dx y f(y) dy dx dx = f(y) dy *7 9
20 4.9. log x dx log x x log x (x log x) = log x +. x log x = log x dx + dx log x dx = x log x x 4.. n = 2, 3,... x (x ) n dx = 2 2n (x ) n. 28. n = 29. x 2 x 2 dx, xe x2 dx 4.. I n (x) = (x ) n dx ( x ) = (x ) n (x ) n 2n x (x ) n+ = 2n (x ) n n (x ) n+ 2 2 ni n+ (x) (2n )I n (x) = x (x ) n 2
21 (recursive reltion) I (x) = x dx = rctn x I 2 (x), I 3 (x), n =, 2,... x n e x dx 4.2. (i) (ii) x 2 dx = rcsin. 4x x 2 2 x2 + A dx = 2 ( x x 2 + A + A log x + x 2 + A Proof. (i) dx = x 2 dx = rcsin. 4x x 2 4 (x 2) 2 2 (ii) x2 + A x x 2 + A ). (x x 2 + A) = x 2 + A + x 2 x2 + A. x x2 + A dx = log + x2 + A x 2 x2 + A = x2 + A A x2 + A = x 2 + A A x2 + A (x x 2 + A) = 2 x 2 + A A x2 + A x2 + A dx 2
22 3. (x 2 x 2 ) 2 x 2 dx = ( x 2 2 x rcsin x ) x 2 t 2 dt * 8 g(x) f(x) dx, deg g < deg f f(x) (x 2 + x + b) m, (x + c) n g(x) f(x) = p(x) (x 2 + x + b) m + q(x) (x + c) n p(x) q(x) p(x) x 2 +x+b x 2 +x+b x 2 +x+b p(x) (αx + β)(x 2 + x + b) k, k < m *8 rtionl function. 22
23 αx + β (x 2 + x + b) l dx, l m q(x) { (x + c) l dx = if l, ( l)(x+c) l log x + c if l = x 2 + x + b = (x + /2) 2 + b 2 /4 y = x + /2 Ay + B (y 2 + C) l dy 4.8, x 3 + dx x 3 + = (x + )(x 2 x + ) x 3 + = x + + bx + c x 2 x +, b, c = /3, b = /3, c = 2/3 x 3 + dx = 3 x + dx x 2 3 x 2 x + dx = 3 log(x + ) 2(x /2) 3 6 (x /2) 2 + 3/4 dx = 3 log(x + ) 6 (x /2) 2 + 3/4 d(x /2)2 + 2 (x /2) 2 + 3/4 dx = 3 log(x + ) 3 6 log(x2 x + ) + 3 rctn((2x )/ 3) 23
24 32. x 3 dx, x 4 dx 33. x 4 + dx e x + e 2x dx t = ex + t + t 2 t dt e x dx + e2x y = f(x) x = ϕ(t), y = ψ(t) t x, y R(x, y) x = ϕ(t) R(x, f(x)) dx = R(x, y) dx = R(ϕ(t), ψ(t))ϕ (t) dt ϕ, ψ t * 9 x 2 + y 2 = (, ) t y = t(x + ), x 2 + y 2 = x = t2 + t 2, y = 2t + t 2 *9 (rtionl curve) 24
25 dx = x 2 y dx t y y = t(x + ) x x = cos θ, y = sin θ t sin θ t cos θdθ = 2 t2 ( + t 2 ) 2 dt cos θ = ( t 2 )/( + t 2 ) dθ = 2dt/( + t 2 ) ( ) t 2 R(cos θ, sin θ) dθ = R + t 2, 2t 2 + t 2 + t 2 dt 35. cos θ = 2 cos 2 θ 2, sin θ = 2 cos θ 2 sin θ 2 t = tn(θ/2) y = x 2 y = t(x + ), y 2 = x 2 25
26 x = + t2 t 2, y = 2t t 2 t 36. x2 dx 5 * 2 f () > f () = f () < f(x) x = f(x) x = f(x) x = f () = f (x) x = y = f(x) x = f() * 2 f(x) f() x x f () f(x) f() + f ()(x ), x *2 *2 locl mximum (locl minimum). 26
27 * 22 f(x) x = (liner pproximtion) f(x) (, f()) 5.. (i) + x + x/2 x x =...5. (ii) sin x x x = 2π/36 sin x f(x) = 4πx 3 /3 x = r f(r + r) f(r) f(r) 3 r r V V 6378Km 7km 6357Km 7Km r r V/V.3%.8% 5.3. x e x e x 2x, log( + x) x (x ) lim x e x e x log( + x) = lim 2x x x = 2. f(x) = f() + f(x) f() f ()(x ) = x x f (t) dt f (t) dt f ()(x ) *22 27
28 x = b d dt (f (t)(b t)) = f (t) + f (t)(b t) t b ( b t ) b b = x f (t) dt f ()(b ) = f(x) = f() + f ()(x ) + b x f (t)(b t) dt f (t)(x t) dt f(x) t x f (t) M x x f (t)(x t) dt f (t) (x t) dt M x (x t) dt = M 2 M x 2 /2 (x )2 37. < x x < 5.4. ( + t) = ( + t) 3/2 /4 t. /4 * (.)2 = sin f (t) x f (t) ( t x) f () f (t) f () x f (t)(x t) dt x f ()(x t) dt = 2 f ()(x ) 2 f(x) f() + f ()(x ) + 2 f ()(x ) 2 (x ) *23 f (t). <.5 28
29 f(x) x = * 24 (qudrtic pproximtion) f () = f () f(x) x = (, f()) f () x = x = f () * 25 y = f(x) x 5.5. f () = (i) f () < f(x) x = (ii) f () > f(x) x = 5.6. (i) cos x x 2 /2 x = = 2π/36 cos (ii) + x x/2 x 2 /8 cos x x 2 /2 (x ) lim x + x x/2 cos x = lim x x 2 /8 x 2 /2 = x e x (x > ) > *24 x *25 f () 29
30 f (x) f (x) x * 26 f(x) f, b f(( t) + tb) ( t)f() + tf(b), t y = f(x) b x 5.7 (Jensen ). f(x) [, b] f (x) f (x) ( x b) t,..., t n t j j t j = {c j } n j= [, b] * 27 n f t j c j j= n t j f(c j ) j= Proof. f(x) = f(c) + f (c)(x c) + x c f (t)(x t) dt f (t) f(x) f(c) + f (c)(x c) c b, x b x = c j, c = t j c j f(c j ) f(c) + f (c)(c j c) *26 *27 j t jc j b t j t j c j t j b j 3
31 t j j t j f(c j ) f(c) + f (c) j j 4. f(x) f(c) + f (c)(x c) t j (c j c) = f(c) 5.8. p = {p j } j n, q = {q j } j n p j >, q j > (reltive entropy) H(p, q) = n j= p j log p j q j log x H(p, q) = p j log q n j log q j p j = log = p j p j f(x) f (x) f (x) = c f (x) j= { f(x) < if x < c, f(x) > if x > c f(x) x < c x > c x = c (point of inflection) 5.9. f(x) f (c) = x = c f (x) (c, f(c)) 5.. f(x) = x 3 x = 3
32 5.. f(x) = e x2 /2σ 2 f (x) = e x2 /2σ 2 x2 σ 2 x < σ x > σ x = ±σ 4. y = f(x) f (c) = f (c) x = c 42. f(x) y = f(x) σ f(x) n n f (n) C n * 28 C 43 (Leibniz Rule). C n f(x), g(x) f(x)g(x) C n d n ( ) dx n f(x)g(x) = n nc k f (k) (x)g (n k) (x). k= f(x) = e x, g(x) = e bx *28 f () (x) = f(x) C 32
33 6.2 (). C n+ f(x) f(x) = f() + f ()(x ) + + n! f (n) ()(x ) n + R n (x), R n (x) = n! x f (n+) (t)(x t) n dt R n (x) (reminder) Proof. n n =, f(x) = f()+f ()(x )+ + (n )! f (n ) ()(x ) n + (n )! d ( f (n) (t)(x t) n) = n! dt (n )! f (n) (x t) n + n! f (n+) (t)(x t) n t (n )! x f (n) (t)(x t) n dt = n! f (n) ()(x ) n + n! x b f (n) (t)(x t) n dt f (n+) (t)(x t) n dt Remrk. B. Tylor (823) Gsprd de Prony (85) de Prony 33
34 6.3. x e x = + x + 2 x2 + + n! xn + e t (x t) n dt. n! sin x = x 3! x3 + + ( ) n (2n )! x2n + ( ) n (2n)! cos x = 2 x2 + + ( ) n (2n)! x2n x (x t) 2n cos t dt. + ( ) n (x t) 2n+ cos t dt. (2n + )! log( + x) = x 2 x2 + + ( ) n+ x n xn + ( ) n+ (x t) n dt. ( + x) n+ ( + x) α α(α ) = + αx + x 2 α(α )... (α n + ) + + x n 2 n! α(α )... (α n) x + ( + x) α n (x t) n dt. n! 44. x 45. g d n x (n )! dx n g(t)(x t) n dt = g(x) 46. ((t )f(t)), ((b t)f(t)), ((b t)(t )f (t)) [, b] b f(t) dt = f() + f(b) (b ) 2 2 b (b t)(t )f (t) dt 34
35 x e x Ce x/2 (x > ) f(x), g(x) C > x f(x) Cg(x) f(x) g(x) f(x) = O(g(x)) * 29 x e x = O(e x/2 ) 6.4. n! n A n (A > ) n log n! log(n!) = log 2 + log log n n log x dx log x dx = n log n n + (n =, 2,... ) A n n! = O((Ae/n) n ) 47. n! = O((n/e) n ) x x x = f(x), g(x) f(x) x = g(x) f(x)/g(x) x = C > f(x) C g(x) x = *29 big O Pul Bchmnn (894) Edmund Lndu (99) little o O Ordnung (order) Omicron 35
36 g(x) f(x) O(g) f(x) f O(g) (nottion) f (x), f 2 (x) f (x) f 2 (x) O(g) f (x) = f 2 (x) + O(g(x)) O(g(x)) f f 2 x f 2 f O(g) f(x) = O(g(x)) 6.5. f(x) = x 2 sin(/x), g(x) = x 2 f(x) g(x) C = x 2 sin(/x) = O(x 2 ) f(x) 48. lim x f(x) = O(g(x)) g(x) t x ( x t ) f (n+) (t) M n+ (x) R n (x) n! M n+(x) x x t n dt = (n + )! M n+(x) x n+ lim x M n+ (x) = f (n+) () n R n (x) = O((x ) n+ ). f() + f ()(x ) + + n! f (n) ()(x ) n ( ) 6.6 ( * 3 ). C n+ f(x) x = f(x) = f() + f ()(x ) + 2 f ()(x ) f (n) () (x ) n + O((x ) n+ ). n! *3 Tylor 36
37 f(x) = c + c (x ) + + c n (x ) n + O((x ) n+ ) c k = f (k) ()/k! ( k n) Proof. b + b (x ) + + b n (x ) n = O((x ) n+ ) b = b = = b n = O b + b (x ) + + b n (x ) n C x n+ x x b = x = x b = 6.7. x = e x = + x + 2 x2 + + n! xn + O(x n+ ). sin x = x 3! x3 + + ( ) n (2n + )! x2n+ + O(x 2n+3 ). cos x = 2 x2 + + ( ) n (2n)! x2n + O(x 2n+2 ). log( + x) = x 2 x2 + + ( ) n+ n xn + O(x n+ ). ( + x) α = + αx + α(α ) x 2 + O(x 3 ) f(x) = tn x x = 5. C n+2 f(x) lim (f(x) x (x ) n+ f() f ()(x ) n! ) f (n) ()(x ) n = 6.8. f(x) = O(x m ), g(x) = O(x n ) α, β m n = min{m, n} (iii) m 37
38 (i) αf(x) + βg(x) = O(x m n ). (ii) f(x)g(x) = O(x m+n ). (iii) g(f(x)) = O(x mn ). 5. (iii) m = 6.9. f(x) = 2x x 2 + O(x 3 ), g(x) = x + 3x 2 + O(x 3 ) (i) f(x) g(x) = + 3x 4x 2 + O(x 3 ). (ii) f(x)g(x) = (2x x 2 + O(x 3 ))( x + 3x 2 + O(x 3 )) = 2x 3x 2 + O(x 3 ). (iii) g(f(x)) = (2x x 2 +O(x 3 ))+3(2x x 2 +O(x 3 )) 2 +O(x 3 ) = 2x+3x 2 +O(x 3 ). 6.. e x sin x e x = + x + 2 x2 + 6 x3 + O(x 4 ), sin x = x 6 x3 + O(x 5 ) e x sin x = x + x x3 + O(x 4 ) 6.. y = cos x = 2 x2 + 4! x4 + O(x 6 ) + y = y + y2 + O(y 3 ) cos x = ( x2 /2+x 4 /4!+O(x 6 ))+( x 2 /2+O(x 4 )) 2 +O(x 6 ) = + 2 x x4 +O(x 6 ) tn x tn x = x + bx 3 + cx 5 + O(x 7 ) cos x = x 2 /2 + x 4 /4! + O(x 6 ) tn x cos x ( x 2 /2+x 4 /4!+O(x 6 ))(x+bx 3 +cx 5 +O(x 7 )) = x+(b /2)x 3 +(c b/2+/4!)x 5 +O(x 7 ) 38
39 sin x = x x 3 /6 + x 5 /5! + O(x 7 ) =, b 2 = 6, c b 2 + 4! = 5! tn x = x + 3 x x5 + O(x 7 ). f(x) = f() + f ()x + 2 f ()x cos x lim x x sin x Proof. cos x x sin x = ( x2 /2 + ) x(x x 3 /3! + ) = x2 /2 x 4 /4! + x 2 x 4 /3! + = /2 + O(x2 ) + O(x 2 ) = 24 = = = 32 ( 2 ) e 53. sin cos x x mc 2 (v/c) 2 (m >, c > ) v 39
40 (i) + t t = cos x (ii) (ii) ( + c x + c 2 x ) 2 = cos x 55. ex e x e x x + e x 6.5. ( lim + n = e n n). Proof. ( n log + ) = n( n n ( ) 2 + ) 2 n = 2 2 n sin x lim (cos x)/x x Proof. cos x = 2 x2 +, x sin x = x 2 3! x4 + lim (cos x x)/x sin x = lim ( 2 ) /x 2 x2 x = e / lim (( + n /n)n e) = Proof. log( + /n) n = n log( + /n) = 2 4 n + 3 n 2 +
41 e ( + /n) n = e /2n+/3n2 + = ( /2n + /3n 2 + ) + ( /2n + /3n 2 + )2 2 + ( /2n + /3n 2 + )3 + 3! = 2n + 24 n 2 + ( + /n) n e e/2n lim n n(( + /n)n e) = e e ( e = lim + ) n n n e n 57. [, ] f(x) ( lim + f(/n) ) ( + f(2/n) ) (... + f(n/n) ) n n n n 58. = e f(t) dt lim rctn x = π/2 x + π 2 rctn x = x + b x 2 + c ( ) x 3 + O x 4, b, c 59. y = tn x x (x ) R n (x) lim R n(x) = n lim n x f (n+) (t)(x t) n dt = n! 4
42 f(x) = f() + f ()(x ) + 2 f ()(x ) n! f (n) ()(x ) n + f(x) x = (Tylor expnsion) (Tylor series) x x (power series), 6.8 ( ). x x < α e x = + x + 2 x2 + 3! x3 +, () sin x = x 3! x3 + 5! x5 7! x7 +, (2) cos x = 2 x2 + 4! x4 6! x6 +, (3) log( + x) = x 2 x2 + 3 x3 4 x4 +, (4) ( + x) α = + αx + α(α ) x α(α )(α 2) x 3 +. (5) 3! Proof. () n! x e t (x t) n dt x n+ e x (n + )! 6.4 R n (x) (2), (3) (4) x R n (x) = ( ) n (x t) n dt ( + t) n+ t x x t x t + t x R n (x) x n+ x (n ) 42
43 (5) n! x f (n+) (t)(x t) n dt = x n! α(α ) (α n) ( + t) α n (x t) n dt x > n α n x ( + t) α n (x t) n dt x < x < x ( ) n x t x ( + t) dt α + t = mx{( x t)/( t); t x } = x x x < (x t) n dt = xn+ n +. ( ) n x t ( t) α dt t ( ) n x t x ( t) α dt x n ( t) α dt. t α(α ) (α n) x n (n ) n! α * 3 α > l < α l l α(α ) (α n) n! = = α(α ) (α l + )(l α) (n α) n! α(α ) (α l + ) l α n α (l )! l n α(α ) (α l + ). (l )! α < l α < l + l (n + )(n + 2) (n + l) (l )! n α l α l + n α n + l α(α ) (α n) n! (n + )(n + 2) (n + l) (l )! *3 Divide nd rule 43
44 n l x n * 32 Remrk. lim n x f (n+) (t)(x t) n dt = n! f(x) = { e /x if x >, if x f f (n) () = (n =,,... ) R n (x) = f(x) x > lim R n(x) = e /x n 6. f (n) (x) = p n (/x)e /x, x > (p n (/x) /x 2n ) f (n) () = 6. (i) t > n =,, 2, e kt k n e nt n! ( e t ) n+. k= (ii) f(x) = k= e kt cos(k 2 x) f C e 2nt f (2n) () (2n)! ( e t ) 2n+. *32 x < x = e 44
45 6.9. f(x) x = (i) g(x) x = f(x)g(x) x = (ii) g(x) x = f() g(f(x)) x = g(x) = /x f() /f(x) x = (iii) f () x = f() f * 33 ( x)( + x + x x n ) = x n x x x = + x + x2 + + x n + xn x. + x = x + x2 + + ( ) n x n + ( ) n xn + x. *33 ymgmi/teching/complex/complex2.pdf (iii) ymgmi/teching/functionl/hilbert2.pdf 45
46 log( + x) = x dt ( t + t 2 + ( ) n t n + ( ) n + t ) = x 2 x2 + 3 x3 + + ( ) n n xn + ( ) n x R n (x) = ( ) n x t n + t dt tn t n + t dt. log( + x) x = t n R n () = + t dt t n dt = (n ) n + log 2 = log( + x) 63., b > x b + x dx = ( ) n n + b n= = 2, b = 64. rctn x x = rctn x = x + t 2 dt. Remrk. B. Tylor Newton Tylor (75) Tylor J. Stirling (77) C. Mclurin (742) 46
47 Newton J. Gregory (Newton ) Tylor Newton (665) (67) Gregory Gregory Newton Tylor J.- L. Lgrnge (772) A. L. Cuchy (82) * 34 Mdhv (Hindu of Sngmgrm (35 425) Kerl Newton-Gregory 7 x /2 dx, x 2 dx improper integrl Remrk. *34 H.N. Jhnke, A History of Anlysis, AMS,
48 7.. x α dx = x α dx = { α if < α <, + otherwise. { α if α >, + otherwise. 65. α > e αx dx 7.2. log x dx y = log x x = log x x log x x log x dx = [x log x x] = x = x log x log x x = x log x x= x = x = > lim log + = /t t + log = log t t (log t << t) 48
49 66. x 2 dx = [ x ] x= x= = 2. y = g(x) x y = g(x) 7.3 ( ). (i) f(x) g(x), x (ii) g(x)dx < + f(x) dx f(x) g(x), x > g(x)dx < + 49 f(x) dx
50 7.4. I n = x n e x dx (n =,, 2, ) Proof. x n e x/2 x x n e x Me x/2, x M > I n+ = (n + )I n, n =,, 2, I = + I n = n! e x dx = 7.5. e x2 dx Proof. x e x2 e x e x2 dx = e x2 dx + e x2 dx + e x2 dx e x dx < ( ). x > Γ(x) = + t x e t dt Γ(x + ) = xγ(x), Γ() =. Γ(n + ) = n!. 5
51 Proof. dt = + dt 7.7. e x2 dx = Γ ( ). 2 * 35 π II Γ(/2) = π 67. > b < ( log x) x b dx 68. ( π 2 rctn x ) dx, > 69. sin x x dx Remrk. f(t) dt < x lim x f(t) dt improper integrl *35 (Gussin integrl) 5
52 7.8. f(x) = { sin(/x) (x ) (x = ) b lim sin(/x) dx + 8 (sequence) { n } n n = n= (series) * 36 n= n S = lim n k n k= * 37, n = + n= *36 sequence sequence series *37 52
53 (geometric series * 38 ) ( x)( + x + x x n ) = x n+ x < n + x + x 2 + = x < < = ( j {,,..., 9}) (deciml expnsion) = k= { k } k = k=2 k k k = 9 ( k 2) k=2 k k k k < 9 k = k=2 = [] * 39 ( ) = 2 = 2 + k=3 k k 2 k = 9 ( k 3) 2 = [ 2 ] *38 geometric sequence rithmetic sequence rithmetic men, geometric men rithmetric *39 [x] x 53
54 k = [ k k k ] { k } k = 9 ( k m) m m k= m k k = k= k k + k=m m 9 k = k= k k + m = m 2 m, m = m Proof. = l/m (l, m ) m m {,,..., m } m n (n-dic system) < < k = n k, k {,,..., n } k= { k } k = [n k n k n k ] n m (n )(n ) = m 2 m, m = m + 54
55 lim n = n 8.3. n α > ζ(α) = n= n α α = ζ(α) < + α >. n= n = +. n+ x dx n k= n k + x dx + 2 lim + + n n log n =, log n k+ 2k(k + ) x k k kx dx 2k 2 γ = lim ( n ) log n n (Euler s constnt) 7. k=n (k + ) 2 k=n k(k + ) = n /2 < γ < γ =
56 8.4. { n } n lim 2n = = lim 2n+ n n lim n = n 8.5 (Leibniz). { n } n = log 2 (N. Merctor, 668) = π 4 (G.W. Leibniz, 682) n n n = lim F N n= k F k Proof. k k F k= k {, 2,..., n} F F + + n k lim k F F N k F n k 7. < + = + 56
57 8.8. {n, n 2,... } 8.9. n n nk = k= n. n= n < + n (bsolutely convergent) f(x) dx f(x) dx < * n c nx n x = x < Proof. lim n c n n = c n n M (n ) n M > x < c n x n = n= c n n x n M n= x n = M x <. n= 8.. (i) x e x = + x + 2 x2 + 3! x3 +, sin x = x 3! x3 + 5! x5 7! x7 +, cos x = 2 x2 + 4! x4 6! x6 + *4 57
58 (ii) x < log( + x) = x 2 x2 + 3 x3 4 x4 +, ( + x) = + x + ( ) x ( )( 2) x 3 + 3! (iii) e x sin x dx (iv) ( * 4 x sin(t 2 ) dt = lim sin(t 2 ) dt x 72. > ( t) Newton t = ± (i) < < (ii) < < ( t) = c t c 2 t 2... c k > (iii) ( n c k = k= lim t k= n c k t k = lim t ( t) k=n+ 8.2 ( ). n n n n n n c k t k ) lim t ( ( t) ) =. Proof. n = b n c n, b n, c n, n = b n + c n *4 Fresnel integrl π/8 58
59 n b n n c n n = n n b n n c n { n } Remrk. (conditionlly convergent) { i } i I i = lim i I i F I i F i < + summble i R i I Remrk. lim F I i F i i. i I i I i I i (supremum) { } sup i ; F I i F 8.3. { i } i I I I = n I n i = i i I n= i I n 8.4. n < +, n= b n < + n= 59
60 I = {i = (m, n); m, n }, c i = m b n c i < + i I ( ) ( ) c i = m b n. i I m= n= m,n m b n 8.5. f(x) = k x k, g(x) = k= b l x l, l= x < r x f(x)g(x) = n c n x n, c n = k b n k n= 8.6. x <, y < (x + y xy) n = (x n + x n y + + xy n + y n ) n= n= k=, 2, 3, 4,
61 + p q n p 2 4 2q (n )p np 2(n )q + 2 2nq (6) + ( np ) 2nq np 2 4 2np 2 ( ) qn γ n = n log n log(2pn) + γ 2pn 2 (log(pn) + γ pn) 2 (log(qn) + γ qn) = log(2p) 2 log p 2 log q + γ 2pn 2 γ pn 2 γ qn n log(2p) 2 log p 2 log q = log(2 p/q) = log 2, = 3 log ( ) ( * 42 ) *42 Bernhrd Riemnn ( ) ( ) 6
62 73 (Chllenging). A A A A Wikipedi quntity vrible * 43 (i) (ii) (iii) (differentil eqution) *43 62
63 * 44 x I(x) x, I x x + I(x) I(x + ) I(x) x I(x) I(x + x) I(x) x x x di dx = I I = I(x) di I dx = I x x x (x x ) = x I = I(x ) di I(x) x I dx dx = I(x ) I di = log I(x) log I(x ) I(x) = I e (x x ) I x (x I I x I x I x, x 2,..., x n I, I 2,..., I n (I j I j ) (x j x j ) I j j j *44 Beer s lw. Bier 63
64 n n di = dx I n di, dx di = dx I, A t h S, V, v t H T h S = S(h) V = h 64 S(x) dx.
65 V (t + t) V (t) Av t dv dt = Av. t t + t m * 45 2 mv2 = mgh v = 2gh. t dv dt = S(h)dh dt V, v dh h dt = 2gA S(h). h t S(h) h dh dt = 2gA h t t t = t = T H S(h) h dh = 2gAT H T S(h) S S H h dh = 2gAT T = 2S 2gA H. *45 m 65
66 y = f(x) y (trctrix) (, ) y x y. (x, y) y = dy dx (x x) + y y (, y dy dxx) ( ) 2 dy x 2 + x 2 = 2 dx dy dx dy dx = ± 2 x 2. x x ( 2 u y = ± 2 du = ± log + 2 x 2 u x 2 x 2 ) u = sin θ rcsin(x/) π/2 rcsin(x/) rcsin(x/) sin θ dθ sin θ dθ = π/2 π/2 sin θ dθ 2 x 2. sin θ = 2t/( + t 2 ), dθ = 2dt/( + t 2 ) θ = π/2 t =, x/ = sin θ t = ( + 2 x 2 )/x rcsin(x/) + 2 x dθ = log t = log 2. π/2 sin θ x B lim Γ(t) = + t + 66
67 x = t x t + x = (t )u u = u + Γ(t + ) = t t+ e t u t e t tu du. g(u) = u t e t tu u = log g(u) u = Tylor log g(u) = t(log u u + ) = t 2 (u )2 + t 3 (u ) t t < u < + t t + < ɛ < + g(u)du t t /2 e t Γ(t) Stirling +ɛ ɛ ɛ t ɛ t e t(u )2 /2 du = ɛ t t e x2 /2 dx n! ( n ) n 2πn e + ɛ t e x2 /2 dx e x2 /2 dx = 2π +ɛ Γ(t + ) t t+ e t e t(x )2 /2 dx t + 67 ɛ
68 + u t e t tu du = + e t(x log(+x)) dx y = { x log( + x) if x, x log( + x) if < x x y, y x x + e t(x log(+x)) dx = y 2 = x log( + x) x dy dx = x 2( + x) y = + dx ty2 e dy dy x 2( + x) x log( + x) > dx lim y dy = lim 2( + x) x log( + x) x x dx lim y + y dy = lim 2( + x) = 2 x + x t + g(y) = dx dy z = ty + u t e t tu du = t + =, ( ) z e z2 g dz t + + ( ) z + lim t u t e t tu du = lim e z2 g dz = e z2 g() dz = πg() t + t + t g(y) = dx dy x < y = ± x (x x 2 /2 + x 3 /3... ) = x x
69 g() = == () dy dx 2( + x) 2 3 x +... x= = 2 ( ) z g = g() + g () t z + 2 t g () z2 t +... e z2 z dz =, e z2 z 2 dz = π 2 t u t e t tu du = ( ) π g() + g () 4 t +... (symptotic expnsion) x = 2y y y g() = 2, g () = 2 3, y = ± 2 x2 3 x x x = 2y y y x = y + by 2 + cy y 69
f(x) x = A = h f( + h) f() h A (differentil coefficient) f(x) f () y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) * t (velo
I 22 7 9 2 2 5 3 7 4 8 5 2 6 26 7 37 8 4 A 49 B 53 big O f(x) x = A = h f( + h) f() h A (differentil coefficient) f(x) f () y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) * t
y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) (velocity) p(t) =(x(t),y(t),z(t)) ( dp dx dt = dt, dy dt, dz ) dt f () > f x
I 5 2 6 3 8 4 Riemnn 9 5 Tylor 8 6 26 7 3 8 34 f(x) x = A = h f( + h) f() h A (differentil coefficient) f f () y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) (velocity) p(t)
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
A S hara/lectures/lectures-j.html ϵ-n 1 ϵ-n lim n a n = α n a n α 2 lim a n = 0 1 n a k n n k= ϵ
A S1-20 http://www2.mth.kyushu-u.c.jp/ hr/lectures/lectures-j.html 1 1 1.1 ϵ-n 1 ϵ-n lim n n = α n n α 2 lim n = 0 1 n k n n k=1 0 1.1.7 ϵ-n 1.1.1 n α n n α lim n n = α ϵ N(ϵ) n > N(ϵ) n α < ϵ (1.1.1)
4................................. 4................................. 4 6................................. 6................................. 9.................................................... 3..3..........................
I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59
() 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.](/thumbs/89/98384840.jpg "() 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 >
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
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
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
- II
- II- - -.................................................................................................... 3.3.............................................. 4 6...........................................
2009 IA I 22, 23, 24, 25, 26, a h f(x) x x a h
009 IA I, 3, 4, 5, 6, 7 7 7 4 5 h fx) x x h 4 5 4 5 1 3 1.1........................... 3 1........................... 4 1.3..................................... 6 1.4.............................. 8 1.4.1..............................
04.dvi
22 I 4-4 ( ) 4, [,b] 4 [,b] R, x =, x n = b, x i < x i+ n + = {x,,x n } [,b], = mx{ x i+ x i } 2 [,b] = {x,,x n }, ξ = {ξ,,ξ n }, x i ξ i x i, [,b] f: S,ξ (f) S,ξ (f) = n i= f(ξ i )(x i x i ) 3 [,b] f:,
f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a
3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (
0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9
1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),
2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p
2012 IA 8 I 1 10 10 29 1. [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.
v er.1/ c /(21)
12 -- 1 1 2009 1 17 1-1 1-2 1-3 1-4 2 2 2 1-5 1 1-6 1 1-7 1-1 1-2 1-3 1-4 1-5 1-6 1-7 c 2011 1/(21) 12 -- 1 -- 1 1--1 1--1--1 1 2009 1 n n α { n } α α { n } lim n = α, n α n n ε n > N n α < ε N {1, 1,
i 6 3 ii 3 7 8 9 3 6 iii 5 8 5 3 7 8 v...................................................... 5.3....................... 7 3........................ 3.................3.......................... 8 3 35
i
i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
(, ) (, ) S = 2 = [, ] ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) ( ) (
![(, ) (, ) S = 2 = [, ] ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) ( ) (](/thumbs/92/107866524.jpg "(, ) (, ) S = 2 = [, ] ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) ( ) (")
B 4 4 4 52 4/ 9/ 3/3 6 9.. y = x 2 x x = (, ) (, ) S = 2 = 2 4 4 [, ] 4 4 4 ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, 4 4 4 4 4 k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) 2 2 + ( ) 3 2 + ( 4 4 4 4 4 4 4 4 4 ( (
() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
IA September 25, 2017 ( ) I = [a, b], f (x) I = (a 0 = a < a 1 < < a m = b) I ( ) (partition) S (, f (x)) = w (I k ) I k a k a k 1 S (, f (x)) = I k 2
![IA September 25, 2017 ( ) I = [a, b], f (x) I = (a 0 = a < a 1 < < a m = b) I ( ) (partition) S (, f (x)) = w (I k ) I k a k a k 1 S (, f (x)) = I k 2](/thumbs/97/133729157.jpg "IA September 25, 2017 ( ) I = [a, b], f (x) I = (a 0 = a < a 1 < < a m = b) I ( ) (partition) S (, f (x)) = w (I k ) I k a k a k 1 S (, f (x)) = I k 2")
IA September 5, 7 I [, b], f x I < < < m b I prtition S, f x w I k I k k k S, f x I k I k [ k, k ] I I I m I k I j m inf f x w I k x I k k m k sup f x w I k x I k inf f x w I S, f x S, f x sup f x w I
DVIOUT
A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -
M3............................................................................................ 3.3................................................... 3 6........................................... 6..........................................
6. Euler x
...............................................................................3......................................... 4.4................................... 5.5......................................
x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n
![x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n](/thumbs/92/108343288.jpg "x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n")
1, R f : R R,.,, b R < b, f(x) [, b] f(x)dx,, [, b] f(x) x ( ) ( 1 ). y y f(x) f(x)dx b x 1: f(x)dx, [, b] f(x) x ( ).,,,,,., f(x)dx,,,, f(x)dx. 1.1 Riemnn,, [, b] f(x) x., x 0 < x 1 < x 2 < < x n 1
1
1 1 7 1.1.................................. 11 2 13 2.1............................ 13 2.2............................ 17 2.3.................................. 19 3 21 3.1.............................
Chap11.dvi
. () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x
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
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
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
1 R n (x (k) = (x (k) 1,, x(k) n )) k 1 lim k,l x(k) x (l) = 0 (x (k) ) 1.1. (i) R n U U, r > 0, r () U (ii) R n F F F (iii) R n S S S = { R n ; r > 0
III 2018 11 7 1 2 2 3 3 6 4 8 5 10 ϵ-δ http://www.mth.ngoy-u.c.jp/ ymgmi/teching/set2018.pdf http://www.mth.ngoy-u.c.jp/ ymgmi/teching/rel2018.pdf n x = (x 1,, x n ) n R n x 0 = (0,, 0) x = (x 1 ) 2 +
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
webkaitou.dvi
( c Akir KANEKO) ).. m. l s = lθ m d s dt = mg sin θ d θ dt = g l sinθ θ l θ mg. d s dt xy t ( d x dt, d y dt ) t ( mg sin θ cos θ, sin θ sin θ). (.) m t ( d x dt, d y dt ) = t ( mg sin θ cos θ, mg sin
Riemann-Stieltjes Poland S. Lojasiewicz [1] An introduction to the theory of real functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,,. Riemann-S
![Riemann-Stieltjes Poland S. Lojasiewicz [1] An introduction to the theory of real functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,,. Riemann-S](/thumbs/94/118569863.jpg "Riemann-Stieltjes Poland S. Lojasiewicz [1] An introduction to the theory of real functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,,. Riemann-S")
Riemnn-Stieltjes Polnd S. Lojsiewicz [1] An introduction to the theory of rel functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,, Riemnn-Stieltjes 1 2 2 5 3 6 4 Jordn 13 5 Riemnn-Stieltjes 15 6 Riemnn-Stieltjes
Fubini
3............................... 3................................ 5.3 Fubini........................... 7.4.............................5..........................6.............................. 3.7..............................
mugensho.dvi
1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y
No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
入試の軌跡
4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29
![[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29](/thumbs/93/114156928.jpg "[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29")
A p./29 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x) + C f(x) A p.2/29 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x)
18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,
2011de.dvi
211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
.1 1,... ( )
1 δ( ε )δ 2 f(b) f(a) slope f (c) = f(b) f(a) b a a c b 1 213 3 21. 2 [e-mail] nobuo@math.kyoto-u.ac.jp, [URL] http://www.math.kyoto-u.ac.jp/ nobuo 1 .1 1,... ( ) 2.1....................................
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............
20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)
![Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)](/thumbs/94/118373946.jpg "Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)")
3 3 22 Z[i] Z[i] π 4, (x) π 4,3 (x) x (x ) 2 log x π m,a (x) x ϕ(m) log x. ( ). π(x) x (a, m) = π m,a (x) x modm a π m,a (x) ϕ(m) π(x) ϕ(m) x log x ϕ(m) m f(x) g(x) (x α) lim f(x)/g(x) = x α mod m (a,
ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
29
9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
(u(x)v(x)) = u (x)v(x) + u(x)v (x) ( ) u(x) = u (x)v(x) u(x)v (x) v(x) v(x) 2 y = g(t), t = f(x) y = g(f(x)) dy dx dy dx = dy dt dt dx., y, f, g y = f (g(x))g (x). ( (f(g(x)). ). [ ] y = e ax+b (a, b )
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
1 I
1 I 3 1 1.1 R x, y R x + y R x y R x, y, z, a, b R (1.1) (x + y) + z = x + (y + z) (1.2) x + y = y + x (1.3) 0 R : 0 + x = x x R (1.4) x R, 1 ( x) R : x + ( x) = 0 (1.5) (x y) z = x (y z) (1.6) x y =
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3) 6 y = g(x) x = 1 g( 1) = 2 ( 1) 3 = 2 ; g 0 ( 1) =
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x
![, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x](/thumbs/93/113428934.jpg ", 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x")
1 1.1 4n 2 x, x 1 2n f n (x) = 4n 2 ( 1 x), 1 x 1 n 2n n, 1 x n n 1 1 f n (x)dx = 1, n = 1, 2,.. 1 lim 1 lim 1 f n (x)dx = 1 lim f n(x) = ( lim f n (x))dx = f n (x)dx 1 ( lim f n (x))dx d dx ( lim f d
i
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED)
rational number p, p, (q ) q ratio 3.14 = 3 + 1 10 + 4 100 ( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED) ( a) ( b) a > b > 0 a < nb n A A B B A A, B B A =
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
( )
18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................
熊本県数学問題正解
00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (
<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>
電気電子数学入門 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/073471 このサンプルページの内容は, 初版 1 刷発行当時のものです. i 14 (tool) [ ] IT ( ) PC (EXCEL) HP() 1 1 4 15 3 010 9 ii 1... 1 1.1 1 1.
..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A
![..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A](/thumbs/93/112175219.jpg "..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A")
.. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.
( ) f a, b n f(b) = f(a) + f (a)(b a) + + f (n 1) (a) (n 1)! (b a)n 1 + R n, R n = b a f (n) (b t)n 1 (t) (n 1)! dt. : R n = b a f (n) (b t
5 1 1.1 ) f, b n fb) = f) + f )b ) + + f n 1) ) n 1)! b )n 1 + R n, R n = f n) b t)n 1 t) n 1)! dt. : R n = f n) b t)n 1 t) n 1)! dt ] b b b t)n 1 + n 1)! = f n 1) b )n 1 ) + R n 1. n 1)! R n = [f n 1)
.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(
06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f
,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)
II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
A 2008 10 (2010 4 ) 1 1 1.1................................. 1 1.2..................................... 1 1.3............................ 3 1.3.1............................. 3 1.3.2..................................
untitled
0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n
ii
ii iii 1 1 1.1..................................... 1 1.2................................... 3 1.3........................... 4 2 9 2.1.................................. 9 2.2...............................
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)
2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x
2014 S hara/lectures/lectures-j.html r 1 S phone: ,
14 S1-1+13 http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 1 S1-1+13 14.4.11. 19 phone: 9-8-4441, e-mail: hara@math.kyushu-u.ac.jp Office hours: 1 4/11 web download. I. 1. ϵ-δ 1. 3.1, 3..
名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト
名古屋工業大の数学 年 ~5 年 大学入試数学動画解説サイト http://mathroom.jugem.jp/ 68 i 4 3 III III 3 5 3 ii 5 6 45 99 5 4 3. () r \= S n = r + r + 3r 3 + + nr n () x > f n (x) = e x + e x + 3e 3x + + ne nx f(x) = lim f n(x) lim
2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =
1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,
30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
body.dvi
..1 f(x) n = 1 b n = 1 f f(x) cos nx dx, n =, 1,,... f(x) sin nx dx, n =1,, 3,... f(x) = + ( n cos nx + b n sin nx) n=1 1 1 5 1.1........................... 5 1.......................... 14 1.3...........................
http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
December 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =
9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =
5 5. 5.. A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d =
meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
ルベーグ積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
ルベーグ積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/005431 このサンプルページの内容は, 初版 1 刷発行時のものです. Lebesgue 1 2 4 4 1 2 5 6 λ a
( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +
(.. C. ( d 5 5 + C ( d d + C + C d ( d + C ( ( + d ( + + + d + + + + C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( + + + + d log( + + + C ( (8 d 7 6 d + 6 + C ( (9 ( d 6 + 8 d
1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,
![1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,](/thumbs/91/105754557.jpg "1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,")
2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)
30 I .............................................2........................................3................................................4.......................................... 2.5..........................................