1.
2.
3.
1. . ` - , . , (, ), . , , , , , , , = , . ` , ` - .
f(x) =0, f(x) . y=f(x) .
- . ` :
1. 1. () , [a;b], y=f(x), f(x) =0.
2. 2. .
.
1) 1) [a;b] , f(a)*f(b)<0. , [a;b] f(x)=0. , y=f(x) a b
, , [a;b] ` . . , , .1 f(a)<0, f(b)>0 [a;b] .
, , , . , [a1;b1] (. .1) ( , .1, f(x)=0. ), f(a1)>0 f(b1)>0.
[a;b] .
2) 2) [a;b] f(x) , [a;b].
1) 2) f(x)=0. .1 , 1) 2) [a;a ], [a;b] .
f(x)=0 [a;b] y=f(x), :
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1) 1) f(a)*f(b)<0;
2) 2) f (x) [a;b];
3) 3) f (x) [a;b].
, , [a;b].
.2 [a;b] 1)-3).
³ [a;b] 1)-3) f(x) , .
, .
f(x)=0 , y=f(x) . . f(x)=0 ()= () (f(x)= ()- ()), y = () = () , , y=f(x). y = () = () (.1).
:
+ -2=0.
`: =2- ()= () =2- . (.2). , ( ), `, . (- <x <0, 0<x < ).
, = ( ), f(a), b , f(b) , f(a). f (x) [a;b]. f (x) [a;b], , [a;b] , , [a;b]. f(c) [a;c] ( f(a)*f(c)<0), [c;b] ( f(c)*f(a)<0). [a ;b ], 䳿, [a;b], , [a ;b ], .
: :
+ -36-20=0.
`: f(x)= + -36-20 . , [0; ].
1. 1. f(0)=-20<0. - , =1, f(1)=-54<0. f(0)*f(1)>0, [0;1] . =b, f(b)>0, + , 36+20. ³, , =4, f(4)=156>0, , [1;4] (f(1)*f(4)<0).
2. 2. f (x)=4 +3x -36=4( -9)+3x , , [1;4] (f (1)=-29<0; f (4)=268>0).
[1;4]. ³, , =3. f(3)=-20<0 f(3)*f(4)<0. ³ [3;4] . . f (3)=99>0, >3, , , . , . [3;4] . ³, f (x)=12x +6x>0 [3;4]. y=f(x) [3;4] , .1.
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, f(x) ( ), , . (;b), f(a)*f(b)<0. . f(x)=0.
, , , f(x)=0 , , (. .1, ). .
:
x-sinx-1=0.
`: f(x)= x-sinx-1, .
1. 1. . f (x)=1-cosx, 1-cosx=0,
cos x=1, =2 n(n=Z),
, (2 n; 2 (n+1)).
2. 2. . [0;2 ], f(0)=-1 <0, f (2 )=2 -1>0 , . , >2 f(x)>0 ( sinx<=1 >sinx+1), <0 f(x)<0 ( sinx>=1 sinx>x-1). ³, . г , [0; 2 ].
3), f (x)=sin x, [0;2 ] . ³, , [0; ],
1) 1) f(0)=-1, f ( )= -1, f(0)* f ( ) <0;
2) 2) [0; ];
3) 3) [0; ];
[a;b] f(x)=0, - , [a;b]. , , . , , , . , < b-a, [a;b]. b-a : ()= b-a. , [a;b], = , , ()= , , f(a), f , f(b) `, . , = , , , =, . , [a;b].
:
f( )-f(x)=f ( )( -x),
- . - , f( )=0 = .
, f () 0 [a;b], m>0, m [a;b], m , , .
, b-a , , , =min .
: , .
`: , [3;4], , b-a=1.
=b=4. f()= f(4)=156. , f (x) f (3)=99, (4)= . , b-a=1<1,58, , (4)=min , 3 4. , , =3,5, f(3,5) 47>0. [3;3,5]. =3,5 . , =3,2 : f(3,2) 2,42>0; [3;3,2]- , , . ³, , =3,2.
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, `, .
2:
1.
2.
3.
4.
5.
6.
1. . , [a;b] , a<x <b. , [a;b], (a<x<b) x f(x)=0. [a;b] , . , [a;b] ( x ), [a; ], [ ;b]. , , (, [a;b]), ( x ) , ., , , (n- ) , - . . . , lim c x
n ,
, . x - .
.
2. . . .
, [a,b] , y=f(x), = , (.1). , (a;f(a)) (b;f(b)):
=0, = ,
=-
.
, [a,b] ( ;0) [a,b] [a;c ] [c ;b], (.1). , , , f(a), f(c ) f(b). .1 , , , >0 (. .1), , , [c ;b], , <0 (. .2), , , [a;c ].
, , ( ) :
( >0),
( <0).
(. .1)
, .
>0 ( ). , . , = .
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: , + -36-20=0, [3;4] (. ). .
`: , f(x)= + -36-20, f (x)=4 +3x -36, f (x)=12x +6x [3;4].
:
1. 1. [3;4] f (x)>0, f (x)>0, > 0, , ==3, =b=4. f()= f(4)=156.
2. 2. .
A-c =1, f(c )=f(3)=-20, f(A)-f(c )=176,
-0,1136. =3-(-0,1136)=3,1136.
- . , =3,1564
=3,1719.
, [3,1719;4].
. , - . ³, . ij, >0 <c <<c << (. .1)), <0 >c >>c >> (. .1)). , . , . f(x), :
,
f( )=0, , f() . , f()=0, [a;b] , , . .
: , m- .
, m= .
>0, , , [c ;b]. >0 >0, , , , m= =f (c ); <0, <0 ( ) , m= =-f .
<0, [a;c ]. ³, (*)
` (*).
0,009; =3,1719 0,009. (. . 1)), 3,172<x <3,181.
: * , >0, , <0, , >0, , <0. , .
3. . , y=f(x) d ( )- (. 1). y=f(x) (x ;f(x )) y-f(x )=f (x-x ). ³, y-f(a)=f (a)(x-a) (a;f(a)), a y-f(b)=f (b)(x-b) B(b;f(b). =0, =d , : .
, (d ;0) . d [a;b] [a;d ] [d ;b], . >0, [a;d ] ( , . .1), <0- [d ;b]( , . .1). , . (d ;f(d )) (. .1) - d : .
d d .. , d d >0, , , =d ( ).
: , , [3;4]. (. ).
`: , f(x)= + -36-20, f (x)=4 +3x -36, f (x)=12x +6x [3;4]. .
1. [3;4] >0, d =b=4.
2). d . d (*) n=1: .
, f(d )=f(4)=156, f =268, 0,5821, d 4-0,5821=3,4179.
2). d - . (*) n=2, . , d =3,2078.
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2). (*) n=3 .
. , . ³, >0 (. . 1), <0 (. 1). n ( ). (*) f(x) f (x), , . , >0, <0.
>0, , . . , =3,1809 .
, 3,1804< <3,1809.
4. . ` . 1 , , , , d , ,- . - : <x < d >0, d < x < <0. ³, d , , x . , , - .
, , = , <E.
5. . . , ` ( ) . , .
. , [a;b] f(x)=0, , f(a)*f(b)<0. .
, [a;b] = ( ) [a;c ] [c ;b], . , , ( ) [a ;c ] [c ;b ], , [a ;b ], , , , [a ], , <E (*).
. (*) , <0.
(*) n , : n> , n= .
6. .
= (0)
[a;b] , , = . [a;b] , = , (1)
. , n , , , <E. , (1), .
.
(0) = = (.1); - ; - y= y=x (. .1,