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

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1 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 ) i (2) n = n = α = f () = d f(x) f() f() = lim x x f(x) = f(x ) + α(x)(x x ), x = x α(x). (3) n 2 α i = h = (,,, h i,,, ) f(x + h) f(x) f(x) = f xi (x) = lim x i h i h i, f, g, u.. () d f(x)g(x) = f (x)g(x) + f(x)g (x) (2) d f(x) g(x) = f (x)g(x) f(x)g (x) f(x)g(x) (3) d f(u(x)) = u (x)f (u(x)) 2 ( ) y = f(x) I = (, b) ( ) x = f (y) d f (x) = f (y)

2 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 (y) y,. (f ) (y ) = α(f (y )) = f (x ) y = exp x, x R y = sin x, x [ π/2, π/2] y = cos x, x [, π] y = tn x, x ( π/2, π/2) 3 () d xn = nx n (3) d sin (x/) = sgn() 2 x 2 x = log y x = sin y x = cos y x = tn y sin x + cos x = π/2, 2 +x 2 (5) d tn (x/) = (7) d log x + A + x 2 = A+x 2 tn x + tn x = sgn(x)π/2 (2) d sin x = cos x (4) d cos (x/) = 2 x 2 (6) d log x = x.2, f(x), g(x). ()f(x) g(x) (2) rcsin(cos x)) (3) rctn ( ) tn x + tn x (4) log(tn x) (5)f(f(f(x))) (6) rcsin f 2 (x).3 () sin x cos x (2) sin x 2 (3) tn x +x 2 5 tn 5x x 2 (4) Arcsecx 2 2 f (x) = d f(x) = f () (x) f (n) (x) = d f (n ) (x) C n (I) = { I f (n) (x) f } n C (I) 2

3 4 () dn n sin x = n sin(x + nπ 2 ) (2) d cos x = n cos(x + nπ (3) dn n e x = n e x 5 Leibnitz d n f(x)g(x) = n ( n n k= k) f (n k) (x)g (k) (x) 2. n () ( x 2 ) (2) x+b cx+d (3) sin 2 x (4) x 3 sin x (5) e x log( + x) (6) sin x 2.2 n 2 ) (4) dn n log x = ( ) n (n )!x n () sin 3 x, (2) (x + )(x + b) ( b), (3)ex sin bx 2.3 H n (x). H n n, n. H n (x) = ( ) n e x2 /2 dn 2 n e x /2 2.4 P n (x). P n n, (, ) n. d n P n (x) = 2 n n! n (x2 ) n 2.5 f(x) C (R). { exp[ /x] x > f(x) = x 3 3 f(x) f(λ + ( λ)b) λf() + ( λ)f(b) λ [, ]] 6 f C 2 (I) f(x) f (x) f(x), λ i, λ i = f x i f( λ i x i ) λ i f(x i ) i. 3

4 3.3 f C (I) f(x) f( + b f() + f(b) ) 2 2. i ( n ) /n n ( i ), ( ) 2. µ i, µ i = exp( µ i x i ) e x i µ i, ( ) 3.5 f(x) I = [, b] f (x) >, f()f(c) <, x (, b) x n+ = x n f(x n) f (x n ), x n f(x) I c (, b) (Newton ) 4,, 7 f(x) [, b], (, b), ξ (, b) f() f(b) b = f (ξ) [ ] f(b) f() F (x) = f(x) (x ) + f() b, F () = F (b) =. F (x), F () = F (b), < ξ < b ξ F (ξ) = F (ξ) =. Q.E.D 8 f(x), g(x) [, b], (, b), f (x) g (x), ξ (, b) f() f(b) g(b) g() = f (ξ) g (ξ) [ ] f(b) f() F (x) = f(x) (g(x) g()) + f() g(b) g(), F () = F (b) =. F (ξ) =. Q.E.D. (, lim n ( )) 4

5 e x = lim( + x n )n = lim n n k= x = lim ( n x k ) n k= n ( x) 2 = ( lim x k ) = n k= n= log( x) = lim ( n x k ) = n k= x k k k! nx n s= x n /n n=. ( s n ) = 9 f(x) I = (, b) n, [, b] < x < b < ξ < x f(x) = f()+f ()(x )+ 2 f ()(x ) (n )! f (n ) ()(x ) n + n! f (n) (ξ)(x ) n R n = n! f (n) (ξ)(x ) n Lngrnge R n = ( θ)n p p(n )! f (n) ( + θ(x ))(x ) n, ξ = + θ(x ),.. () ϕ(t) = f(x) n= [ n ] k! f (k) (t)(x t) k + A (x t)n n! k= ϕ(x) =. A { n! A = (x ) n f(x) [ n k= ]} k! f (k) ()(x ) k, ϕ(x) = ϕ() =. ξ ϕ (ξ) =., ϕ (t) =, A = f (n ) (ξ). (2) (n )! f (n) (t)(x t) n A (x t)n (n )! n F (x) = f(x) k! f (k) ()(x ) k, G(x) = (n!) (x ) n k= F (k) () = G (k) () =, k =,,, n, F (x) F (x) F () = G(x) G(x) G() = F (ξ ) G (ξ ) = F (ξ ) F () G (ξ ) G () = F (ξ 2 ) G (ξ 2 ) = = F (n) (ξ n ) G (n) (ξ n ) = f (n) (ξ n ) F (x) = G(x)f (n) (ξ n ), (ξ = ξ n ). x n n! 5

6 (3) f(x) f() = x f (t)dt = = (x )f () x x (x t) f (t)dt = [(x t)f (t)] x + [ ] (x t)2 f (t)dt 2 = (x )f () + 2 (x )2 f () x x [ ] (x t)3 f (t)dt 3! (x t)f (t)dt n R n =, (n )! x (x t) n f (n) (t)dt 4 x, g i+ (x) = o(g i (x)). f(x) {g (x), g (x), } x = f(x) = g (x) + g (x) + g 2 (x) + + g n (x) + o(g n (x)) f(x) {, x, (x ) 2, } f(x) = + (x ) + 2 (x ) n (x ) n + o((x ) n ), k = b k, k =,,, n. = b + b (x ) + b 2 (x ) b n (x ) n + o((x ) n ) {, x, (x ) 2, } (sin, cos, R 2n+, R 2n cos(ξ). ) 4.. cos x = x 2 /2 + x 4 /4! + + ( )n cos ξ x 2n (2n)! sin x = x x 3 /6 + x 5 /5! + + ( )n cos ξ x 2n+, (2n + )! e x = + x + x 2 /2 + + eξ n! xn log( + x) = x x 2 /2 + + ( )n n( + ξ) n xn, ( ) ( + x) p = + px + p(p )x 2 p /2 + + ( + ξ) p n x n n 6

7 4.2 () f(x), g(x), g(x), b f(x)g(x) = f(c) b g(x) < c < b ). (2), Lgrnge 4.3 z/(e z ) z : z e z = z n b n n! n= b n. ( ) n n b k = k k= b =, b = /2, b 2n+ = 4.4 b n, B n = ( ) n b n.. 2 2n (2 2n )B n x 2n tn x = (2n)! n=, log cos x. 4.5 () Tn x = rctn x. (2) π/4 = 4 rctn(/5) rctn(/239) (. (3) 4. 5, Archimedes Newton, Leibnitz, 5 Riemnn = { = x < x < < x n = b}, {ξ} = {ξ; x i ξ i x i+ } S(, {ξ}) = (x i+ x i )f(ξ i ) I i = [x i, x i+ ], M i = mx ζ Ii f(ζ), m i = min ζ Ii f(ζ), S mx ( ) = (x i+ x i )M i, S min ( ) = (x i+ x i )m i 7

8 2 () S min ( ) S mx ( ) (2) I = [, b], ( ) S min ( ) S min ( ) S mx ( ) S mx ( ) (3) Σ U = {S mx ( )}, Σ L = {S min ( )} Σ L Σ U. 6 Σ U = {S mx ( )}, Σ L = {S min ( )}, b f(x) = inf Σ U b f(x) = sup Σ L, 3 () I = [, b], f(x) (), (2) (2) f(x). d x f(t)dt = f(x) 5. lim lim n () lim n n k= + k/n [ (2) lim n n exp log(n + k) n k= ] 5.. sin, cos, e x 2. f(g(x))g (x) = f(g)dg 3. f (x)g(x) = f(x)g(x) f(x)g (x) 8

9 ., x = Tn (x/) = Sin (x/) 2 x x 2 = log(x x 2 ) I = 2 + x 2 = x 2 + x 2 (x ) x 2 = x 2 + x 2 I + 2 log(x + x ) I = (/2)(x 2 + x log(x + x ). 5.. x ( ), tn, log (7, ). P (x) = x n + n x n + + {α i }, {β i }, (p i, q j ). P (x) = l m (x α i ) p i (x 2 2Reβ j x + β j 2 ) q i i= j= (x α)(x β) = ( α β x α ) x β P (x) = i ( pi ) C (k) i (x α i ) k + j k= ( qj k= A (k) j x + B (k) ) j (x 2 2Reβ j x + β j 2 ) k (A, B, C ) Q(x)/P (x) Q P ) A, B, C,, x. x 4 + = Ax + B x 2 + 2x + + Cx + D x 2 2x + (x ) x, A+C =, x = B+D =, x = i, (A C) i(b D) = 2, B = D = /2, A = C = 2 2 9

10 Q(x)/P (x), log, tn (Leibnitz ): Ax + B (A/2)(2x + 2p) + (B pa) (x 2 + 2px + q) n = (x 2 + 2px + q) n A = 2( n + ) (x2 + 2px + q) n+ + (B pa) (x 2 + 2px + q) n I n = (x ) n = x (x ) n + 2nx 2 (x ) n+ x = (x ) n + 2nI n 2n 2 I n+ R(X, Y ) X, Y, () R(cos x, sin x), tn(x/2) = t t (2) R(x, n n (x + b)/(cx + d), (x + b)/(cx + d) = t t (3) R(x, x x + bx + c), > x x + bx + c = t x t 3, b, c,, x x = tn θ, x 2 2 x = sec θ, 2 x 2 x = sin θ sin, cos, t = tn(θ/2), I = = log(x + + x 2 ) + x 2 + x 2 = t x x = tn θ, = (cos 2 θ) dθ dθ I = cos θ = dθ sin(θ + π/2) = dθ/2 cos 2 = log tn(θ/2 + π/4) (θ/2 + π/4) tn(θ/2 + π/4) tn(θ/2 + π/4) = + tn(θ/2) tn(θ/2) = tn θ + cos θ = x + + x I() = π log( cos x) = mx{, π log( 2 )} > ) d π [ d I() = 2 + cos x 2π ] cos x = π + ( ) cos x 2π [ ] [ ] = 2π + ( ) 2 = π = 2π, I() = π log( 2 ) + C. C = I() = π log 2.

11 I() = I( ) 2 cos 2 θ = + cos 2θ I() = 2 = 2 π π log(( + 2 ) 4 2 cos 2 x) = 2 π log( cos x) = 2 I(2 ) log( cos 2x) I() = 2 I(2 ) = = 2 n I( 2n ) < lim n n = I() =, > 2 n log( + 2 2n + 2 2n cos x) 2 n log( 2 2n ) = log( 2 ), x n = n i= x + exp[2πik n ] [ = exp ( log + x 2 + 2x cos(2πk/n) )] [ lim n xn /n = exp 2π 2π ] log( + x 2 2x cos θ)dθ x <, x > x, π/2 log sin x = π log 2 2,. 5.2 π + cos x = π x y Fubini. ( b ) d ( d ) b f(x, y)dy = f(x, y) dy c c., f(x, y)., f(x, y) f(x, y ) < ε (x, y) (x, y ) δ ( ).

12 5.2 f(x) [, ), lim x f(x) < α b f (αx) f(x) f(bx) = [f( ) f()] log(b/) x. x b x ( ) b b ( log x = x y dy = ) b x y dy = + b dy = log + y [ e x2 = e x2 y 2 dy] /2 = ( π ) /2 e r2 2rdr = π /2, dy rdrdθ, r θ. exp[ x 2 y 2 ]. 5.4 sin x x = lim R [ R e xt sin xdt] = lim R [ R R = lim R + t 2 dt = [tn ] = π/2 e xt sin xdt], exp[ xt] sin x. R e xt sin x = [ ] R cos x + t sin x + t 2 e xt = + t 2 O()e Rt 6, n k= k! f (k) ()(x ) k = f(x) R n+ (x) 4 lim n R n = lim n n k= s n = n k= k. k! f (k) ()(x ) k = f(x) 2

13 7 s n = n k= k. lim s n = s n s.. n k = s k= 5 () n. lim p+ + p q = p,q (2) n lim n = 6 () n, n. (2) n b n. b n n, n b n. 8 k= k. n k= k. 7, ,.. () (2) n n log p (n + ), p > (3) ( ) n n n= n= n= (4) ( ) n f(n), f(x) >, lim x f(x) = (5) n= n= n, n n ( = 2, 2 = 3, 3 = 5, ) 6.3 n > n.. () n + n (2) n + n 2 n 6.4 n > n.,.. () n + 2 n (2) n + n n (2) n n () n >, n, n, lim n n = (2) n n, n. 3

14 n= nx n (, ). p = lim n n+, p 2 = lim n n /n, n, p = p 2. p, ρ = p.. p = lim n sup{ k /k ; k > n} ( ), ρ = /p. 6.6 p, p 2, p 8 x < ρ n x n. x > ρ n x n. x < ρ n nx n. 9 ρ. x < ρ d n x n = d nx n = n nx n n= () n! x n (2) (n!) 2 (2n)! xn (3) ( n k= ( + k ) ) x n 6.9 (4) n!x n! (5) log(n!)x n (6) (n!) /n x n ( > ) ( )n = log 2 + R n, R n = O(/n) n, p, q, ( 2p ) ( + 2q 2p ) 4p 6. n >, < x < 2π. 6. () n cos nx (2) n sin nx n sin nx + n ( ) n= x x. n 2 4

15 7 7. f(x) f (n ) (x) [, b], (, b), x (, b) f(x) = f()+f ()(x )+ 2! f (2) ()(x ) (n )! f (n ) ()(x ) n + n! f (n) (ξ)(x ) n ξ (, b) () log(cos x) (2) tn x (3) e x x (4) e x sin x 7.3 () log(x + + x 2 ) (2) log( + x + x 2 ) (3) rcsin x (4) sin 2 x (5) cosh x (6) rctn x 7.4 () π + cos x (2) + x ( )n = log 2 + R n, R n = O(/n) n, p, q, p ( q ) + ( 2p p ) 7.6 I = [, ] < r < (r) = {r n, r n,, r = r, r = } n r, lim r r n = n = ( r) 2 r = / n r n S r = n k= rpk (r k r k+ ), lim r S(r). 7.7 k b k ( k p) /p ( bk q) /q < p <, p + q = 5

() 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. () 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 >