3.1.
- , .
1. ( )
f (a,b). , f() , , (a,b) :
► . f() (a,b), 0 . h, 0 + h (a,b) : = = = 0; , .
. () (a,b). 0 (a,b) : f(0) . , 0 . , 0 f (. . 2.2.); ξ, - , f() f(0)= ( 0). , ξ (a,b), = 0. , f() =f(0)=; : . ◄
2. ( )
f (a,b). , () , , () (a,b).
► . 0 (a,b). , , , = .
.
= . h > 0; f , f(x0+h) f(x0) ≥ 0; . (. 1, .4.5) : ≥ 0. 0 (a,b). , .
: .
. 1 2, ≤ 1 < 2 ≤ b, - , . [ 1 , 2 ] f . : f(2) f(1) = (2 - 1), 1 <ξ < 2.
(a,b). ≥ 0, , , f(1) ≤ f(2). , 1 2, ≤ 1 < 2 ≤ b, f(1) ≤ f(2), .. f .
≤ 0 (a,b) f(2) f(1) = (2 - 1) : f(1) ≥ f(2), , f . ◄
3. ( ) f (a,b). (a,b) > 0 ( < 0), f () .
► 1 2, ≤ 1 < 2 ≤ b, - , . [ 1 , 2 ] f . : f(2) f(1) = (2 - 1), 1 <ξ < 2.
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(a,b). > 0, , , f(1) < < f(2). , 1 2, ≤ 1 < 2 ≤ b, f(1) < f(2), .. f .
< 0 (a,b) f(2) f(1) = (2 - 1) : f . ◄
3.2.
: f (a,b), ; .
. 2.1. . f 0, 0 .
. 0 ( ) f, δ > 0 , , (9- δ, 0) (0, 0 + δ) f() < f(0) (f() > f(0)).
(. .2.1), f 0 (a,b), 0, 0 . , f, (a,b), , . , . , . , .
1. f(x) =x . : ()= 0 =0. , f(x) , (. . 7).
, , , , . .
f 0 . , 0 + -, δ > 0 , f() > 0 (0 δ, 0) f()< 0 (0, 0 + +δ). 0 + 0 , 0 = 0 f(x) =x + (. . 7), .
4. ( ) f 0 , 0 f , ,. :
1) 0 + -, 0 f;
2) 0 +, 0 f;
3) 0 , 0 f.
► 1) δ > 0 , f() > 0 (0 δ, 0) f()< 0 (0, 0 + δ). f (0 δ, 0 ] [ 0, 0 + δ); , , .. 0 - .
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2) .
3) , δ > 0 , f() > 0 (0 δ, 0), (0, 0 + δ). , f (0 δ, 0 ], [ 0, 0 + δ); .. f (0 δ, 0 + δ); 0 . ◄
2. f(x) =x . , 9 =0 : (0) = 0. () = 2 +; , 9 =0 .
3. f(x) = | x |. , 9 =0; () ≡ -1 (-∞, 0) () ≡1 (0,+∞). , 9 =0 , - ( ), +; , 9 =0 .
5. ( ) f n , n> 1, 9,
, . :
1) n , 9 , , ;
2) n , 9 .
► -
, ,
,
, (13) 0. δ > 0 , , 0< |x-x 0| < δ, | | < . (0 - δ, 0) (0 , 0 + δ) (. (13)) .
n - . < 0, (13) , (0 - - δ, 0) (0 , 0 + δ) , . . . , , 0 - (. ). 0,, ; , 0 .
n - . (13) 0 ; f(x) f(0 ). , 0 . ◄
4. f(x) =x . 0 = 0 : , . , 0 = 0 .
5. f(x) =x . 0 = 0 : , . , 0 = 0 .
3.3.
f [ x1,x2 ], x1 < x2,
= (14) , A(1,f(x1)) B (2,f(x2)). (x1,x2) l(x) ≤ f(x) (l(x) ≥ f(x)), f(x) (x1, x2) ( ) . (x1,x2) l(x) < f(x) (l(x) > f(x)), f(x) (x1, x2) ( ) (. .8).
f - .
. , f ( ), x1 x2, x1 < x2, , l(x) ≤ f(x) (l(x) ≥ f(x)) , (x1,x2). , - (x1,x2). , .. l(x) < f(x) (l(x) > f(x)), , f ( ) .
.9 , .
6. ( ) f (a,b). , ( ) , , f ′ () (a,b).
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:
(f ) (f ′ (a,b))
► . f , .. x1 x2, x1 < x2, , l(x) > f(x) , x1 < < <x2 . l(x) (. (14)), l(x) > f(x) :
(15) (x1 , . (15) : (x1 , g(x) < g(x2). , x2 , <x2, (x1 , ; , g(x) (x1 , . , - ([1], . 67) , ..
f 1, f ′(1), . f ′(1) , l(x) (14), . , x1 x2, x1 < x2, f ′(1) , .. f ′ .
. f ′ . x1 x2, x1 < x2, - , : x1 < <x2 . [ x1, ] [ , x2 ] f , , ,
f(x) - f(x1) = f ′ (x x1) f(x2) f(x) = f ′ (x2 x). : f ′ = ; f ′ = . , ,
f ′ , f ′ < f ′ ; , < :
( 1)(2 - ) , : x2 x1 :
= .
, , x1 < <x2 , f(x) < , x1 x2, x1 < x2, - , f .◄
(f ) (f ′ (a,b))
.
. f (a,b). (a,b) (), f ( ) .
► , f ′, (a,b) (), (. 3) f ′ () (a,b). ◄
6. f(x) =x . : (- ∞, +∞); , (- ∞, +∞).
7. f(x) =x . : ; (- ∞, 0) (0, +∞). , (- ∞, 0) (0, +∞), . . 7.
3.4.
f 0 , .
. 0 f, - δ > 0 , (0 δ, 0) (0, 0 + δ) , .
, - , , . . 10 0 , .
7. ( ) f U 0 , , . , 0 , , δ > 0 , (0 δ, 0) (0, 0 + δ) f ′ , .
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( 6), .
. .
8. 0 , , f. f , .
► 0 f ′. f 0 , f ′ ; , f ′, .. , 0 0. ◄
. f 0, , 0 f.
f 0 , . 0 , , , . 8 , . , , , .
8. f(x) =x . : . , 0 = 0 , . , (- ∞, 0), (0, +∞); .
9. ( ) 0 , . , f, f ..
1) 0 , 0 ;
2) 0 , 0 ;
► 1) , , (0 δ, 0) (0, 0 + δ), δ . , . (. ) , ; 0 - .
2) . ◄
9. f(x) =x . : , . , 0 = 0 , , 0 , 0 = 0 .
10. f(x) =x . : . , 0 = 0 , . 0 = 0 ; , 0 = 0 .
3.5.
, f
1. = 0 f →0.
f =(0, 0 + δ) ( = (0 δ, 0)), δ .
2. (), = 0 f →0 + 0 ( →0 - 0).
. 11 ) . → 0 ∞, = 0 ( ) → 0. . 11 ) . , =-1 = 1 → -1+0 → 1=0 .
f (- ∞, ) - (, +∞), . L , OY, y = kx + b .
3. (), L f → - ∞ ( → +∞).
11. a b , . f (- ∞, - ) (, +∞), .11. L , . , L f → + ∞. , .
, - f → - ∞.
10. ( - ) f (- ∞, ). 1) , → - ∞, ,
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. (16) 2) (16) , , y = kx + +b f → - ∞.
► 1) . f → - ∞ , y = kx + b : . : . , → - ∞, , .. (16) . ([1], . 50) : , .. (16) .
. , L - , y = kx + b. : , , L → - ∞.
2) , . 1), . ◄
11. f (, + ∞).
1) , → + ∞, ,
.
2) , , y = kx + +b f → + ∞.
.
3.6. ,
, , .
. 13
(, , ). , , (.1, .5.5). = f(), f() = .
, t ( ), : = f(). ,
. (17)
(17) f() ; t . = f(), (17) t: t = , : = () = f(x).
12.
[0, π], a> 0. [0, π], [0, π] = [- , ]; y = f(x) = [- , ]. , f(x), t : t = = arccos = ; , :
y = f(x) = a sin (arccos ) = .
13.
[0, 2π]. (18)
: , ′ > 0 (0, 2π); , [0, 2π] 0 2π . y = f (x), [0, 2π ]. f (x), . t. , , , , f(x).
, . 13, , , y = f(x). : (17) f, .. ; , , f: ? ? ? .
1. (), f (a,b) ().
, (. 1, 5.4), () ; (a,b). (.1, . 5.5) (a,b). (. 1, .5.2) , .., f, (a,b).
2. (), ′ (t) (), f (a,b), f ′
t (). (19)
, (.1.3, 5) (a,b), ′() = = . (.1.3, 4) (a,b),
.. : , t (). - , y′= , .. f ′.
3. (), ′ (t) (), f (a,b), f ″
t (), . , 2, (19) f ′.
14. f , (18). [0, 2π ] (. 13). 1. : f (0, 2π ). : f [0, 2π ]. , [0, 2π], , [0, 2π ] ([1], . 1); f . 2. 3. : f (0, 2π ),
f ′: t (0, 2π); f ″: t (0, 2π);
f . t (0, π) x =a(t sint) (0, π), y =a ( 1 cost) (0,2 ), a . , f ′ (x) > 0 (0, π); f [0, π] 2 . t (π, 2π) , [ π, 2π ] f 2 , , 0 = π f, f ( π) = 2 . t (0, 2π)
,
, f ″ () < 0 (0,2 ), f [0, 2π ]