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 (velocity) r(t) = (x(t), y(t), z(t)) ( dr dx dt = dt, dy dt, dz ) dt. ( ) = x x 2 2. x =, x = *2 o(h) = f( + h) f() Ah o(h) h h ( ) o(h) f( + h) f() h h = 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) * geometry geo *2 Weierstrss 2
h o 3 (h) f( + h)g( + h) = f()g() + (f ()g() + f()g ())h + o 3 (h), o 3 (h) = h h (f(x)g(x)) = f (x)g(x) + f(x)g (x). ( ). (i) (ii) *3 (f(x)g(x)) = f (x)g(x) + f(x)g (x). {f(g(x))} = f (g(x))g (x). 3. (ii) o(h) =, x =, ( ) = x x 2. x x+h x = x h h 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. *3 *4 bse *5 (nturl logrithm) ln x 3
(ii) x x (x > ) x = e log x d dx x = e log x ( log x) = x. 4. > y = x y = x 5. x x (x > ) 6. x < (log( x)) = x (e x ) = e x, (log x ) = x (x ), (xα ) = αx α (x > )..3. ( log x + ) x 2 + = x2 +. 7. x ( ) ( ).4. f(x) = (f(x)) ( ) = (f(x)) 2 f (x) = f (x) f(x) f(x) 2. 8. ( ) 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 h = cos h h sin x + sin h h cos h sin h, h h h cos x x = y = cos x x = y = *6 (sin h h tn h cos x *6 (rdin).7 rd rd 4
y h x (sin x) = cos x (cos x) = sin x (sin x) = cos x, (cos x) = sin x, (tn x) = 9. tn x (cos x) 2 = + (tn x)2.. rdin. 2 ( > ) log x, x, e x x x e x x e bx (b > ) f(x) = x e bx, x > f(x) f (x) = x e bx ( bx) x = /b x /b f (x) + f(x) 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 5
y C /b x *7 x + e x/2 = x + x e x = log x x t = log x log x x + x = t + t e t = t + ( ) t / = e t x + t + log x << x << e x (x + ) 2. >, 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 2. 2.. x x/x log x << x 3. x + xx log x x log(x/x ) = x x =. x x/x =. 4. > x < n n x n = *7 6
5. (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 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 *8 sin x sin x sin x (sin x) 7
6. rccos x + rcsin x = π ( x π/2) 2 7. < < π/2 sin x = sin x 8. 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) h = k k h f(x + h) f(x) = f (x). y = sin x x = rcsin y 9. rctn x rcsin (y) = 2. f(x) = ex e x 2 (sin x) = cos x = y 2 (i) (ii) g(y) (iii) g(y) y 4 b b (f(x) + g(x)) dx = f(x) dx = n n k= b f(x) dx + b g(x) dx f( + k(b )/n) b n *9 8
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 ] 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 Remrk. x j x j dx [, b] I(t) t V = Q = b b S(x) dx. I(t) dt. * B. Riemnn (826 866) A.-L. Cuchy (789 857) 9
L = b (dx ) 2 + 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 r = f(θ) (θ j θ θ j ) S 2 n f(ξ j ) 2 (θ j θ j ) j= S = 2 β * α f(θ) 2 dθ 2. 22. r = ( + cos θ) ( π θ π) (crdioid) 4. ( ). (i) b f(x) dx + c f(x) dx = c b f(x) dx. * 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 j
(ii) f(x) g(x) ( x b) b f(x) dx b g(x) dx. b f(x) dx b f(x) dx. Remrk. x b f(x) dx = b f(y) dy =. 23. n ( x) n sin x dx n + n+. 24. log x = t dt log(xy) = log x + log y 25. b > > b b b t dt > b > x > b t dt > 2 dt t dt = +, x t dt =, x + t * 2 4.2 (Dirichlet). f(x) f(x) = { x x [, b] *2 e x
f(x) x = f(x), x 4.3. f(x) x + (i) (x < ) f(x) = (x = ) 2 (x > ) x = (ii) x = g(x) = { sin(/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 = j= f(ξ j )(x j x j ) *3 *4 2
x = c ( < c < b) c [x k, x k ],,, f(ξ j )(x j x j ) = c j f(x) dx, 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 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 ( ). 4.5 (). f(x) [, b] d dx f(t) dt = f(x). 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) = M h = m h = f(x). h h h h *5 3
M h m h S(x + h) S(x) x x + h 4.6 ( * 6 ). f(x) f(x) F (x) b Proof. f(x)dx = F (b) F () F (b) F () = [ ] b F (x) ( d x ) F (t) dt F (x) = F (x) F (x) = dx C 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 = b f(x) dx 27. f(x) f() = f (t) dt f(x) ( < x < b) *6 4
(i) f (x) ( < x < b) f (, b) (incresing) (ii) f (x) > ( < x < b) f (, b) (strictly incresing) * 7 2 x 2 dx = rcsin x, x 2 + 2 dx = rctn x. x2 + A dx = log(x + x 2 + A). 4.7. 4.8 ( ). x 2 + dx = π 4. (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 4.9. log x dx log x x log x (x log x) = log x +. *7 5
x log x = log x dx + dx log x dx = x log x x 4.. n = 2, 3,... x (x 2 + 2 ) n dx = 2 2n 28. n = (x 2 + 2 ) n. 29. x 2 x 2 dx, xe x2 dx 4.. I n (x) = (x 2 + 2 ) n dx ( ) x (x 2 + 2 ) n = (x 2 + 2 ) n 2n x2 + 2 2 (x 2 + 2 ) n+ = 2n (x 2 + 2 ) n + 2 2 n (x 2 + 2 ) n+ 2 2 ni n+ (x) (2n )I n (x) = x (x 2 + 2 ) n (recursive reltion) I (x) = x 2 + 2 dx = rctn x I 2 (x), I 3 (x),... 3. n =, 2,... x n e x dx 4.2. x 2 (i) dx = rcsin. 4x x 2 2 x2 (ii) + A dx = ( x x 2 2 + A + A log x + ) x 2 + A. 6
Proof. (i) dx = x 2 dx = rcsin. 4x x 2 4 (x 2) 2 2 (ii) x 2 + 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 3. (x 2 x 2 ) 2 x 2 dx = ( x 2 2 x 2 + 2 rcsin x ) 2 t 2 dt * 8 g(x) f(x) dx, deg g < deg f * 9 *8 rtionl function. *9 deg f f (degree) 7
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 (prtil frction decomposition) 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 q(x) αx + β (x 2 + x + b) l dx, l m (x + c) l dx = { ( l)(x+c) l if 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, 4.9 4.3. x 3 + = (x + )(x 2 x + ) x 3 + dx 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) 8
32. 33. x 3 dx, x 4 + dx x 4 dx 4.4. + e x + e 2x dx t = ex 34. + 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 * 2 x 2 + y 2 = (, ) t y = t(x + ), x 2 + y 2 = x = t2 + t 2, y = 2t + t 2 x 2 dx = y dx t x = cos θ, y = sin θ t sin θ t cos θdθ = 2 t2 ( + t 2 ) 2 dt *2 (rtionl curve) 9
y y = t(x + ) x cos θ = ( t 2 )/( + t 2 ) dθ = 2dt/( + t 2 ) R(cos θ, sin θ) dθ = 35. t = tn(θ/2) ( ) t 2 R + t 2, 2t 2 + t 2 + t 2 dt cos θ = 2 cos 2 θ 2, sin θ = 2 cos θ 2 sin θ 2 y = x 2 y = t(x + ), y 2 = x 2 x = + t2 t 2, y = 2t t 2 t 36. x2 dx 5 * 2 *2 2
f () > f(x) x = f () = f(x) x = f () < f(x) x = f () = f (x) x = y = f(x) x = f() * 22 x f () f(x) f() x f(x) f() + f ()(x ), x * 23 f(x) x = (liner pproximtion) f(x) (, f()) 5.. (i) + x + x/2 x x =...5. (ii) sin x x x = 2π/36 sin.7. 5.2. x f(x) = 4πx 3 /3 x = r f(r + r) f(r) f(r) 3 r r V V 6378Km 7km 6357Km 7Km 7 6378 r r 7 6357 V/V.3%.8% 5.3. x e x e x 2x, log( + x) x (x ) x e x e x log( + x) = 2x x x = 2. *22 locl mximum (locl minimum). *23 2
f(x) = f() + f(x) f() f ()(x ) = f (t) dt f (t) dt f ()(x ) 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 f (t)(b t) dt f (t)(x t) dt f(x) t x f (t) M f (t)(x t) dt f (t) (x t) dt M M x 2 /2 (x t) dt = M (x )2 2 37. < x x < 5.4. ( + t) = ( + t) 3/2 /4 t. /4 * 24 38. sin..5 8 (.)2 =.25 7 f (t) x f (t) ( t x) f () f (t) f () f (t)(x t) dt f ()(x t) dt = 2 f ()(x ) 2 *24 f (t). <.5 22
f(x) f() + f ()(x ) + 2 f ()(x ) 2 (x ) f(x) x = * 25 (qudrtic pproximtion) f () = f () f(x) x = (, f()) f () x = x = f () * 26 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.99986 (ii) + x x/2 x 2 /8 cos x x 2 /2 (x ) + x x/2 x 2 /8 = x cos x x x 2 /2 = 4. 39. x e x (x > ) > f (x) f (x) x * 27 f(x) *25 x *26 f () *27 23
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] n n f t j c j t j f(c j ) * 28 j= j= Proof. f (t) f(x) = f(c) + f (c)(x c) + f(x) f(c) + f (c)(x c) c f (t)(x t) dt c b, x b x = c j, c = t j c j f(c j ) f(c) + f (c)(c j c) t j j t j f(c j ) f(c) + f (c) j j t j (c j c) = f(c) 4. f(x) f(c) + f (c)(x c) *28 j t jc j b t j t j c j t j b j 24
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 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) p j 5.9. f(x) f (c) = x = c f (x) (c, f(c)) 5.. f(x) = x 3 x = 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 4 25
6 6.. f(x) n n f (n) C n * 29 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 6.2 (). C n+ f(x) f(x) = f() + f ()(x ) + + n! f (n) ()(x ) n + R n (x), R n (x) = n! R n (x) (reminder) f (n+) (t)(x t) n dt Proof. n n =, f(x) = f() + f ()(x ) + + (n )! f (n ) ()(x ) n + (n )! b f (n) (t)(x t) n dt d ( f (n) (t)(x t) n) = n! dt (n )! f (n) (x t) n + n! f (n+) (t)(x t) n t (n )! f (n) (t)(x t) n dt = n! f (n) ()(x ) n + n! f (n+) (t)(x t) n dt Remrk. B. Tylor (823) Gsprd de Prony (85) de Prony *29 f () (x) = f(x) C 26
6.3. 44. 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 t) 2n cos t dt. + ( ) n (x t) 2n+ cos t dt. (2n + )! log( + x) = x 2 x2 + + ( ) n+ n xn + ( ) n+ (x t) n dt. ( + t) n+ ( + x) α α(α ) = + αx + x 2 α(α )... (α n + ) + + x n 2 n! α(α )... (α n) x + ( + t) α n (x t) n dt. n! 45. g 46. (n )! dx n d n g(t)(x t) n dt = g(x) ((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 x e x Ce x/2 (x > ) f(x), g(x) C > x f(x) Cg(x) 27
f(x) g(x) f(x) = O(g(x)) * 3 x e x = O(e x/2 ) 6.4. n! n A n (A > ) log(n!) = log 2 + log 3 + + log n n log x dx log n! n 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 = 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. x f(x) = O(g(x)) g(x) *3 big O Pul Bchmnn (894) Edmund Lndu (99) little o O Ordnung (order) Omicron 28
t x ( x t ) f (n+) (t) M n+ (x) R n (x) n! M n+(x) x t n dt = (n + )! M n+(x) x n+ 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 ) 2 + + f (n) () (x ) n + O((x ) n+ ). n! 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 ). 2 *3 Tylor 29
49. f(x) = tn x x = 5. C n+2 f(x) x (f(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 (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) (ii) f(x) g(x) = + 3x 4x 2 + O(x 3 ). 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 2 + 3 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 x2 + 5 24 x4 + O(x 6 ). 3
6.2. 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 ) sin x = x x 3 /6 + x 5 /5! + O(x 7 ) =, b 2 = 6, c b 2 + 4! = 5! tn x = x + 3 x3 + 2 5 x5 + O(x 7 ). f(x) = f() + f ()x + 2 f ()x 2 + 6.3. Proof. cos x x x sin x cos x x sin x = ( x2 /2 + ) x(x x 3 = x2 /2 x 4 /4! + /3! + ) x 2 x 4 = /2 + O(x2 ) /3! + + O(x 2 ) 2. 6.4. 2 = 24 = 24 24 = 2 5 24 ( 24 = 32 2 52. e ) 24 24 +... 53. sin 6 54. cos x x ) v (i) + t t = cos x (ii) (ii) ( + c x + c 2 x 2 +... ) 2 = cos x 55. ex e x e x x + e x mc 2 (v/c) 2 (m >, c > 3
6.5. Proof. ( + n = e n n). ( n log + ) = n( n n ( ) 2 + ) 2 n = 2 2 n +. 6.6. sin x (cos x)/x x Proof. cos x = 2 x2 +, x sin x = x 2 3! x4 + (cos x x)/x sin x = ( 2 ) /x 2 x2 = e /2. x 6.7. (( + n /n)n e) = Proof. log( + /n) n = n log( + /n) = 2 n + 3 n 2 + 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 n n(( + /n)n e) = e 2 32
56. e ( e = + n n e n 57. [, ] f(x) ( + f(/n) ) ( + f(2/n) n n n 58. ) n ) (... + f(n/n) ) n = e f(t) dt 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 ( xπ/2) R n (x) R n(x) = n n n! f (n+) (t)(x t) n dt = f(x) = f() + f ()(x ) + 2 f ()(x ) 2 + + 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 2 + 2 α(α )(α 2) x 3 +. (5) 3! 33
Proof. () n! e t (x t) n dt x n+ e x (n + )! 6.4 R n (x) (2), (3) (4) R n (x) = ( ) n (x t) n dt ( + t) n+ t x x t x t + t x R n (x) x n+ (n ) x (5) n! f (n+) (t)(x t) n dt = n! α(α ) (α n) ( + t) α n (x t) n dt x > n α n < x < ( + t) α n (x t) n dt ( ) n x t ( + t) dt α + t = x 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 * 32 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 )! *32 Divide nd rule 34
α < l α < l + l (n + )(n + 2) (n + l) (l )! n α α l l + n α (n + )(n + 2) (n + l) n + l (l )! α(α ) (α n) n! n l x n * 33 Remrk. n n! f (n+) (t)(x t) n dt = f(x) = { e /x if x >, if x f f (n) () = (n =,,... ) R n (x) = f(x) x > 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+. *33 x < x = e 35
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 * 34 x x log( + x) = ( x)( + x + x 2 + + x n ) = x n x = + x + x2 + + x n + xn x. + x = x + x2 + + ( ) n x n + ( ) n xn + x. dt ( t + t 2 + ( ) n t n + ( ) n + t ) = x 2 x2 + 3 x3 + + ( ) n n xn + ( ) n R n (x) = ( ) n t n + t dt tn t n + t dt. log( + x) x = t n R n () = + t dt t n dt = (n ) n + *34 http://www.mth.ngoy-u.c.jp/ ymgmi/teching/complex/complex2.pdf (iii) http://www.mth.ngoyu.c.jp/ ymgmi/teching/functionl/hilbert2.pdf 36
log 2 = 2 + 3 4 + 62. log( + x) 63., b > x b + x dx = ( ) n n + b n= = 2, b = 64. rctn x x = rctn x = + t 2 dt. Remrk. B. Tylor Newton Tylor (75) Tylor J. Stirling (77) C. Mclurin (742) Newton J. Gregory (Newton ) Tylor Newton (665) (67) Gregory Gregory Newton Tylor J.-L. Lgrnge (772) A. L. Cuchy (82) * 35 Mdhv (Hindu of Sngmgrm (35 425) Kerl Newton-Gregory 7 x /2 dx, x 2 dx improper integrl *35 H.N. Jhnke, A History of Anlysis, AMS, 23. 37
Remrk. 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 = > + log = /t t + log = log t t (log t << t) 66. [ x 2 dx = ] x= = 2. x x= 38
y = g(x) x y = g(x) 7.3 ( ). (i) f(x) g(x), g(x)dx < + x f(x) dx (ii) f(x) g(x), x > g(x)dx < + f(x) dx 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 > 39
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 = 7.6 ( ). x > Γ(x) = e x2 dx + e x2 dx + + t x e t dt Γ(x + ) = xγ(x), Γ() =. Γ(n + ) = n!. e x2 dx e x dx < + Proof. dt = + dt 7.7. e x2 dx = Γ ( ). 2 * 36 π II Γ(/2) = π 67. > b < ( log x) dx x b *36 (Gussin integrl) 4
68. ( π 2 rctn x ) dx, > 69. Remrk. sin 2 x x dx f(t) dt < x f(t) dt improper integrl 7.8. f(x) = { sin(/x) (x ) (x = ) b + sin(/x) dx 8 (sequence) { n } n n = + 2 + 3 + n= (series) * 37 n= n S = n k n k= *37 sequence sequence series 4
* 38, n = + n= (geometric series * 39 ) ( x)( + x + x 2 + + x n ) = x n+ x < n + x + x 2 + = x < < =. 2... ( j {,,..., 9}) (deciml expnsion) = k= { k } k = k = 9 ( k 2) k=2 = [] * 4 k=2 k k k k k k < 9 k = k=2 ( ) = 2 = 2 + k = 9 ( k 3) 2 = [ 2 ] k=3 k k 2 *38 *39 geometric sequence rithmetic sequence rithmetic men, geometric men rithmetric *4 [x] x 42
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 =. 2... m 2 m, m = m + 8.. 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. 2... m (n )(n ) =. 2... m 2 m, m = m + 8.2. + 2 + 3 + n = n 43
8.3. n + 2 + 3 + α > ζ(α) = n= n α α = ζ(α) < + α >. n= n = +. n+ x dx n k= + 2 + + n n log n log n n k + x dx =, k+ 2k(k + ) x k k kx dx 2k 2 γ = ( + 2 + + n ) log n n (Euler s constnt) 7. k=n (k + ) 2 k(k + ) = n k=n /2 < γ < γ =.5772... * 4 8.4. {c n } n c 2n = c = c 2n+ n n c n = c n 8.5 (Leibniz). { n } n 2 + 3 4 +... *4 44
Proof. {c n } n c =, c 2 = 2, c 3 = 2 + 3, c 4 = 2 + 3 4,... c 2n c 2n+ c 2 c 4 c 6 c 5 c 3 c c 2n+ c 2n = 2n+ n c n 8.6. c 2n = c 2n+ n n 2 + 3 + = log 2 (N. Merctor, 668). 4 3 + 5 7 + = π 4 (G.W. Leibniz, 682). 8.7. n n n = F N n= k F k Proof. k k F k= {, 2,..., n} F F + + n k k F F N k F n k k 7. < + = + 8.8. {n, n 2,... } nk = n. 8.9. n n k= n= n < + n (bsolutely convergent) f(x) dx f(x) dx < 45
* 42 8.. n c nx n x = x < Proof. n c n n = c n n M (n ) n M > x < c n x n = c n n x n M x n = M x <. n= n= n= 8.. (i) x (ii) 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 + (iii) log( + x) = x 2 x2 + 3 x3 4 x4 +, ( + x) = + x + e x sin x dx (iv) ( * 43 ( ) x 2 + 2 sin(t 2 ) dt = x ( )( 2) x 3 + 3! sin(t 2 ) dt 72. > ( t) Newton t = ± (i) < < (ii) < < ( t) = c t c 2 t 2... c k > (iii) ( n c k = k= t k= n c k t k = t ( t) k=n+ c k t k ) t ( ( t) ) =. *42 http://www.mth.ngoy-u.c.jp/~ymgmi/teching/teching.html *43 Fresnel integrl π/8 46
8.2 ( ). n n n n n n Proof. n = b n c n, b n, c n, n = b n + c n n b n n c n n = n n b n n c n { n } Remrk. (conditionlly convergent) { i } i I i = i I i F I i F i < + summble i R i I i I Remrk. F I i F i i. i I i I i (supremum) { } sup i ; F I i F 8.3. { i } i I I I = I n n i = i i I n= i I n 47
8.4. n < +, n= b n < + n= 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) = b l x l, k= l= x < r x n f(x)g(x) = c n x n, c n = k b n k n= n= n= k= 8.6. x <, y < (x + y xy) n = (x n + x n y + + xy n + y n ), 2, 3, 4, 2 + 3 4 + + p q n + 3 + + 2p 2 4 2q +... + 2(n )p + + + 2np 2(n )q + 2 2nq (6) 48
+ ( 3 + + 2np 2 + 4 + + ) 2nq + 2 + + 2np 2 4 2np 2 ( + 2 + + ) qn γ n = + 2 + + 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) 8.7. 2 + 3 + = log 2, 4 + 3 2 + 5 + 7 4 + = 3 log 2. 2 2 + 3 4 + ( ) ( * 44 ) 73 (Chllenging). A A A A Wikipedi quntity vrible * 45 *44 Bernhrd Riemnn (826 866) (828 877) *45 49
(i) (ii) (iii) (differentil eqution) * 46 x I(x) x, I x x + x I(x) I(x + 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 ) = I = I(x ) x I di dx dx = I(x) I(x ) I(x) = I e (x x ) I di = log I(x) log I(x ) I x (x I I x I x I *46 Beer s lw. Bier 5
x, x 2,..., x n I, I 2,..., I n j I j (I j I j ) j (x j x j ) 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 S(x) dx. V (t + t) V (t) Av t dv dt = Av. 5
t t + t m * 47 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 H S dh = 2gAT h T = 2S 2gA H. 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 *47 m 52
( 2 u y = ± 2 du = ± log + 2 x 2 u x 2 x 2 ) u = sin θ rcsin(x/) π/2 sin θ dθ rcsin(x/) π/2 rcsin(x/) sin θ dθ = 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 Γ(t) = + t + 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 )3 +.... t t < u < + t t + < ϵ < + g(u)du +ϵ ϵ e t(u )2 /2 du = ϵ t t ϵ t e x2 /2 dx t t /2 e t Γ(t) ϵ t e x2 /2 dx + ϵ t e x2 /2 dx = 2π 53
n! ( n ) n 2πn e Stirling +ϵ Γ(t + ) t t+ e t e t(x )2 /2 dx t + ϵ + y = u t e t tu du = + e t(x log(+x)) dx { 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 y dy = 2( + x) x log( + x) =, x x dx y + y dy = 2( + x) = 2 x + x t + g(y) = dx dy z = ty + u t e t tu du = + ( ) z e z2 g dz t t t + + t u t e t tu du = t + + ( ) z e z2 g dz = t + e z2 g() dz = πg() g(y) = dx dy x < y = ± x (x x 2 /2 + x 3 /3... ) = x 2 2 3 x +... 54
g() = == 2( + x) 2 () 3 x= x +... = 2 dy dx ( ) z g = g() + g () t z + 2 t g () z2 t +... e z2 z dz =, π e z2 z 2 dz = 2 t u t e t tu du = ( π g() + g () 4 ) t +... (symptotic expnsion) x = 2y + 2 3 y2 + 2 8 y3 +... 74. g() = 2, g () = 2 3,... y = ± 2 x2 3 x3 +... x x = 2y + 2 3 y2 + 2 8 y3 +... x = y + by 2 + cy 3 +... y 55