I y = f(x) a I a x I x = a + x 1 f(x) f(a) x a = f(a + x) f(a) x (11.1) x a x 0 f(x) f(a) f(a + x) f(a) lim = lim x a x a x 0 x (11.2) f(x) x

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1 I y = a I a x I x = a + 1 f(a) x a = f(a +) f(a) (11.1) x a 0 f(a) f(a +) f(a) = x a x a 0 (11.) x = a a f (a) d df f(a) (a) I dx dx I I I f (x) d df dx dx (x) [a, b] x a ( 0) x a (a, b) () [a, b] a x>a x a b x<b b x a+ () a f(a) = f x a+ x a +(a) (11.3) a x b () b f(b) = f x b x b (b) (11.4) 1 x x 100

2 f(a +) Q f(a) P 0 a a + x 11.1: b f (a) f +(a) f (a) 11.1 x a ( > 0) < 0 P (a, f(a)) Q(a +) f(a) f(a +) f(a) = x a α x a α a 0 ( ) P 11.1 a α ε(x, a) x a =f(a)+α(x a)+ε(x, a) (11.5) x a ε(x, a) x a =0 ( ) a α = f (a) ε(x, a) x a x a = f(a) { α} = α α =0 x a x a ε(x, a) f(a) ε(x, a) =0 = α + x a x a x a x a f(a) = α x a x a x = a f (a) =α x = a x a f(a) x a α x a f(a) = f(a) (x a) α 0=0 x a 101

3 f(a) (x a) x = a y = x x = , g(x) I λ (λ ) ±g(x) g(x) (λ) = λf (x) (11.6) ( ± g(x)) = f (x) ± g (x) (11.7) (g(x)) = f (x)g(x)+g (x) (11.8) g(x) 0 g(x) { } = f (x)g(x) g (x) g(x) {g(x)} (11.9) ( ) (11.6) (11.7) (11.8) {g(x)} f(x +)g(x +) g(x) = 0 (11.9) {f(x +) }g(x +)+{g(x +) g(x)} = 0 = f (x)g(x)+g (x) = = f(x +) g(x +) g(x) 1 1 {f(x +)g(x) g(x +)} g(x +)g(x) 1 1 [g(x){f(x +) } {g(x +) g(x)}] g(x +)g(x) 0 { } = f (x)g(x) g (x) g(x) {g(x)} 10

4 (λ) = 0 (λ ) (11.10) (λx a ) = λax a 1 (λ, a ) (11.11) =λx n (λ n ) (x +) n = ( n 0 ) x n + ( ) n n! = k k!(n k)!. ( ) n x n df dx = λ 0 = λnx n 1 df dx = λ(x +) n λx n 0 ( ) ( ) n n x n () + + x() n 1 + n 1 {( ) n x n ( ) n x n + + ( } n )() n n ( ) n () n n (ln x) = 1 x (x>0) (11.1) =lnx df dx = ln(x +) ln x = 0 0 t = x 0 t ln ( ) 1+ x ( df dx = t t x ln 1+ 1 ) ( 1 = t t x ln 1+ 1 ) t t = 1 x ln e = 1 x y = e ax (e ax ) = ae ax (a ) (11.13) ln y = ax ln e = ax x 1 y dy dx = a dy = ay = aeax dx 103

5 (a x ) = (lna)a x (a>0) (11.14) (log a x) = 1 1 ln a x (x>0,a>0,a 1) (11.15) (sin x) = cosx (11.16) (sin x) sin(x +) sin x = 0 = sin cos ( x + = 0 ) =cosx cos ( ) x + sin (cos x) = sin x (11.17) ( cos x =sin x + π ) y =sinu, u = x + π ( (cos x) =(cosu) 1=cos x + π ) = sin x (tan x) = 1 cos x, (cot x) = 1 sin x (11.18) (tan x) = (cot x) = ( ) sin x = (sin x) cos x sin x(cos x) cos x cos x ( cos x ) (cos x) sin x cos x(sin x) = sin x sin x = 1 cos x = 1 sin x ( (Rolle) ) [a, b] (a, b) f(a) =f(b) f (c) = 0 (11.19) c (a, b) ( 11. ) ( )[a, b] c (a, b) f (c) =0 [a, b] 10.9 [a, b] f(a) =f(b) f(a) 104

6 f(a) = f(b) 0 a c c b x 11.: x c a <c<b a c + b f(c) f(c +) f(c) 0 <0 f(c +) f(c) 0 f(c +) f(c) >0 0 0 f (c) 0 0+ f +(c) 0. f (c) = ( ) [a, b] (a, b) ( a <b) f (c) = f(b) f(a) b a (11.0) c (a, b) ( 11.3 ) ( ) F (x) = f(a) f(b) f(a) (x a) b a F (x) [a, b] ( 10.6 ) (a, b) ( 11. ) F (a) =F (b) = c (a, b) F (c) =f (c) f(b) f(a) b a 11.1 [a, b] (a, b) θ (0, 1) =0 f(b) =f(a)+(b a)f (a + θ(b a)) (11.1) 105

7 P 0 a c b x 11.3: ( ) θ = c a b a 11.5 [a, b] (a, b) [a, b] ( ) x (a, b) f (x) 0 ( 0) ( ) ( ) x (a, b) >0 f(x +) 0 ( 0) f (x) =f +(x) = f(x +) 0 ( 0) 0+ x 1 x a x 1 <x b 11.4 f (c) = f(x ) f(x 1 ) x x 1 c (x 1,x ) (a, b) f (x) 0 ( 0) f (c) 0 ( 0) f(x ) f(x 1 ) 0 ( 0) 11.6 ( (Cauchy) ), g(x) [a, b] (a, b) x (a, b) g (x) 0 f (c) f(b) f(a) g = (c) g(b) g(a) (11.) c (a, b) 106

8 ( ) F (x) =g(b) g(a) g(x)f(b) f(a) F (x) [a, b] (a, b) F (a) =F (b) =f(a)g(b) f(b)g(a) F (c) =f (c)g(b) g(a) g (c)f(b) f(a) =0 c (a, b) g (c) 0. g(x) g (c) = g(b) g(a) b a c (a, b) x (a, b) g (x) 0 g(b) g(a) 0. F (c) =0 g (c)g(b) g(a) 0 f (c) g (c) = f(b) f(a) g(b) g(a) g(x) =x f(b) =f(a) 11.4 =α, x c α β x c 11.7 ( (L hospital) ) x c g(x) =β g(x), g(x) [a, b] c (a, b) (a, b) c x c g (x) 0 x c g(x) = f (x) x c g (x) (11.3) ( ) i) 0/0 f(c) =g(c) =0 x (a, c) f, g [x, c] (x, c) f(c) = g(x) g(x) g(c) = f (d) g (d) d (x, c) x c d c x c g(x) = f (d) d c g (d) = f (x) x c g (x) x c f (x) g (x) 107

9 x c+ ii) / = cg(x) = x 0 (a, c) x (x 0,c) x c x f, g [x 0,x] (x 0,x) f(x 0 ) g(x) g(x 0 ) = f (d) g (d) d (x 0,x) f(x 0 ) g(x) g(x 0 ) = 1 f(x 0 )/ g(x) 1 g(x 0 )/g(x) g(x) = f (d) 1 g(x 0 )/g(x) g (d) 1 f(x 0 )/ f (x) x c g (x) = α (11.4) g(x) = f (d) g (d) α 1 g(x 0)/g(x) 1 f(x 0 )/ + α 1 g(x 0)/g(x) 1 f(x 0 )/ (11.4) (11.5) ε>0 δ x (c δ, c) f (x) g (x) α<ε x 0 c f (d) g (d) α<ε x 0 x c g(x) (11.5) x c (11.6) 1 g(x 0 )/g(x) x c 1 f(x 0 )/ = 1 (11.6) x c g(x) = α = f (x) x c g (x) x c+ x ( ), [a, ) ( a>0) x g(x) (a, ) = g(x) =0 x x g(x) = f(1/x) x 0+ g(1/x) f(1/x), g(1/x) (0, 1/a) [0, 1/a] f(1/x) x 0+ g(1/x) ( 1/x )f (1/x) = x 0+ ( 1/x )g (1/x) f (1/x) = x 0+ g (1/x) = f (x) x g (x) 108

10 y x a 0 a : x a R n A = i 0 f(a 1,...,a i 1,x i,a i+1,...,a n ) f(a) x i a i x i a i f(a 1,...,a i 1,a i + i,a i+1,...,a n ) f(a) i (11.7) a x i f i (a), f(a) f (a) a x i x i x i a i (round d) a x i a ( 11.4 ) A x i (i =1,...,n) x i A x i A x i i f i (x) f, (x) x i x i a x i a i 109

11 y x a 0 a 1 x : 11.4 ( 11.1 ) 11.6 a (n 3 ) 11.5 x 1, x y = f(x 1,x ) a =(a 1,a ) (a,f(a)) (a,f(a)) (x 1,x, ) f(a 1,a )+f 1 (a 1,a )(x 1 a 1 )+f (a 1,a )(x a ) a R n U(a) α 1,...,α n n =f(a)+ α i (x i a i )+ε(x, a) (11.8) i=1 ε(x a) x a x a = 0 (11.9) 110

12 a n =1 ( 1 ) a a x j = a j (j i) (11.8) x i a i (11.9) α i = f(a 1,...,a i 1,x i,a i+1,...,a n ) f(a) ε(x, a) + x i a i x i a i α i = f x i (a) α i (i =1,...,n) i ( 11.1 ) 11.8 a U(a) x i (i =1,...,n) f i (x) a a ( ) =( 1,..., n ) f(a +) f(a) = f(a +) f(a 1 + 1,...,a n 1+ n 1,a n ) +f(a 1 + 1,...,a n 1+ n 1,a n ) f(a 1 + 1,...,a n + n,a n 1,a n ) +f(a 1 + 1,a +,a3,...,a n ) f(a 1 + 1,a,...,a n ) +f(a 1 + 1,a,...,a n ) f(a) a U(a) x i (i =1,...,n) 0 <θ i < 1 θ i (i =1,...,n) f(a+) f(a 1 + 1,...,a n 1+ n 1,a n ) = f n (a 1 + 1,...,a n 1 + n 1,a n + θ n n ) n f(a 1 + 1,...,a n 1+ n 1,a n ) f(a 1 + 1,...,a n + n,a n 1,a n ) = f n 1 (a 1 + 1,...,a n + n,a n 1 + θ n 1 n 1,a n ) n 1 f(a 1 + 1,a +,a 3,...,a n ) f(a 1 + 1,a,...,a n ) = f (a 1 + 1,a + θ,a 3,...,a n ) f(a 1 + 1,a,...,a n ) f(a) = f 1 (a 1 + θ 1 1,a,...,a n ) 1 3 f(a +) f(a) = f n (a 1 + 1,...,a n 1 + n 1,a n + θ n n ) n +f n 1 (a 1 + 1,...,a n + n,a n 1 + θ n 1 n 1,a n ) n f 1 (a 1 + θ 1 1,a,...,a n ) 1 f i n 1 f i 3 f i (a +) x n f i (a 1 + 1,...,a i + i,a i+1,...,a n) x i 111

13 0 a f i (i =1,...,n) a f i (a) (i =1,...,n) n f(a +) f(a) f i (a) i i=1 ε(, a) = 0 [ 0 = n 0 {f n(a 1 + 1,...,a n 1 + n 1,a n + θ n n ) f n (a)} + n 1 {f n 1(a 1 + 1,...,a n 1 + θ n 1 n 1,a n ) f n 1 (a)} + + {f (a 1 + 1,a + θ,a 3,...,a n ) f (a)} ] + 1 {f 1(a 1 + θ 1 1,a,...,a n ) f 1 (a)} = f(x, y) xy f(x, y) = x + y ((x, y) (0, 0) ) 0 ((x, y) =(0, 0) ) f(, 0) f(0, 0) 0/() 0 f(0, 0) = = =0 x 0 0 f(0, 0) = 0 y f(x, y) (x, y) =(0, 0) x y y = x 1/ (x 0 ) f(x, y) = 0 (x =0 ) y = x f(x, y) x =0 (1) =x 3 3x +x 3 () = x +1 (3) = x +1 x 1 (4) =e x sin x 1 x (5) =x x (6) = 1+ x x I f (x) =0 x =0 11

14 x (x>0) (1) = 0 (x 0) xe 1/x (x 0) () = 0 (x =0) (3) = x 3 113

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f ,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)