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

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1 I Riemnn 9 5 Tylor f(x) x = A = h f( + h) f() h A (differentil coefficient) f f ()

2 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 = f () = f x = f () < f x = f () =f (x) x = f x =. ( ) = x x 2 o(h) =f( + h) f() Ah f(x) =f() +A(x ) +o(x ) o(h) h h o(h) h h = 2

3 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) h h = (f(x)g(x)) = f (x)g(x)+f(x)g (x). (). (i) (ii) (iii) (f(x)g(x)) = f (x)g(x)+f(x)g (x). {f(g(x))} = f (g(x))g (x). ( ) = f (x) f(x) f(x). 2 =, x =, (x ) = x 2. 3

4 x x+h x h h = x h h h A x A y = x x = A A + A = > e Npier (e x ) = e x e log x e x (log x) =/x (x >).2. (i) > y = x x = e x log = e log d dx x = e x log log = x log (ii) > x x x = e log x d dx x = x (x > ) 2. > y = x y = x 3. x< (log( x)) = x y = x α log y = α log x x y y = α x 4

5 y h x y = αx α (e x ) = e x, (log x ) = x (x ), (xα ) = αx α (x>)..3. ( log x + x 2 ) + = x x sin(x + h) sin x h h = cos h h sin x + sin h h cos h sin h, h h h cos x cos x x = y =cosx x = y = (sin x x tn x (sin x) =cosx (cos x) = sin x (sin x) =cosx, (cos x) = sin x, (tn x) = 5. tn x 5 (cos x) 2.

6 6. 2 log x, x n, e x. (the rpidity of increse) x n e x x n e x (>) f(x) =x n e x,x f(x) f (x) =x n e x (n x) x n/ f (x) + f(x) x = n/ y C n= x x + f(x) = C> f(x) C C> f(x) x n e x x n e x x n e x/2 e x/2 6

7 =/2 x n e x/2 C C> x n e x Ce x/2 x + e x/2 = x + xn e x = log x x n t =logx log x x + x n = t + te nt = x + t + log x<<x n << e x (x + ) 7. >, n> x n << e x (x + ) Remrk. x n << e x x x n <e x x n =4, = x =2 2 n =6> 3 2 >e log n<<n n n/n log n n log(n/n ) = n n = n n/n = 7

8 8. x + xx 3 (inverse function) y = f(x) y = f(x) y + k y x x + h x = g(y) 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, rctnx. sin x sin x sin x (sin x) 8

9 9. rccos x +rcsinx = π ( x π/2) 2 (rcsin x) =, (rctn x 2 x) = +x. 2 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). 4 Riemnn [, b] f(x) = x <x < x 2 < <x n = b ξ j [x j,xi j ] n (x j x j )f(ξ j ), =mx{x x,,x n x n } j= ξ j f(x) [, b] (integrble) (integrl) b f(x) dx f (integrnd) 4. (). 9

10 b x (i) b f(x) dx + c f(x) dx = c b f(x) dx. (ii) b b f(x) dx f(x) dx ( <b). Remrk. x. b f(x) dx = log x = b f(y) dy =. x t dt log(xy) =logx +logy (). f(x) [, b] d x f(t) dt = f(x). dx

11 x x + h 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) () f(x) f(x) F (x) b f(x)dx =[F (x)] b = F (b) F () 2. f(x) f() = x f (t) dt f(x) (<x<b)

12 (i) f (x) (<x<b) f (, b) (incresing) (ii) f (x) > (<x<b) f (, b) (strictly incresing) 2 x dx =rcsinx 2, x 2 + dx = 2 rctn x x 2 + dx = π 4. (integrtion by prts) f(x)g(x) = f (x)g(x)dx + f(x)g (x)dx. (chnge of vribles) 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. y = g(x), g (x) =dy/dx y f(y) dy dx dx = f(y) dy 2

13 4.7. log xdx log x x log x (x log x) =logx +. x log x = log xdx+ dx log xdx= x log x x 4.8. n =2, 3,... x (x ) dx = n 2 2n 3. n = (x ). n 4. x 2 x 2 dx, xe x2 dx 4.9. I n (x) = ( x (x ) n ) = (x ) n dx (x ) n 2nx (x ) n+ = 2n (x ) n n (x ) n+ 3

14 2 2 ni n+ (x) (2n )I n (x) = x (x ) n (recursive reltion) I (x) = x 2 + dx = 2 rctn x I 2 (x),i 3 (x), n =, 2,... x n e x dx (i) dx =rcsinx. 4x x (ii) x 2 dx = ( x 2 2 x rcsin x ). (iii) x2 + A dx =log(x + x 2 + A). g(x) f(x) dx, 4 deg g<deg f

15 f(x) (x 2 + x + b) m, (x + c) n g(x) f(x) = p(x) (x 2 + x + b) + q(x) m (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 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 dx 4.. x 3 + dx 5

16 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 (x /2) 2 +3/4 dx = 3 log(x +) 3 (x /2) 2 +3/4 d(x /2)2 + 3 (x /2) 2 +3/4 dx = 3 log(x +) 3 log(x2 x +)+2 3 rctn(2x/ 3 3) x 3 dx, x 4 + dx x 4 dx y = f(x) x = ϕ(t), y = ψ(t) t x = ϕ(t) f(x) dx = ydx= ψ(t)ϕ (t) dt ϕ, ψ t x 2 +y 2 = (, ) y = t(x +), x 2 + y 2 = 6

17 x = t2 +t 2, y = 2t +t 2 dx = x 2 y dx t x =cosθ, y =sinθ t y y = t(x +) x y = x 2 y = t(x +), y 2 = x 2 x = +t2 t 2, y = 2t t 2 t 8. x2 dx 7

18 5 Tylor x x log( + x) = ( x)( + x + x x n )= x n+ x =+x + x2 + + x n + xn+ x. +x = x + x2 + +( ) n x n n+ xn+ +( ) +x. x dt ( t + t 2 +( ) n t n n+ tn+ +( ) +t ) = x 2 x2 + 3 x3 + +( ) n n + xn+ +( ) n+ x ( )n+ x < x +t dt tn+ dt t n+ = n +2 xn+2. log( + x) =x 2 x2 + 3 x3 4 x4 +. x x (power series) 5.. x dt tn+ +t. log(.) =. 2 (.)2 +r =..5+r =.995+r, <r 3 (.)3 log x x f(x) =c + c (x )+c 2 (x ) 2 + 8

19 n x = c n = f (n) ()/n! f(x) =f()+f ()(x )+ 2 f ()(x ) 2 + (C. Mclurin) b f(b) =f()+ f (t) dt f(b) =f()+f ()(b )+ b f (t) dt b b (f (t)(b t)) = f (t)+f (t)(b t) f (t)dt = f ()(b )+ f(b) =f()+f ()(b )+ b b f (t)(b t)dt f (t)(b t)dt. ( f (t)(b t) 2) = f (t)(b t)+ 2 2 f (t)(b t) 2 f (t)(b t)dt = 2 f ()(b ) b f(b) =f()+f ()(b )+ 2 f ()(b ) f (t)(b t) 2 dt b f (t)(b t) 2 dt. f(b) =f()+f ()(b )+ + (n )! f (n ) ()(b ) n + (n )! 9 b f (n) (t)(b t) n dt

20 ( f (n) (t)(b t) n) = n! (n )! f (n) (b t) n + n! f (n+) (t)(b t) n (n )! b f (n) (t)(b t) n dt = n! f (n) ()(b ) n + n! b f (n+) (t)(b t) n dt 5.2 (Johnn Bernoulli ). f(x) f(b) =f()+f ()(b )+ + n! f (n) ()(b ) n + n! b f (n+) (t)(b t) n dt (reminder) f(x) n n! x f (n+) (t)(x t) n dt = Tylor-Mclurin Tylor, Tylor expnsion Remrk. Johnn Bernoulli B. Tylor Newton Tylor C. Mclulin 2

21 9. g 5.3. d n (n )! dx n x (i) e x =+x + 2 x2 + + xn +. n! g(t)(x t) n dt = g(x) (ii) sin x = x 3! x3 + 5! x5. (iii) cos x = 2 x2 + 4! x4. (iv) ( + x) α =+αx + α(α ) x 2 + ( x < ). 2 Proof. (iv) 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 ( ) n x t x ( + t) dt α +t = = x x (x t) n dt t n dt = xn+ n +. ( ) n x t ( t) α dt t sup{( x t)/( t); t x } = x x ( ) n x t x ( t) α dt x n ( t) α dt t α(α ) (α n) x n n! 2

22 n α 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 n l n (l )! α(α ) (α l +). (l )! α < l α< l + l (n +)(n +2) (n + l) α α n α (l )! l l + n + l n α(α ) (α n) n! (n +)(n +2) (n + l) (l )! n x n x < x = e Remrk. 2. e = 24 = = ( 24 =32 ) Newton 22

23 5.5. log( + sin x) x log( + y) =y y2 2 + y y =sinx = x x3 3! +... x ) log( + sin x) = (x x = x ( 2 x (x x ) x ) 2 + ) 3 (x x cos x x 4 m v v / cos x x tn x x x f(x) = = x g(x) f(x), g(x) f(x) x g(x) = f(x) g(x) (infinitesiml of higher order) f(x) =o(g(x)) Lndu log( + x) =x 2 x2 + +( ) n n xn +( ) n x x < x t n +t dt x x 23 t n dt = n + x x n+ t n +t dt

24 ( ) n x dt tn /( + t) = x x n log( + x) x + 2 x2 ( ) n n xn = o(x n ) log( + x) =x 2 x2 + +( ) n n xn + o(x n ) e x = +x + 2 x2 + + n! xn + o(x n ) sin x = x 3! x3 + +( ) n (2n +)! x2n+ + o(x 2n+ ) cos x = 2 x2 + +( ) n (2n)! x2n + o(x 2n ) ( + x) α = +αx + α(α ) x 2 + o(x 2 ) cos x x x sin x Proof. cos x x sin x = ( x2 /2+ ) = x2 /2 x 4 /4! + x(x x 3 /3! + ) x 2 x 4 /3! ( + n = e n n). 24

25 Proof. ( n log + ) = n( n n ( ) 2 + ) 2 n = n sin x (cos x)/x x Proof. cos x = 2 x2 +, xsin x = x 2 3! x4 + (cos x x)/x sin x = ( 2 ) /x 2 x2 x = e / (( + 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

26 ( + /n) n e e/2n n n(( + /n)n e) = e Npier e ( e = + ) n n n e n 25. x + rctn x = π/2 π 2 rctn x /x /x 6 x /2 dx, x 2 dx improper integrls 26

27 6.. x α dx = { α if <α<, + otherwise. x α dx = { α if α>, + otherwise. 26. α> e αx dx 6.2. log xdx y =logx x = log x x log x x log xdx=[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) 27

28 27. x 2 dx = [ x ] x= x= = 2. g(x) g(x) 6.3 (). (i) f(x) g(x), x g(x)dx < + f(x) dx (ii) f(x) g(x), x > + g(x)dx < + f(x) dx 28

29 6.4. 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, 6.5. I = + I n = n! e x dx = e x2 dx Proof. x e x2 e x e x2 dx = e x2 dx + e x2 dx + e x2 dx e x dx < (Γ ). x> Γ(x) = + t x e t dt Γ(x +)=xγ(x), Γ() =. Γ(n +)=n!. 29

30 Proof. dt = e x2 dx = 2 Γ ( 2 π/2 IIΓ(/2) = π dt ). 28. ( π 2 rctn x ) dx 7 f () f(x) x = f () = f(x) =f()+ 2 f ()(x ) 2 f () f () x = x = (locl mximum, locl minimum) f () f () 3

31 29. x e x y = f(x) x f (x) f (x) x f(x) f, b f(( t) + tb) ( t)f()+tf(b), t y = f(x) b 7.. f(x) [, b] f (x) f (x) ( x b) 3

32 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 ) j= (Jensen ) Proof. Tylor j= f(x) =f(c)+f (c)(x c)+ f (t) x f(x) f(c)+f (c)(x c) c f (t)(x t) dt c, 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) t j (c j c) =f(c) j j 3. f(x) f(c)+f (c)(x c) 7.2. p = {p j } j n, q = {q j } j n p j >, q j > (reltive entropy) n H(p, q) = p j log p j q j log x H(p, q) = ( n ) p j log q j q j log p j =log= p j p j j= 32 j=

33 f(x) f (x) f (x) = c f (x) { f(x) < if x<c, f(x) > if x>c f(x) x <cx >c x = c (point of inflection) 7.3. f(x) f (c) = x = c f (x) (c, f(c)) 7.4. f(x) =x 3 x = 7.5. f(x) =e x2 /2σ 2 f (x) =e x2 /2σ 2 x2 σ 2 x <σ x >σx = ±σ 3. y = f(x) f (c) = f (c) x = c σ 4 33

34 8 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 ) t t <u<+ t t + <ɛ< + g(u)du t t /2 e t Γ(t) +ɛ ɛ ɛ t ɛ t e t(u )2 /2 du = ɛ t t e x2 /2 dx + n! 2πn n+/2 e n 34 ɛ t e x2 /2 dx e x2 /2 dx = 2π

35 Stirling +ɛ Γ(t +) t t+ e t e t(x )2 /2 dx t u t e t tu du = e t(x log(+x)) dx { x log( + x) if x, y = x log( + x) if <x x y, y x x + + e t(x log(+x)) dx ty2 dx = e dy dy ɛ y 2 = x log( + x) x dy dx = x 2( + x) y = x 2( + x) x log( + x) > dx y dy = 2( + x) x log( + x) x x y + dx y dy = x + 2( + x) x =2 =, t + g(y) = dx dy z = ty + u t e t tu du = + ( ) z e z2 g dz t t 35

36 t + t + u t e t tu du = t + + ( ) z e z2 g dz = t + g(y) = dx dy x < e z2 g() dz = πg() y = ± x (x x 2 /2+x 3 /3...)= x x +... g() = == () dy dx 2( + x) 2 3 x +... x= = 2 ( ) z g = g() + g () t z + 2 t g () z2 t +... e z2 zdz=, e z2 z 2 dz = π 2 t u t e t tu du = ( π g() + g () 4 ) t +... (symptotic expnsion) x = 2y y2 + 8 y g() = 2, g () = 2 3,... 36

37 32. x y = ± 2 x2 3 x x = 2y y y x = y + by 2 + cy y 37

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