. -; . , .
- . : . . . , p/q p q . . . , . .
. . . , 1,2n . , 1, 2 n , . .
- , , .
- , , - . , - S(x), - ={½ - S(x)}.
- 2 - - - - , - - , , - - - - , , - - . Ì . =- - .
- ={! . , } - .
Ç ={½ Î Î } - .
\ ={½ Î , Ï }. - -
-
R,N,Z,Q - - . . (,)= {½ <<} R ( , .. )
[,] . . -.
(,] .
- - - , .
- . .
- . (), - , - ³ (³ ). - .(.) - . -, . -
- 1 -.
X=R+ - , , .
- , - , .. , - *=maxX. - * , min -
=[0,1) max[0,1) $ . min [0,1)=0
* - . , - , - . . -, - * . . -.
. supX=x*, . infX=x*
. () . - () .
. - $ , - - . .
. n xn, - 1,2, ,n, - {xn}, - - -, - n - - .
! - . , 2- - . -.
. -:
) , - n - n, .. xn=f(n), f- - . -.
) , -.
:
) xn=5n x1=5, x2=10
) x1=-2 xn=4n-1 3, n=2,3 2=-11, 3=-47
- {xn} - . (), - {xn} M(m) xn£ M " n (xn³ m " n) - - ., . .
- {xn} - ., - - n -, ½ xn½ >.
- : - - - () . - - - - .
e >0 N, n >N:½ xn-a½ < e
- - , - .
.
- xn , - - xn=a+a n, - {a n}® 0, .. /.
-
) , xn® a - a n . xn=a+a n. a n=xn-a, n® ¥ ½ xn-a½ xn ® 0 => a n / a n xn=a+a n.
/
{xn},{yn}- -, {xn+yn}, - xn+yn. , .
- - /
) {xn}{yn}-/ -, /
1) , /
2) . - / /
! , .. / /.
- {xn} . /, >0 - N n>N ½ xn½ >c.
! / : - ., /.
1,1/2,3,1/4,5,1/6,7 . ., .. , - . .
- xn , .
- ( )
{xn} . a b, ¹ b. . - - xn - b. e = (b-a)/2, .. , - . . -.
- {xn}® e > N:" n>N½ xn-a½ <e -e <xn<a+e " n>N => - ½ xn½ £ c = max {½ a-e ½ ,½ a+e ½ ,½ xn½ ,,½ xn-1½ }
- {xn}® a,{yn}® b - - , :
) lim(n® ¥ )(xn± yn)=a± b
) lim(n® ¥ )(xn* yn)=a* b
) lim(n® ¥ )(xn/yn)=a/b, b¹ 0
-:
)xn± yn=(+a n)± (b+b n)=(a± b)+(a n± b n) a+b / -, xn+yn a± b. . -.
) xn* yn=(+a n)* (b+b n)=ab+a nb+ab n+a nb n
a n* b const /
* b n® 0, a nb n® 0, /.
=> * b+ / -. - - / - xn* yn a* b
, . - . lim -
- {xn} - ., x1<<xn<xn+1<;
, x1£ x2£ £ xn£ xn+1£ ; , x1>x2>>xn>xn+1>; ., x1³ x2³ ³ xn³ xn+1³
- - . . . -
- , . , 1 , .
- - , .. .
- - {xn} . . X - - - . - ., . . supX xn® supX ( supX *). .. * . , xn£ x* " n. " e >0 - - $ xm( m- n ):xm>x*-e " n>m => 2- x*-e £ xn£ x*+e n>m ½ xn-x*½ <e n>m. , x* . -.
- . - xn=(1+1/n)^n ( n)(1) . , - (1) -, - - , - - » 2,7128
- - (1)
- - - y=(1+x)^1/x, x>0 . x=1,1/2,1/3,,1/n, - y - (1).
- - . => . - (1) . lg x - ., . - . , - lgy, 1/lg(1+x) (2) -, .. 0<x1<x2, 1/x1* lg(1+x1)>1/x2* * lg(1+x2) (3). . $ M:1/xlg(1+x)£ lgM " x>0 (4). . - y=kx lg(1+x) x>0.
tga 1=(lg(1+x1))/x1 a 1>a 2=>tga 1>tga 2
tga 2=(lg(1+x2))/x2
a 1>a 2, tga 1>tga 2, (3). y>lg(1+x) " x>0 => kx>
>lg(1+x) " x>0
- - xn . - : - y=e^x .
-: - % r % m (r- ) n- - - . % m .
Sn=P(1+r/m)^mn (5) % - , .. - -. - - , (5) -, - Xm, - - - lim(n® ¥ )P(1+r/m)^mn=Pe^rn
Lg(e)x . lnx.
- [a1,b1],[a2,b2],,[an,bn],
- . .:
1) - , .. [an+1,bn+1]Ì [an,bn], " n=1,2,;
2) ® 0 n, .. lim(n® ¥ )(bn-an)=0. - - - .
- - - , .
- {an}-- . b1.
{bn}-- , - . , .. - 1=lim(n® ¥ )an 2=lim(n® ¥ )bn => c1=c2 => c - . lim(n® ¥ )(bn-an)= lim(n® ¥ )(bn)- lim(n® ¥ )(an) 2) o= lim(n® ¥ )(bn-an)=2-1=> 1=2=
. , " n an£ c£ bn. .
$ . , - {an},{bn} - - (.. an bn ). - -.
-. - . - Î - , .
-. {an} - . . . b1; - {bn} . 1, - ., .. $ c1=lim(n® ¥ )an c2=lim(n® ¥ )bn.
1=2 - . . . lim(n® ¥ )(bn-an)= lim(n® ¥ )bn® lim(n® ¥ )an=c2-c1=c " n an£ c£ bn. - ( ). c¹ c . . , {an}, {bn}, - - , . , .. an bn® c c . . - -.
. - 1 . , - 1- .
Y=f(x); x . ., y- . .
X=Df=D(f) y={y;y=f(x),xÎ X} x1Î X1, y1=f(x1)
1) . ; 2) ;
3) ;
4)Min max -: - f(x) , . - , .. $ m,M: m£ f(x)£ M " xÎ X
m£ f(x) " xÎ X => . .; f(x)£ M, " xÎ X=> . .
yÎ Y ® . . , y=f(x), , - Y - - f(x) - x=f^-1(y).
y=f(x) X
. " {xn} Ì X, xn® x0
f(xn)® A,=> f(x) . x0 ( , xn® x0) =
=lim(x® x0)f(x) f(x)® A x® x0
- x0 Î Ï - .
1) - -,
2) 0 - f(x) lim(x® x0)f(x)=A
lim(x® x0)g(x)£ B=> - $ , , . 2- -.
) lim(x® x0)(f(x)± g(x))=A± B
) lim(x® x0)(f(x)* g(x))=A* B
) lim(x® x0)(f(x):g(x))=A/B
) lim(x® x0)C=C
) lim(x® x0)C* f(x)=C* A
- xn® x0, $ lim(x® x0)f(x)=A . f(xn)® A {f(xn)}
. - - f(x) 0, f(x)® A ® 0, x>x0
, - {xn}® x0, - xn>x0, f(x)® A. f(x0+0) f(x0+) lim(x® x0+0)f(x)®
.
- f(x) - 0 ., - - f . . (f(x0+)=f(x0-) (1), -.
-. f(x) - 0 , f(x)® A 0 0 (1)
- - - 0 " e >0 >0, 0 (-0)<0 ½ f(x)-A½ <e
" e >0 ½ -0½ <d
½ f(x)-x0½ <e , d =e , ½ -0½ <d => ½ f(x)-x0½ <e
- f(x) - 0 - . .
- f(x) - * 0Î 0Ï .
. - - f(x) =0, " e >0 $ d >0 , Î , ¹ 0, . ½ -0½ <e , ½ f(x)-A½ <e .
, - - f(x)=C(C- ) =0(0- ) , , .. lim (x® x0)C=C
e >0. d >0 ½ f(x)-C½ =½ C-C½ =0<e , => lim(x® x0)C=C
. -.
. - f(x) g(x) - 0 . - f(x)± g(x),f(x)g(x) f(x)/g(x) ( ¹ 0) - 0 , ± , * , /, .. lim[f(x)± g(x)]= B± C, lim[f(x)* g(x)]= B* C, lim[f(x)/g(x)]= B/C
0 . + ¥ , - ¥ , ¥
. - f(x) - =0, - , .. lim(x® x0)f(x)=f(x0)
- f(x) g(x) - 0. - f(x)± g(x), f(x)* g(x) f(x)/g(x) -.
. - f(x) - 0, $ (0):" xÎ (x0) f(x)Î .
.
. - f(x) .0( f(x0)) f(x)® A ® 0, >x0
, - 0 xn>x0 f(xn)® A
: f(x0+o), f(x0+ ). lim(x® x0+o)f(x) x® x0+o 0 - > 0.
. - f(x0-o);f(x0-)
. - f(x) 0 - - (f(x0+)=f(x0-)) -, .. f(x0+)=
f(x0-)=lim(x® x0)f(x)=A
-
) - 0 , f(x)® , 0 > x0 <, 1.
) - f(x0+)=f(x0-) , $ . {xn}® 0 .
1. - 0 {xn};
2. - 0 {n};
xn® x0-o xn® x0+o, .. $ , f(xn)® A f(xn)® A - - {f(xn)} . :
1){f(xn)} {f(xn)} f(xn)® A
- .
. - f(x) x® +¥ " {xn} ® +¥ {f(xn)}® A lim(x® +¥ )f(x)=A. -¥ .
. - f(x) x® ¥ {f(xn)}
, $ -.
- : lim(x® o+)(1/x)
-, .. " {xn}® + - {f(xn)}={1/xn}, . - +¥ .
lim(x® o+)1/x=+¥ - 0.
-¥ .
+¥ -¥ - - , {xn}® x0 {f(xn)}® ± ¥ ,¥
1) lim(x® 0)sin/x=1
2) . -. . :
lim(n® ¥ )(1+1/n)^n=e (1)
lim(n® 0)(1+x)^1/x=e (2)
t=1/x => ® 0 t® ¥ (2) => lim(x® ¥ ) (1+1/x)^x=e (3)
-
1)x® +¥ n x:n=[x] => n£ x<n+1 => 1/(n+1)<1/x<1/n
- -, - , (1/(n+1))^n£ (1+1/n)^x£ (1+1/n)^(n+1) (4)
- . . (® +¥ , n® ¥ )
lim(n® ¥ )(1+1/(n+1))=lim(n® ¥ )(1+1/(n+1))^n+1-1= lim(n® ¥ )(1+1/(n+1))^n+1* lim(n® ¥ )1/(1+1/(n+1))=e
lim(n® ¥ )(1+1/n)^n+1= lim(n® ¥ )(1+1/n)^n* lim(n® ¥ )(1+1/n)=e* 1=e
2) x® -¥ . . y=-x => y® +¥ , x® -¥ .
lim(x® -¥ )(1+1/x)^x=lim(y® +¥ )(1-1/y)^-y= lim(y® +¥ )((y-1)/y)^y=lim(y® +¥ )(1+1/(y-1))^y=e
3) x® ¥ - xn ® ¥ (3) lim(x® ¥ )(1+1/xn)^xn=e (5)
5~3, . 3 -. - xn 2 -: {xn}® +¥ ,
{xn}® -¥ . - .1 .2 5 xn® xnxn. -
. - a () - / - 0 - / -:
) / - / -.
) / - - / -, .. a ()® 0 ® 0, f(x) ($ :½ j ()½ £ )=> j ()a ()® 0 ® 0
/ 0 . :
1) 2- / a ()/b ()® 0 ® 0 / a b .
2) a ()/b ()® A¹ 0 ® 0 (A-), a () b () - / .
3) a ()/b ()® 1 , a () b () - / (a ()~b ()), ® 0.
4) a ()/b ^n()® ¹ 0, a () - / n- b ().
: ® 0-, ® 0+, ® -¥ , ® +¥ ® ¥ .
. f(x) 0 - - . - -, .. lim(x® x0)f(x)=f(x0)- - -. - - . . - -. lim(x® x0)x=x0 (1). . - -. - - 0 - . D -, .. D =f(x0+D x)-f(x0) ( - . 0). D - .
- 0 , . , - 0 . lim(D x® 0)D y=0~ D ® 0 (1). - 0 - , - ® 0 .
f(x) - 0 <º > D y® 0 D ® 0.
. - . .
. f(x) - 0(=f(x0+)) - - - 0, .. f(x0+)=lim(x® x0,x>x0)f(x)=f(x0), - f(x) - . - 0.
- . f(x0-)=lim(x® x0, x<x0)f(x)=f(x0), - - . . 0.
. - f(x) . - , . -, , . f(x0-)=f(x0+)=f(x0)
. - f(x) - D, - - , - D -, . . . - -.
- . -
Q=f(k)=k^1/2 Q- , . D(f)=R+=>f(0)=0 f(0+) $ 0 => - . . -. - - - . . - , - (D Q® 0 D k® 0). - . . - . - - . . - -
- - 3 : . -; - 1- , 2- .
) - 0 $ , f(x0+)= f(x0-), ¹ f(x0), - - -.
0 - -, - f . - 0. - f - . f(x0)= f(x0-)=f(x0+) . . -, . f.
) - 0 $ 1- f(x0± ), f(x0+)¹ f(x0-), 0 - - - .
) - 0 1 - $ , 0 - - - 2- .
. - . - - . :
1) - . - => . - . - - -.
2) - , .. . ., . - - .
3) - . -. - . - . -:
. - - D ( -) - - .
I) - . - 0 -.(- . -)
- - e d . f . - 0 e >0 d >0 ½ f(x)-f(x0)½ <e ½ -0½ <d ~ f(x0)-e <f(x)<f(x0)+e - 0.
II) - f(x) . - 0 f(x0)¹ 0 $ - - 0.
III) . - f(x) . [a,b] f(a)=A, f(b)=B A¹ B => CÎ (A,B) $ cÎ (a,b):f(c)=C f(c)=f(c)=f(c).
IV) . . - 0. f(x) . (a,b) f(a) f(b), $ - Î (a,b).
- - - f(x0)=0 . f(d)=0 c=d - .
f(d)¹ 0 [a,d] [d,b] - f . - [a,d] [a1,b1]. 2 - - d [a2,d2] - [a1,b1]>[a2,b2] (a-b)/2^n® 0, - - - . - . :f(c)=0. , f(c)¹ 0 - . d , - f f(c) [an,bn] N f .
f . - 0 => f . - 0 f(x0)¹ 0 => f . [a,b] f(x)* f(b)=0 (f(x)* f(b)>0 - 0) => $ Î (a,b). f(c)=0 - 2 - . - .
- 1( . . - ). f(x) . [a,b], f(x) . , .. $ >0:½ f(x)½ £ c " xÎ (a,b).
- 2( $ . . - .). f(x) . [a,b], . , .. $ - max X*:f(x*)³ f(x) " xÎ [a,b], - min X_:f(x_)£ f(x) " xÎ [a,b].
. . -
1. f(x)=1/2 (0;1] ® f . (0;1] .
2. f(x)=x; (0;1) f(x) . inf(xÎ (0;1))x=0, - x_Î (0;1):f(x_)=0, - x*, sup(xÎ (0;1))x=1
- - 1. . ; f . [a,b], , .. [a;c][c;b] f(x) .
. [a1,b1] . [a2,b2], f-. . . . [an;bn] . . - d (d=c ) [a,b], . f(x) . - - d . [an,bn], . f . [a,b] => - d - . d. . d => . - - d - [an;bn] 0.
- - 2. E(f) - f(x) . [a,b] . - - . - supE(f)=supf(x)=( Î [a,b])=M(<¥ ). InfE(f)= inff(x)=m(m>-¥ ). . [a,b] f(x) . [a,b], .. $ *:f(x)=M. , - $ - f(x)<M " xÎ [a,b] . - g(x)=1/(M-f(x) Î [a,b]. g(x) . 2- . - . 0 - 1 g(x)- . .. $ c>0
!0<g(x)£ c g³ 0, [a,b] 1/(M-f(x))£ c => 1£ c(M-f(x)) => f(x) £ M-1/c " xÎ [a,b]
- ., .. - . f [a,b] C
: f(x) . [a,b] . . max min, .. E(f)=[m;M], m M max min f .
. - - - ( - ) - - y=kx+b . -; =k => k>0 . , k<0- . , k=0 -
-
1) - y=f(x) - - 0, D -. . - - 0. D y=D f(x0)=f(x0+D x)-f(x0)
D y/D x=D f(x0)/D x (1) ( . - D , .. 0-, D ® 0 . 0/0).
. - - y=f(x) - 1 ( $ ), D ® 0. - ., - ® 0. df(x0)/dx f(x0), ( - 0 - - . f(x0)=lim(D x® 0) (f(x0+D x)-f(x0))/D x (2)
- f(x) - 0 -, .. (2) $ , f(x) . - 0.
2)
-. - f(x) . - 0 -, D f - 0 D f(x0)=f(x0+D x)-f(x0)= f(x0)D x+a (D x)D x (3), a (D x)-/ - D ® 0
-. , (3) , D ® 0 D f(x0)® 0, => - 0 - . - - (3). - $ (2) / =>, $ / - a (D ) D f(x0)/D x=f(x0)+a (D x) - (3) - D x.
.
1)- - 0, .. y=c=const " x, y=0 " . D y/D x -, - 0, => - = 0.
2)- -, =^k, y=kx^(k-1) " kÎ N. - =0 - -. " - D D /D =(+D )^2-x^2/D x=2+ D => lim(D x® 0)D y/D x=2x=y. - - - - .
3)- - -, =^x => y=e^x. D y/D x=(e^x+D x-e^x)/D x=e^x(e^D x-1)/ D x. = 1.
4)y=f(x)=½ x½ =(x, x>0;-x,x<0). " ¹ 0 -, y=1 x>0 y=-1 x<0. - x=0 - $ . - . . - . - -. . [-1,+1], . - 2 $ x0=0. D x>0 D y/D x=D x/D x=1=>lim(D x® 0,D x>0)D y/D x=1 - - 1. .. . . - $ . $ . -.
. () - - - 0, - lim (2) . D ® 0+(D ® 0-).
., f(x) . - 0, . - $ f(x0-) f(x0+) $ - f(x0) , . . - . . .
- f(x) ; f(x) ; f(x)-; fn(x)=(f(n-1)(x)). - - - . .
dy= f(x)dx . - f(x) d^2y, .. d^2y=f(x)(dx)^2. . d(d^(n-1)y) . d^(n-1)y - . n- - f(x) . d^ny.
. - f(x) (a,b) - 0 . - 0 $ -, = 0, f(x0)=0.
2) . [a,b] - f(x) : f(x) [a,b]; f(x) . (a,b); f(a)=f(b). $ - Î (a,b), f(c)=0.
3) . [a,b] f(x), : f(x) . [a,b]; f(x) . [a,b]. $ - cÎ (a,b) , - (f(b)-f(a))/b-a= f(c).
4) . - f(x) g(x) . [a,b] . (a,b). , g`(x)¹ 0. $ - Î (a,b) , . - (f(b)-f(a))/(g(b)-g(a))=f(c)/g(c).
0/0. 1- . lim(x® a)f(x)= lim(x® a)g(x), lim(x® a)f(x)/g(x)= lim(x® a)f(x)/g(x), $ .
¥ /¥ . .
lim(x® a)f(x)= lim(x® a)g(x)=¥ , lim(x® a)f(x)/g(x)= lim(x® a)f(x)/g(x). , x® ¥ ,x® -¥ ,x® +¥ ,x® a-,x® a+.
- 0¥ , ¥ -¥ , 0^0, 1^¥ , ¥ ^0.
. 0¥ , ¥ -¥ 0/0 ¥ /¥ . . 0^0, 1^¥ , ¥ ^0 f(x)^g(x)=e^g(x)lnf(x) 0
- . . -, - - . - - , . .
-. - - - - , . - , . -. . . - - . f(x) . x>0. . (0,a) - . . -, , , . . . . , . - . (¥ ,a) - . . . . . . . . . . . . -. () . f(x)>0 $ x³ 0, 0 (0;) f(x) . (0;¥ ) f ., - -max. (0;) f(x)³ 0 (f-), (a;¥ ) f(x)£ 0 (f-).
. f(x) . - (a,b), :
1) - f(x) () (a,b), 2- - , .. f(x)³ 0 (f(x)£ 0) (a,b)
2) 1 - - 2- -, - - () (a,b)
. - . - - . 0 - , f(x0)=0 2- - 0=> - f(x) .
. - - . - - .
. - . () (a,b) . - - - , () . -.
y=y0+f(x0)(x-x0)=f(x0)+f(x0)(x-x0) - , f(x) f(x) - () . f(x)³ f(x0)+ f(x0)(x-x0) " x,x0Î (a;b) f (,b). (), . - (.) - kx+b, , . . .
- {xn} - /, " - $ N , n>N - - ½ xn½ >A
>0. ½ xn½ =½ n½ >A n>A. N³ , " n>N - ½ xn½ >A, .. - {xn} /.
. / - . . . - /. 1,2,1,3,1,,1,n . / >0 - ½ xn½ >A " xn . .
. - f(x) . , .. f $ . - F(x)=f(j (x))* j (x) (4). - (4) y=(lnf(a))=f(x)/f(x) (5) -. (- f(x)) - . - . . - y f(x). , P(t) -. -, t=R. ¹ .
- y=e^a x. . f/f= =a e^a x/e^a x=a . - .
. - y=f(x) . . . f(x)>0 => . -. - - f(x) - - - - . -.
Ef(x)=x* f(x)/f(x)=x(lnf(x)) (6). . (6) - D f(x0)/D x Ef(x)» x(D f(x)/D x)/f(x)=(D f(x)/f(x))/(D x/x). . - f - x, . . . => - - % - y=f(x) . 1%. - 1- .
-. - - , D=f(p)=-aP+b - , >0. . Ed(P)=P* D/D=P* (-a)/(-aP+b)=aP/(aP-b)=> - -
- -, , - -.
- . . (a,b) f(x) - 0 , - - 0, .. f(x0)=0 (8). . . ., .
. - - - f(x) 0 - . - f(x). - => . -.
- . - f(x) g(x) [a,b] . (a,b). , g(x)¹ 0, $ - cÎ (a,b) , - (f(b)-f(a))/(g(b)-g(a))=f(c)/g(c)
-. f(x) . (a,b), . f(x) . () (a,b) , f(x)³ 0 (a,b) f(x)>0 (f(x)<0), . () (a,b).
Î . , f(x1)<0 x2 .
- . - f(x) . [a,b] . (a,b), " . x+D x Î [a,b] $ - +D - (f(x+D x)-f(x))=f(c)* D x (7) => - - . , (7) . -, - - . (a,b) .
- (7) => x=a x+D x=b+> - (7)=(f(b)-f(a))/(b-a)=f(c) (7) - .
(f(b)-f(a))/(b-a)=f(c) (1)
- - . - . - g(x)=f(x)-f(a)-(f(b)-f(a))/(b-a) * (x-a)
- g(x) . . - [a,b]
) [a,b]
) . (a,b)
) g(a)=g(b)=0
. , $ - (a,b) g(c)=0 g(c)=f(x)-(f(b)-f(a))/(b-a). - (1) - - .
- . - f(x) . . .
) [a,b]
) . (a,b)
) f(a)=f(b), (a,b) $ - f(c)=0, .. -. -.
-. - , , f(x) [a,b] (f(a)=f(b)), f(x)=0 $ x Î (a,b), - - . f¹ const [a,b], .. . , - . max min. f . . -, 1- . max min . -. Î (a,b) ( f=const), - , f(c)=0, -.
- . -
. - f(x) - - n+1. - , ¹ . - - e , - . f(x)=f(a)+f(a)/1!(x+a)+ f(a)/2!(x+a)^2+f^(n)()/n!+f^(n+1)(e )/(n+1)!(x-a)^(n+1).
-. . g(x).
g(x)=f(x)-f(a)-f(x)(x-a)--1/n!* f^n(x)(x-a)^n-1/(n+1)!(x-a)^n+1* l . - $ - (a,b), g(c)=0 l =f^(n+1)(c)
- f(x) g(x) . - 0 - f g - 0. lim(® D )=lim(x® D x)g(x)=0 f(x)/g(x) x® x0 0/0. lim(x® x0)f(x)/g(x) $ (4), - lim(x® x0)f(x)/g(x)= lim(x® x0)f(x)/g(x) (5)
-.
" - >0 [x0;x] - . t
h(t)=f(t)-Ag(t), tÎ [x0;x], .. . - - - 0, - 0. - h [x0;x], lim(t® x0)h(t)=lim(t® x0)[f(t)-Ag(t)]=lim(t® x0)-A lim(t® x0)g(t)=0=h(0)=> . t=x0 - (x0,x)$ c:h(c)=0
-. . - -, , - - - -.
-. - y=f(x) . yx=f(x)¹ 0.
D ¹ 0 D - x=j (y). : D x/D y=1:D y/D x (2) - (2) D ® 0 , D ® 0, : lim(D y® 0)D x/D y=1:lim(D x® 0)D y/D x => xy=1/yx. - -.
-. . - -, , - - - -.
-. - y=f(x) . yx=f(x)¹ 0.
D ¹ 0 D - x=j (y). : D x/D y=1:D y/D x (2) - (2) D ® 0 , D ® 0, : lim(D y® 0)D x/D y=1:lim(D x® 0)D y/D x => xy=1/yx. - -.
- . - . -.
-
1. - , $ m M, " m£ xn£ M, " n.
D 1=[m,M] , - -. . - - -.
D 2 , - -. . . D 2 - - -. - D 3. D 3 .. - , 0. - , $ . - , . . D 1, - - D n1. D 2 - xn2, n2>n1. D 3 .. - - xnkÎ D k.
- - - [a,b] - , $ - Ì (a,b) - 0.
-
- - [a,b], f(x)<0. - . Î [a,b], , . c=supx. a£ c£ b a<c<b - . , c¹ a, c¹ b. f(c)=0, , $ - - , , .. - - . f()=0.
- .
- - . - n, .. - , xnÎ [a,b], ½ f(xn)½ >n. - - xn. - - - xn - xnk$ ® x0. - .
a£ xnk£ b a£ x0£ b x0Î [a,b]
- xnk x0, f(xnk) f(x0)
½ f(xnk)½ >nk, a nk® ¥ Þ ½ f(xnk)½ ® ¥ , .. f(xnk) / -.
f(xnk) . , . ¥ , , .. , - . .
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