2.1.
f 0, 0 .
. 0 ( ) f, δ > 0 , , (9- δ, 0+ δ) f()≤ f(0) (f()≥ f(0)).
. 4 , 1 , 2 .
.
1. ( ) , .
► 0 f, f . , , . δ > 0 , , (0 - δ, 0+ δ) f() ≤ f(0). , (0 - δ, 0) ≥ 0, (0 , 0+ δ) ≤ 0. (.1, . 4.5) : = = . ; , , 0.
. ◄
. f 0 , : 0. 0 .
2.2. ,
2 ( ) f [ a,b ], a<b, (a,b), : f(a) =f(b). (a,b) , .
► f [ a,b ], , [ a,b ] , f( ) = m, f( ) = M, (. 1, . 5.3). , m≤ M. m= M, [ a,b ] , (a,b) ; , m= M . m < M. f(a) =f(b), , .. (a,b). , , - ; .◄
. . 5 f, . , a b . , , . 0 (0,0) - , (. .1.5).
3. ( ) f g [ a,b ], a<b, (a,b), . (a,b) ξ ,
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.
► : g(b) ≠ g(a). , g(b) = g(a), g . , (a,b), .
: , F(x) = f(x) - g(x). , F(x) [ a,b ] . , (a,b) ξ, : 0. : .◄
4. ( ) f [ a,b ], a<b, (a,b), (a,b) ξ, f(b) f(a) = (b a).
► g () = [ a,b ]. , f g ; , (a,b) - ξ , , .. , . ◄
1.
f(b) f(a) = (b a)
. . , (,f()), B(b,f(b)), (ξ, f( ξ )) (. 6). , . , : , , .
2. 0 , h ≠ 0, f , 0 0 + h ( h> 0, h< 0) , . f(0 + h) f(0) = h, ξ , 0 0 + h. : Δ f(h) = h. ξ 0 0 + h, θ, 0 < θ < 1, , ξ = 0 + θ h; Δ f(h) = + θ h) h.
2.3.
. , , .
5. α β (a, b), a<b, β′() ≠ 0 (a, b), .
, , +∞, - ∞ ∞, .
► , (a, b). [ ,b ] :
, [ ,b ] ; , (,b) ξ () - , , .. . (,b), < ξ () < b, ξ () b . :
. ◄
6. α β (a, b), a<b, β′() ≠ 0 (a, b), . , , +∞, - ∞ ∞, .
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5.
7. α β 0 , β′() ≠ 0 , . , , +∞, - ∞ ∞, .
► (a, 0) (0 , b), a b - , a < 0 < b. , . (a, 0) 5, (0 , b) 6, : ; , . ◄
1. .
► = (-1,0) (0,1) α() = tgx x β(x) = arcsinx ln(1+x) , β′(x) 0: (-1,0) (0,1)
, , 7 . ◄
5 (a, b) (a,+∞), , , ([3], 12, .1). .
2. .
► α() = π arctgx β(x) = ln arctgx (0,+∞), β′(x) = - 0 . , ,
.
, π. ◄
, 6 7. , 6 (a, b) (- ∞, b ), . 7 .
( ) . . , , α β . , 5: α β (a, b), a<b, β′() ≠ 0 (a, b), . , , +∞, - ∞ ∞, .
3. , a > 1, μ > 0.
► → +∞. (0, +∞), 0. : . . 0 < μ ≤ 1, , , ; , 0 < μ ≤ 1 = 0. μ > 1, , - : = . 1 < μ ≤ 2, ; ( ), 0 < μ≤ ≤ 2 = 0. μ > 2, , .
, μ > 0, n+1 , n μ, : = 0. ◄
. , , , .
4. .
► : = . . , : . : . , = = = 1. ◄
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2.4.
n , , 0 .
. , f n 0, , n- 1 0 .
, , ,, (. .1.2). , f n 0, n- 2 , n- 1 0.. , n - f 0 n .
, , 0 (. .1.2), .
f n , n , 0, : t0 = f(x0); tk = , k = 1, , n;
Tn(x) = t0 + t1(x-x0) ++ tn(x-x0) .
tk, k = 0, 1, , n, f. , Tn(x) n; f. , 0 Tn(x) n f :
Tn(x0) = f(0); T n(x0) = k = 1, , n. (8) .
8. ( -) f n , n , 0 , :
f () = Tn(x) + ((-0) ), →0. (9)
► n= 1, .. f 0 (.1.2):
Δ f(h) = f(x0+h) f(x0) = + o(h). h = x x0, : f(x) f(x0) = (-0)+ o(-0). , T1(x) = f(x0) + (-0), : f ()=T1(x) + ((-0)), →0 . n= 1.
n> 1. : rn (x) = f () - Tn(x). , rn (x) = ((-0) ), →0, .. .: f () n , rn (x) , (8)
rn (x0) = 0; rn (x0) = 0 k = 1, , n. (10)
: α() = rn (x), β() = (-0) . , x0 7; , , +∞, - ∞ ∞, = , .., , = . α() = r′n (x) β() = n (-0) 7: = , = , , = .
n- 1 , : ,
= . (11)
α() = 0, α()- α(0) = = α′(0) (-0) +(-0), .. - = . = = 0 (. (10)); , = . (11), :
= = 0, . ◄
(9) f 0; ((-0) ) . n c .
, f () Tn(x) →0 , n. , n Tn(x). , .
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9. ( ). f - -, n 0, n (x) n.
f() = n (x) + ((-0) ). n (x) Tn(x) f().
► f() = n (x) + ((-0) ). , - f () = Tn(x) + ((-0) ). , : Tn(x) - n (x) = ((-0) ). : →0
Tn(x) - n (x) → 0 0, j = 1,2, , n. (12) : Tn(x) = , n (x)= ;
Tn(x) - n (x) = (t0 p0) +(t1 p1) (x-x0) + (t2 p2) (x-x0) + + (tn -pn) (x-x0) .
Tn(x) - n (x) → 0 : t0 p0 = 0, .. t0 = p0 . ,
Tn(x) -n (x)= (t1 p1) (x-x0) + (t2 p2) (x-x0) +(t3 p3)(x-x0) ++ (tn -pn) (x-x0) ;
(t1 p1) + (t2 p2) (x-x0) +(t3 p3)(x-x0) + (tn -pn) (x-x0) . 0, t1 p1 = 0, .. t1 = p1. ,
(t2 p2) (x-x0) +(t3 p3)(x-x0) + + (tn -pn) (x-x0) ;
(t2 p2) +(t3 p3)(x-x0) ++ (tn -pn) (x-x0) . , , t2 =p2..
(12), tk = pk, k = 1,2, , n. , Tn(x)≡ n (x). ◄
0 = 0. () .
5, f()= , 0 = 0.
; (9) n. k - ; : f ()= . : t0 = f( 0 )= =1; k tk = .
n n: Tn(x) = = . , n f()= 0 = 0 :
= .
6. f()= sinx, 0 = 0.
k (., , [3], . 11.1). : t0 = f( 0 )= sin 0 = 0; k ≥ 1
tk =
n . 2 n sinx: sinx = + o(), ..
sinx = .
7. f()=cos x, 0 = 0.
k . : t0 = f( 0 )= cos 0 = 1; k ≥ 1
tk =
n . 2 n+ 1 cosx: cosx = + o(), ..
cosx = .
8. f()=ln(1+x), 0 = 0.
: k . , t0 = f( 0 ) = 0, k tk = . n . n: ln(1+x) = + o(), ..
ln(1+x) = .
9. f()=(1+x) , μ , 0 = 0.
: k . : t0 = f( 0 )= 1; k ≥ 1 tk = . n . n: (1+x) = + o(), ..
(1+x) = .
f n 0 , f(0) = 0. f → 0 , t0 = f( 0 )= 0, (8) :
f (x) = t1(-0) + t2(x-x0) ++ tn(x-x0) + ((-0) ). t1, t2, , tn ; , . , f → 0 , : , 1≤ ≤ n, , tk = 0, k = 0,1, , p- 1, a t ≠ 0, t1, t2, , tn .
f (x) = t(-0) + t+1(x-x0) ++ tn(x-x0) + ((-0) )=
= t(-0) + ((-0) ); , → 0 f (x) , t(-0) . , , f (x) n. , .
10. .
► 6: . : ; , - . 8: ; : . , . , :
= . ◄
. (9) , , 0, f(x) Tn(x); , Tn(x) f(x). , (9) f(x) ≈ Tn(x). f 8 , , .
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10. ( - ) f n + 1 0, 0 . , , ≠ 0, ξ, 0, : f () - Tn(x) = .
► , > 0 . [ 0, ] φ ψ: z
.
[ 0, ] ( 3). , ξ [ 0, ] , . .. .
z :
φ′ (z) = - = - , ; : φ′ (z) = - . , φ′ (ξ) = - . : . φ′ (ξ) ψ′ (ξ) (. ), : f () - Tn(x) = .
< 0 . ◄
.
11. . f(x)= = , = = 8,12, 0 = 8, - n= 1: , .. , ξ , 8 < ξ < 8,12. . : , .. = 2 + . , ≈ ≈ (. 12, 1). . : , .. | - | = . , 8 < ξ < 8,12: | - | = 0,00005. , ≈ 0,00005.
2.5.
f 0, (h) = h + o(h), A = f′(x0) (. .1.4). , f n, n> 1, 0 , A1, A2, , An , (h) = .
, f = (α, β), α < < <β, n, n> 1, 0. n:
. h = = x - x0, x = x0 + h; (α, β), = (α - , β - ). f(x0) , :
(h) = , . 9 . , df(h).
. k, k = 2,3, , n, f 0 , h 0: .
.
k f 0 , (h). ,
(h) . k = 2,3, n. df(h) = h , (h). n : .
: f g n, n> 1, 0 ;
1.
2. 3. ( f(x0) g(x0) ),
.