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1 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:, S(f), ε >, δ >, < δ [,b] S,ξ (f), S(f) S,ξ (f) < ε, S(f) f [,b], S(f) = b f(x)x 42, [,b] f x,, x i x i, f(ξ i ) f(ξ i )(x i x i ),, S,,,,, x i x i x i, n S,ξ (f) = f(ξ i ) x i i=,, x x, S,ξ (f) = n f(ξ i ) x i b i= f(x)x = S(f), x, x, 43 [,b] f: R Version:, My 8, 22

2 22 I [,b] f: R, [,b] = {x,,x n }, s (f) = S (f) = n m i (x i x i ), m i = inf{f(x) : x [x i,x i ]}, i= n M i (x i x i ), M i = sup{f(x) : x [x i,x i ]} i=, {s (f)}, {S (f)}, sups (f) inf S (f) m(b ) s (f) S (f) M(b ), m = inff(x), M = supf(x), {s (f)}, {S (f)},,, s (f) S (f),, s (f) s (f) S (f) S (f), sups (f) inf S (f),, S(f) = sups (f), S(f) = inf S (f) 45 [,b] f: R, f [,b] 2 S(f) = S(f) 3 ε >, S (f) s (f) < ε 45, f, ε > S (f) s (f) < ε, M i = mx f(x), m i = min f(x), [x i,x i+ ] [x i,x i+ ] S (f) s (f) = (M i m i )(x i+ x i ), f, ε >, δ >, x x < δ f(x) f(x ) < ε, < δ, S (f) s (f) = (M i m i )(x i+ x i ) ε(b ), f 46 [,b] f, g, Version:, My 8, 22

3 22 I 4-3 b f(x) = k, kx = k(b ) b 2 f +g, (f(x)+g(x))x = b f(x)x+ 3 k R, kf, kf(x)x = k 4 fg b b g(x)x b f(x)x 5 x [,b] f(x), f(x)x b b b 6 f, f(x)x b 7 c [,b], f(x)x = b, f(x)x = b c f(x) x f(x)x+ b c f(x)x f(x)x,, b, c f(x) = k, s (f) = S (f) = k(b ) S(f) = S(f) = k(b ) = S(f), 2, ( 4) S(f +g) = S(f)+S(g), S(f +g) = S(f)+S(g),, S(f +g) = S(f)+S(g) 3 4,, f f 2, = {x,,x n }, S (f 2 ) s (f 2 ) C(S (f) s (f)) fg = 4 ( f +g 2 f g 2 ), 2, 6, fg 5, = {x,,x n }, f(ξ i )(x i x i ) (ξ i [x i,x i ]), S (f), s (f), S(f), S(f) n n 6, f(ξ i )(x i x i ) f(ξ i ) (x i x i ) i= i=, S(f) S( f ), S(f) S( f ), f f, 5, I j, sup f(x) inf f(x) supf(x) inff(x) I j I j I j I j, S ( f ) s ( f ) S (f) s (f) f 7,, c [,b] 7 Version:, My 8, 22

4 22 I [,] f n lim f(k/n) = n n k= f(x)x [,] = {,/n,2/n,,(n )/n,} ξ = {,/n,,(n n )/n}, f(k/n) S,ξ (f) n k= n, n lim f(k/n) f [,b] n n k= n n 48 lim n n 2 +k = 2 k= +x x, f(x) = 2 +x 2, f(k/n) = +(k/n) = n2 2 n 2 +k 2, n n 2 n n 2 +k = n f(k/n) 2 n 49 f [,b] [,b], f,, { x Q, f(x) = x R\Q, [,], inf S =, sup k= s = 4 ( ) [,b] f: R, b b ξ [,b] f(x)x = f(ξ) m = inf{f(x) : x [,b]}, M = sup{f(x) : x [,b]}, m f(x) M, m b f(x)x M b, f [,b], µ [m,m], f(ξ) = µ ξ [,b], µ = b b f(x)x = µ = f(ξ) b b k= f(x)x, Version:, My 8, 22

5 22 I R f: X, f(+h) f() lim h h,, f (), x f(), f x (), f,, x, f (x) f, f, f 42, h = (+h), x, f(+h) f() f x = f = f(+h) f(), f lim h x f x 43 f x =, x =, x = f x =, ε >, δ >, h < δ f(+h) f() < h(ε+ f () ), f x =, x = f(x) = x 44 f, g x =, f+g x =, x f()+ x g() x (f+g)() = 2 k, f x =, kf x =, x (kf)() = k x f() 3 f, g x =, fg x =, ( ) ( ) x (fg)() = x f() g()+f() x g() 4 ( f, ) g x = (, g(), f/g x =, f () = g() x g (g()) 2 x f() f() ) x g() Version:, My 8, 22

6 22 I f x =, g x = f(), g f x =, (g f)() = x x g(f()) x f() 6 f x =, f x =, f ( ) y = f() = b, x f (b) = x f() 4, g = /f, /g(+h) /g() h = g() g(+h) g(+h)g() h (g()) 2 x g() 3 5 f x =, f(+h) = f()+hf ()+ε h, h ε, f x = g(b+k) = g(b)+kg (b)+ε 2 k, b = f(), b+k = f(+h), g(f(+h)) g(f()) = (f(+h) f())(g (f())+ε 2 ) g f(+h) g f() h = g(f(+h)) g(f()) f(+h) f() f(+h) f() h = (g (f())+ε 2 )(f ()+ε ) g (f())f () 6, f f(x) = x, 5 45 y = f(x), x = f (y), = y x x y y x xy, x y = ( ) y x,, y = f(x), z = g(y), z x = z y y x z x = z y yx Version:, My 8, 22

7 22 I F 44, f F x f F F, 44, 47 f f, f x =, f 2 (), 2 f x 2f(), x 2(), x f (x) f,,n, f (n) n (), n f x nf(), x n() f n, f (n), f C n, f, f, C 48 (Leibniz rule) f, g n, n ( ) n (fg) (n) (x) = f (k) (x)g (n k) (x) k k= (Rolle ) [,b] f, (,b), f() = f(b) =, ξ (,b), f (ξ) = f, f [,b], M, m, f() < M m < f(), f(b) = f() < M, M = f(ξ) ξ (,b), f (ξ) =, h, ξ, f(ξ +h) f(ξ), f(ξ +h) f(ξ), h >, h f(ξ +h) f(ξ), h <, h, f (ξ) = 42 ( ) [,b] f, (,b), ξ (,b), f(b) f() b = f (ξ) Version:, My 8, 22

8 22 I 4-8 k = f(b) f(), F(x) = f(b) f(x) k(b x), F(b) = b F() =, F (x) = f (x)+k, F Rolle, ξ F (ξ) = k = f (ξ) 42 [,b] f, c (,b) f (c) >, f (,b), c (,b) f (c), f (,b) ( 42) x, x 2 (,b), ξ (x,x 2 ), f(x 2 ) f(x ) = f (ξ)(x 2 x ) f (ξ) > x 2 > x f(x 2 ) > f(x ) 422 (Cuchy ) [,b] f, g (,b), ξ (,b), f(b) f() g(b) g() = f (ξ) g (ξ) k = f(b) f(), F(x) = f(b) f(x) k(g(b) g(x)) g(b) g(), F Rolle 423 (e l Hopitl ) f, g x =, lim lim x g(x) =, f (x) lim x g (x) = α lim f(x) x g(x) = α x f(x) = Cuchy ( 422), f, g x = f() = g() =,, f() =, g() =, x = f, g, b, b <, ξ (,b), < b, ξ (b,), f(b) g(b) = f(b) f() g(b) g() = f (ξ) g (ξ) f(b), b ξ, lim b g(b) = lim ξ f (ξ) g (ξ) Version:, My 8, 22

9 22 I ( ) f [,b], F(x) = x f(t)t, F [,b] 2 f [,b], F [,b], F(x) = f(x) x, F f, f, x [,b] f(x) M, x, y [,b], x < y x y y F(x) F(y) = f(t)t f(t)t = y f(t)t f(t) t M(y x) x y, x, y [,b] F(x) F(y) M x y, F [,b], ( 4) f, x, y [,b] (x < y), ξ (,b), x x F(y) F(x) y x = y x y x f(t)t = f(ξ) y < x, x, y [,b], x y, ξ (,b), F(y) F(x) = f(ξ) y x, y x, f f(ξ) f(x), F(x) = f(x) x, 425 [,b] f, f F, F f 426, [,b] f, F(x) = x f(t)t f,, f F, C G(x) = F(x) + C, G = F = f, G f, G f x, (G(x) F(x)) = x, G(x) = F(x)+ t = F(x)+C = Version:, My 8, 22

10 22 I 4- x f(t)t+c, f ( ) f(x) =, 427 f f, f(x)x 428 f [,b], F f, c, [,b], c f(x)x = F() F(c) =: [F(x)] c 426, F f F(x) = f(t)t+c, ( ) ( c ) F() F(c) = f(t)t+c f(t)t+c = x c f(t)t 429 f [,b], c, [,b], x f(t)t = f(x) f() t f x f, C, x C = 2 n N x xn = nx n 3 sin(x) = cos(x) x 4 x ex = e x 5 x log x = x 6 x Arcsin(x) = x 2 Version:, My 8, 22

11 22 I 4-43 f, x Dom(f) f(x) >, g(x) = logf(x),, x g(x) = x logf(x) = f(x) x f(x) α R, x > f(x) = x α, x (xα ) = αx α x (logxα ) = α x logx = α x, x (xα ) = (x α ) x (logxα ) = αx α 2 x > x x x xx = x x (+logx) 432 (f(x)g(x)) = f(x) x ( x g(x) ) + ( ) x f(x) g(x), x ( ) x ( ) f(t) t g(t) t = [f(t)g(t)] x t f(t) g(t) t, ( ) ( ) f(t) t g(t) t = f(t)g(t) t f(t) g(t) t,, logxx = (x) logxx = xlogx x(logx) x = xlogx x+c 433 ( ) f [,b], f, g f([,b]), G(x) = x α f(b) f() b g(x)x = g(x)x = x, g(f(x))f (x)x, g(f(x))f (x)x, g(t)t, G (x) = g(x), G(f(x)) x G(f(x)) = G (f(x))f (x) = g(f(x))f (x) Version:, My 8, 22

12 22 I 4-2, b g(f(x))f (x)x = G(f(b)) G(f()) = f(b) f() g(x) x g 434, x = g y yx x, g g x x = y yx x 435, x 2 + x = tn(t), x t = cos 2 (t), x 2 + x = x tn 2 (t)+ t t = t = t+c = Arctn(x)+C 2 x 2 x = sin(t), x t = cos(t), x2 x = sin 2 (t)cos(t)t = cos 2 (t)t = 2 (t+cos(t)sin(t))+c = (Arcsin(x)+x ) x 2 2 +C, x2 x /4, S S = 4 x2 x = 2[ Arcsin(x)+x x 2 ] = π 3 x > x x 2 () t = x, x x 2 x = t 2 t = Arcsin(t)+C = Arcsin( x )+C (b) t = x 2, x x 2 x = t t 2 + = Arctn(t)+C = Arctn( x 2 )+C Version:, My 8, 22

13 22 I 4-3 (c) t = / x 2, x x 2 x = +t t = Arctn(t)+C = Arctn(/ x 2 )+C 2 x 436 +x, Arctn(x), 2 x +x = 2 [Arctn(x)] = π/2,, (Arctn(/x)) = /(+x 2 x ), +x = 2 [Arctn(/x)] = π/2,, Arctn(/x) x = x 4 3x 2 +6, R R(x) = ( x 6 5x 4 +5x ) 2 +4 Mthemtic x(x 2 3) (Version 7), Arctn x 2 2 x = ± 2 Arctn((x 5 3x 3 +x)/2)+arctn(x 3 )+ Arctn(x) , F,,,, F [,b] f, F(x) = x f(t)t f, f F, F F F, F, F F, f F F 439 P(x), Q(x) R(x) = P(x) Q(x), P(x) Q(x), P(x) Q(x), P(x) = Q(x)q(x)+r(x), r(x) Q(x), 44,, P(x), P(x) = A(x α ) i (x α k ) i k (x 2 +β x+γ ) j (x 2 +β l x+γ l ) j l, A, α i, β i, γ i R, βi 2 4γ i < Version:, My 8, 22

14 22 I 4-4, 44 ( ) n, n 44, 442 ( ) Q(x) α k ) i k, (x 2 + β l x + γ l ) j l, R(x) = P(x) Q(x), Bx+C 44 (x A (x α k ) ( n i k) n (x 2 +β l x+γ l ) ( m j l), m P(x) x Q(x) x, x x x n, (x 2 +) n, (x 2 +) x n, x 443, x, l = (x 2 +) n log(x2 +), l > 2( n)(x 2 +) n x (x 2 +) n, ( 46 ) 444, R(x) = x (x ) 2 R(x) = x + (x ) 2, x R(x)x = x + x (x ) = log x + 2 x +C 2 R(x) = x 2 +3x+ (x+)(x ) 2 (x 2 +)(x 2 +x+) 2 R(x) = 8x x (x ) 3 x 2 4x x (x+ 2 ) x 9 (x+ 4 2 ) x ( ) 3 4 (x+ 2 ) , 445 ( ) R(x,y) R(cos(x), sin(x)), t = tn( x 2 ), t Version:, My 8, 22

15 22 I 4-5 tn 2 (x) + = /cos 2 (x), t = tn(x/2), t x = 2cos 2 (x/2) = t2 +, cos(x) = t2 2 +t2, sin(x) = 2t +t 2, ( ) t 2 2t x R(cos(x), sin(x)) x = R +t 2, +t 2 t t ( ) t 2 2t 2 = R +t 2, +t 2 +t t 2, t 446 R(cos(x),sin(x)), 445, 445,, f(x) = x sin(x)+bcos(x) = sin(x)+bcos(x) 445, 2 2t+b( t 2 ) t = 2 +b log bt + 2 +b 2 2 bt 2 +b 2 +C, sin(x)+bcos(x) = Asin(x+α), t = x sin(x)+bcos(x) = t A sin(t) = u A u = 2 2A log u +u ( ) x+b 447 R(x,y), R n cx+ x+b, bc, t = n cx+, t 448, x 2n e x2 x, n = n N n =,, π = e x2 x, R Version:, My 8, 22

16 22 I 4-6 sin(x)+bcos(x)x, 2 sin 2 (x)+bcos 2 (x)x, x, sin(x)+bcos(x) x sin 2 (x)+bcos 2 (x), ( ) f (,b], [,b), (,], [, ), b b f(x)x = lim α + f(x)x = lim α α f(x)x, α f(x)x, b f(x)x = lim β b f(x)x = lim β α α f(x)x, f(x)x,, 45 α R f(x) = x α [, ), α < 2 α R f(x) = x α (,], α > 45 ( ) f (,b), (, ),,, η (,b) η R, b η f(x)x = lim α + f(x)x = lim α α η β f(x)x+ lim β b α η f(x)x, β f(x)x+ lim f(x)x, β η, 452 f(x) = x 2 (,), x = lim x 2 α + α x β + lim x 2 β x x 2 = lim Arcsin(β) lim Arcsin(α) = π β α + 2 ( π 2 ) = π Version:, My 8, 22

17 22 I f(x) = +x 2 R, x +x 2 = lim α α x β +x + lim 2 β x +x 2 = lim Arctn(β) lim Arctn(α) = π β α + 2 ( π 2 ) = π 3 f(x) = x (,), x α x = lim x α x + lim x β + β x = lim log α lim log β α β +,, 453 [, ) f: R, f(x) >, n N, n f(k +) k= n n f(x)x f(k), [,n] = {,,,n}, [k,k+] n n f(k+) f(x) f(k), s = f(k+), S = f(k), nα, α >, < α n= α =, log(n+) = n+ k= k= k= x n x n k + x x = +log(n) k= 4 sec(x), 2 cosec(x), 3 cot(x), 4 sinh(x), 5 cosh(x), 6 tnh(x), 7 Arccos(x) 8 Arctn(x) 9 Arctn(/x) Arcsinh(x) cos(karccos(x)) 2 e tn(x) 42 Version:, My 8, 22

18 22 I 4-8 x x x +x 4 sin(x) 5 +btn(x) 6 x 3 7 x x 2, 8 Arctn( x), 9 x(arctn(x)) 2, Arccos(x) Arctn(x) 2 Arctn(/x) 43 [,b], (,b) f, f (x), f n N, x 2n+ e x2 x x 46 I n = (x 2 +), I n n+ = 2n+ 2n I n x 2n(x 2 +) n,, n N π 47 sin n (x)x 48 n α n= 49 [, ) f, s > lim x xs f(x) = A I = f(x)x, < s, lim x xs f(x) = A I 4, α, β 3 sinx x 2 x β e αx sin(x)x 4 4 p >, Γ(p) = sin(x) x x x α ( x) β x pγ(p) e x x p x, Γ() =, Γ(p+) = 42, /k, n = n k, 39 k=, n Version:, My 8, 22

19 22 I 4-9 Version (My 8, 22): 46, 2, 423, 424 Version:, My 8, 22

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