I 2 7 4 2 2 6 3 8 4 5 26 6 32 7 47 8 52 A 62 B 66 big
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
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
(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. > 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 + = x2 +. 8. 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
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
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
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 2. 2.. 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
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
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
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 (826 866) A.-L. Cuchy (789 857)
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
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 * 22. 23. (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
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 + 4.3. (i) (x < ) f(x) = (x = ) 2 (x > ) x = *2 e x 4
(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
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
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
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
* 7 2 x 2 dx = rcsin x, x 2 + 2 dx = rctn x. x2 + A dx = log(x + x 2 + A). 4.7. x 2 + dx = π 4. 4.8 ( ). (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
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 2 + 2 ) n dx = 2 2n (x 2 + 2 ) n. 28. 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 2
(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. (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
3. (x 2 x 2 ) 2 x 2 dx = ( x 2 2 x 2 + 2 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
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, 4.9 4.3. 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
32. x 3 dx, x 4 dx 33. x 4 + dx 4.4. + e x + e 2x dx t = ex + t + t 2 t dt 34. + 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
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
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
* 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.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 ) 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
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 * 23..5 8 (.)2 =.25 7 38. 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
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.99986 (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 = 4. 39. x e x (x > ) > *24 x *25 f () 29
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
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
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 6 6.. 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
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
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
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 3 + + 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
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 ) 2 + + f (n) () (x ) n + O((x ) n+ ). n! *3 Tylor 36
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 49. 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
(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 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 ). 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 ) 38
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. 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 ) 2. 6.4. 2 = 24 = 24 24 = 2 5 24 24 = 32 ( 2 ) 24 24 +... 52. e 53. sin 6 54. cos x x mc 2 (v/c) 2 (m >, c > ) v 39
(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 6.5. ( lim + n = e n n). Proof. ( n log + ) = n( n n ( ) 2 + ) 2 n = 2 2 n +. 6.6. 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 /2. 6.7. lim (( + n /n)n e) = Proof. log( + /n) n = n log( + /n) = 2 4 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 lim n n(( + /n)n e) = e 2 56. 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
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! 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
(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
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
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 2 + + x n ) = x n x x x = + x + x2 + + x n + xn x. + x = x + x2 + + ( ) n x n + ( ) n xn + x. *33 http://www.mth.ngoy-u.c.jp/ ymgmi/teching/complex/complex2.pdf (iii) http://www.mth.ngoy-u.c.jp/ ymgmi/teching/functionl/hilbert2.pdf 45
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 = 2 + 3 4 + 62. 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
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, 23. 47
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
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
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 < + 7.6 ( ). x > Γ(x) = + t x e t dt Γ(x + ) = xγ(x), Γ() =. Γ(n + ) = n!. 5
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
7.8. f(x) = { sin(/x) (x ) (x = ) b lim sin(/x) dx + 8 (sequence) { n } n n = + 2 + 3 + n= (series) * 36 n= n S = lim n k n k= * 37, n = + n= *36 sequence sequence series *37 52
(geometric series * 38 ) ( x)( + x + x 2 + + x n ) = x n+ x < n + x + x 2 + = x < < =. 2... ( 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
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 + 54
8.2. + 2 + 3 + lim n = n 8.3. n + 2 + 3 + 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 ( + 2 + + n ) log n n (Euler s constnt) 7. k=n (k + ) 2 k=n k(k + ) = n /2 < γ < γ =.5772... 55
8.4. { n } n lim 2n = = lim 2n+ n n lim n = n 8.5 (Leibniz). { n } n 2 + 3 4 +... 8.6. 2 + 3 4 + = log 2 (N. Merctor, 668). 3 + 5 7 + = π 4 (G.W. Leibniz, 682). 8.7. 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
8.8. {n, n 2,... } 8.9. n n nk = k= n. n= n < + n (bsolutely convergent) f(x) dx f(x) dx < * 4 8.. 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 http://www.mth.ngoy-u.c.jp/~ymgmi/teching/teching.html 57
(ii) x < log( + x) = x 2 x2 + 3 x3 4 x4 +, ( + x) = + x + ( ) x 2 + 2 ( )( 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
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
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, 2 + 3 4 + 6
+ p q n + 3 + + 2p 2 4 2q +... + 2(n )p + + + 2np 2(n )q + 2 2nq (6) + ( 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 4 + = log 2, + 3 2 + 5 + 7 4 + = 3 log 2. 2 2 + 3 4 + ( ) ( * 42 ) *42 Bernhrd Riemnn (826 866) (828 877) 6
73 (Chllenging). A A A A Wikipedi quntity vrible * 43 (i) (ii) (iii) (differentil eqution) *43 62
* 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
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.
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
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
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 + < 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
+ 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 2 2 3 x +... 68
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 + 2 3 y2 + 2 8 y3 +... g() = 2, g () = 2 3,... 74. y = ± 2 x2 3 x3 +... x x = 2y + 2 3 y2 + 2 8 y3 +... x = y + by 2 + cy 3 +... y 69