: . . ::

. . II 100

Il .. 20

r

.

.

,

, , , ,J'! , .

, , . /(,

( ) u , (. .

) u , (.. ).

, , . . ,

, , .

. , .

:

, , , .

. , R .

.

.

.

. , , . ,

.

.

. .

:

- ( ), ( / ), , ( .. ), ( / /),

( / /), .- . (

) ., , , .

, 396, 153 . .

1992

1.0

, . .

, Cantor, . . , , (2).

. ... . . . . , . . . , . . :

.xeA .

: , .

= : ' , . . .

AUB : . . :

AUB=! / j

7

AnB: , - . :

[ AnB== ( / J

c.,~iI . , . n == e , .

, : .

\=={ / xiB J . .

== [ (,) / J (,) . ' = , ~oA - \ ..

IR2 == ( (,) / xEIR YEIR j ,

fJ IR .

- ==( 0,1,2, ... ., ... j, ,- == ( ... , - 2, - 1,0,1,2,3, ... j ,- Q== [ /=7' , - * J - ; .J2, 3ff ,

= 3,1415 .. ., e == 2,718 ...

, , Q IR , , . . * == \ {}.

:

8

.

,

.

, IR, , IR :(,) =! xEIR /

; ;

(): 2' >

]0

()

= , 2 ' > 1,

= 2, 22>2,

= 3 , 23>3 . () ; . , ,

.

; (), * , , ?' , :) (1) ,

) () ( + 1) ,

() .

, 2V > , :- =] , 2 ' > 1- = *, 2 >

. = + 1, 2 ' + ' > +] .

, 2' > 2 "1 = 2'2 >K2~ K +] 2V > * .

, :

> - ] * Bernoull i:(1 + ) " ~ 1 + .

(11 , ; ; ; : !* :(i) l ) = ! ' ,() , + ,

1I

= ; , ( + ) 1 ~ 1 + l. V=K EIN*, . ( +) '~1 + . = + ,

( + ) " ~ + ( + l)a. , ( + ) ' ~ + 1+ > ,

(1 + ) " 1 = ( + ) ' ( + ) ~ (1 + )(1 + )= 1 + ( + ) + K '~ + ( + ), K~ ~ O.

, , BernouJli *.

;

1. 2 ' > 8ernouII i =.2. () ; p(l )

( ) > , () ~ . ,

ii ) / \2- 3 \ > 3 J) / + 31< 2 J

-~--

1. ; :

) / - ! J2. ; :

) [ / jx - < : ' !"> j

) lx /X ~-5 4v ) 7"+ 3">8v ) 7 -'< - '-6

2

8' ~* :

( -- 1)) ] -- 2 +...+ = ---'--'-'---'--.::..!-2

2. . =~- , , = , * :i) , , " > *

,

ii)

f(A)=! YEIRj , y=f(x) J

- 00

+00

- 00 ()

f(x)+00

, f, .

, f(A), .

, f(x) = . [()

( .2) .

.2

f(x) , . . [(x)=x + '~ -

f f(x) = +..JXi=T , () =+ ' 2 - 1 - x+~.

14

, , f IR f(x) . .. f(x) = + v!JZ2-=-l = ( - , - 1] U [1, + ).

G(f)=! (x,f(x)/ j, IR2 , f.

G(f) , f.

(,) f, , = f(x). , t

, .. Dirichlet

f(x) = [ , , , ,

' ; f, f ,

1. f ; f ' (.3).

2. f(x) 4.

.3

! . .' . ,

.4

15

3. f . ', ;

f ' (.5). ;

' )' f ( .6) .

4-

f(A) .... ---- - -

4 - - - - - -

.5 .6

f , f . f ,, f.

, f(x o) f '

1.

2x-l. f(x) = + 2 '* , '* - 2, +2 = IR \ - 2} .. f(x) = '5= 5 - ;::0, :55,

= ( - 00 ,5].

2.

. f(x) =__ = IR \ {- 2}. -+2 , ,

f(x) = , , IR \ {- 2}.

-- = =- +2= =- (1-)=2 (1)+2

16

- = (1) =2, . 16! f(A)- * 1 (1) =~ , ffi. \ 1- 21 . -

]-

, = -2, ~ = -2 = 2=2-2 = 0= -2, - . f f(A) = IR \ (] J.

. 2 + - 3 . f(x) = 2 = IR, ++2 2 + + 2 > 0 xEffi..

, f(x) = , , IR.

2 + - 3 2 2 22 2 == ++2= +-3 = (-) +(- J)x+2y+3=0.(2) ++

- = (2) 2 + + 5 = , . i f(A).

- *] (2) IR, ~ ,

13( _ 1)2_ 4(- 1)(2 + 3)~O = (y-l)( -7-13)~O = --- :5Y:51.7

] i f(A) , f f(A) = [ - ' 1).

'

f(x + h) - f (x) . .1. :h

:

i) f(x) =2x -1

-2) f( x) =-2- 2""-'--s--'x=--+- 3-

) f(x) = 2 2 - 3 + 1

) f(x)= J~

1) f(x)= - - .~x2- 4

) f(x) = + _ 1_

.J--.:]ii) f( x) =---'-'----=--

-2

~-x) f( x) = - - +217

~ ~ ; ~ f :

'---_.:......._------_._---- - - _ ._---- ----- - -._---

)

3 - - - --- - - -

2

-1

-1

)

-2

!_______ ______1

18

k ; :

i) f(x)='16-2xl i) f(x) = 3xlx - I1-

i) f(x) = 1-2 2+Sx-71 ) f(x) = -+ 2 +3

,'1 , , ; :

22+-3) f () = --=..:---c-.:.-=--..:'---2 -1

) f(x) = _ 2_ -

ii) f(x) = - 2

-3) f(x)=--

6. :

i) f(x)=~-

8 g(X)=-

) [(x)=~+~

. , ' , . Ohm , . E=]R. = 10 Volt "

, . , ,2S < R < S -! ; , . 11

9. V , S - ; . 10. Boyle, V - ' = 800. 100s V s 200 , . ; L. . . . . . ,. l

i) f(x) = 2 - 3 + 2 g(X) =~;h ~ :

, ) f(x) = 3 - i) f(x) = 32:5 - 2 ) f(x) = 2 + 2-2 -4i

19

11. R. ad ,

.

12. ; f f - . . .

-q'

. Nq :'() f ( ) 1 1 =--- +---

>-.

2. ~ :.'.".~ J''() = 5 + vx_1 f(x) = ..}4-.,JX- 5 .

3. f(x) =2' + + g(X) =( 2+ 2) 1- 3 + 2- 6, } . . f,g } =2.

, 4. f xf(x) + ( - ) = IR,) f) f .

S. ' - cm 324 cm J 2 . cm" . cm'.

.

h

6 . = 3, = 7 == 4.

= , .

r'--- - - - - --- - - - ------- - - - _.._--- -- - - - - - - -'

20

1.2 .

l ' /1 iJ

() = +_ -+ ... ++, . , , ... ,avEIR . IR. *0,

.

:

yt / ~-.11" /V Q

> QO

f(x) = ', *0

a

0>0

f()=J , :#: : IR, : IR

uyv:f1. 'i.HJYOPTillJEif; , , . ;

.*.

~x) = x , 011

;fX / =F + ~;i-' EZ I : 1R

f(x) = : IR : [- 1,1]

f(x) = : IR

: [- 1,1 ]

)!

.7

.

:

fx) = lx l= [ , ( -, 9.

f IR [, + (0).

[ 2 -

; .;

1f(x)=- = !

>

1.3

f(x) _ 3 - - 2 + + 1 g(X)= - "

= IR . :

. ; f,g f = g.

" [= g

f(x) = g(X)

f,g f(x) = g(X), .

, (' .1. ; f(x) = 2 g(X)= x lx l, IR , . [, + 00) , [, + 00) f(x) = g(X).

. . f ( ) 2 - 1 = - g(X) = + 1 , f IR \ !11 g IR . IR \ 111 .

, g *- 0

f + g, f - g, f g

;

(f + g)(X) = f(x) + g(X)(f - g)(X) = f(x) - g(X)(f g)(X) = f (x) ' g(X)

(_r )(x)=~g g(X)

fg

25

n , ---.L g n 13 g(x).

, f(x) = - 2 + 3 - 5 g(X)= 2 2 - IR, :

(f' +g)(x) =nx)+g(x)=(-x2+3x-5)+(2x2- x)=x2+2x-5, x EIR,(' , g)(X) = nX)g(X) = (- 2 + 3 - 5)(2 2 - ) = - 2 + 7x J -13x2+ 5 , xEIR,

(---.L ) ( )= () = -x2~3x-5 , xEIR \(0,1- J, g(O)=g(1-)=O,g g(X) 2 - . 2 2(.~ )()=~ = 2~2_x , xEIR, x EIR f(x):;tO f(x) - + 3 - 5

; f(x) = 2 , g(X) = '=-2 , IR [ - J , 1] ,

(f + g)(x) = 2 +v'=X- , (f - g)(x) = 2 - '=- (fg)(x) = 2'!-

[- 1, ] = Rn [- , 1]. (: )()= b

(- , ), g( - ) = g(1) = ,

' (~f )() .f17 ' .. [- 1,0)U(O,I] f(O)=O .

2. :

1. :

-) f(x)=-- +2/ - /

---.-----'1

) f(x)=x lxl

3. f, :

2.....-----

) )

-2 -1

i) . )

4. f =g. f :;:g, IR, f(x) = g(x).

) f (X)= X22+ lxl - g(x) -~- -1 ' - 3g(x )=-

+2

i) f(x) = g(X) = ;;+ 1- - .;;:-,X+T+-JX

~ ~ -) f( x) = -JX g(X) =--

S. f + g, f .g _f_ g

..) f( ) g(X) -- IXI - 311 = - - . .2- lxl 2 - 4

. 2 + 2 g(x)=-- -, - 2 +

-) f( x)

__________ _ _ _ ____ _ _____ _ _____J

- g(X)=--

8'

. ; f(x) = + 21 + - , .

(3+ , :52 [- +3 , < - 2. f(x) = 2 4 2 g(X)= _ 3 > - 1+, > + , _ f +g -_ . ~1.4 f : - IR g : - IR . f(),,

- , , f(x) f(A) ;

- - f(x), g, g(f( x))EIR.

->

+>

. 1

, g(f(x)) . f g gof. '

28

f : -+ , g : - IR, [(): f g

gof: -+

-+ (got)(x) = g( f(x)

. h(x) = 5 2 - 3, IR,

[()=, g(X)=5x 2-3x, xEIR, f(lR) = [ - 1,1] IR, gof IR

(gof)(X)= g( f(x) = g( ) = 5( )2 - 3( ) = 5 2 - 3 = h(x)

_. f(x) = 2 - g(X)= 2 + 2 = IR = IR . :i) gof = IR IR

(got)(X) = g(f(x) = g(2x - ) = (2 - l + 2 = 4 2 - 4 + 3i) fog = IR IR

(fog)(x) = f(g(x) = f(x 2+ 2) = 2( 2 + 2) -} = 2 2 + 3., f,g , fog gof, gof = fog.

f() , ' = / f(X)EB).

i) ' * 0 , gof g( f(x) '.

ii) = 0 , f g. ,

f(x)= JX -} g(X)=~ = [, + 00) = [ - , ] , ' = XE[O, +oo) / J x -IE[-I,I] 1=1 XE[O, +oo ) /-I$. JX"-I$.1 1=[0,4J 0 . ' [0,4] :

(gof)(x) =g(f(X) = g(Jx - ) = .J)=-(V'; - 1)2= .J2JX-~;. f, g, h :: ho(gof)

(hog)of : ho(gof) = (hog)of ; ; ; f() , ' .

29

(fog)(O).

(gof)(4)

[ ~ , f,,) " >+ 5 g(x) " \\.

) [()= g(x)=3x 2 ,

' fog gof, i) f(x) = 2 - 3 = 4 - , ) f(x) = 22 - 1 g(X) =

/. gof,

i) f(x) =.;;:+l g(X) =~ . 1 g(X) =----9

f(x) =_1_ l()=, fof=I;

.

~$'

1.5 f : - IR EC .

f , , 2,

, .

, ; ; :1. f(x) = + :

IR, >0

IR, , IR. ( - 00,0] [, + 00).

3. f(x) = ) : IR, >

IR, , :

(- 00,0)

(, + 00) f JR*,

. . ] ] ' <

:- [, + 00) .

-XI ~+XZ ~ -

- , ( - , ], =

f (- 00,0].

[ ) . X: ~-x , ~- , 1 ,+ 00 , = . - X~

-

, g(x) = 2 - 1, (.2), g

, ; - 1, IR

g(x)=x 2 - 1~ -1., g - 1.

.2, , f(x) = (.3) - 1

1,

.

:

.3

f : - IR : , ,

f(x) :;;; , mEIR ,

f(x)~m , m, M EIR ,

m :;;; f(x) :;;;

, , f, m, ; , f.

f(x):;;; f .

f f(E) .

34

, f(x) = -1-1 (.4).-

-

(- 00,1) =,

( - 00,1) _1_ 0-

- , .

:

.4

f : - IR == , , > , f(x) ::5 .

- f(x) ::5 , -K::5f(x)::5K, f .

- f t , m~f(x)~M. , Im\ ~K \ ! ~K :

- K::5m::5K - ::5::5

:

- ::5m:5 f(X):5 :5 - :5 f(X)::5 \ f(x) \ :5. [(X)=~2 , xEIR

. +

:

; f , > , f(xM.

.35

1.7 f ,

f , r f(xo) '

' ; f ,

f , f(xo) ' f , f

. :- f(x) = - ' + ( . , 1.6) =

, J.- f(x) = x ~ - (.2, 1.6) = !

- 1. . ,

' . , 5 . ' ' , 2 ' -2- . .. , 3 7-- --

2 ' 2 , , , . , .

. . ~ , xoEIR*, . .

_________ --!

.

i) f(x ) =_ 2_ -

36

ii) f(x) ={x ~ ,

\") f,g

f(x) >0 g(XO, [. g .

i) F(x) = 2

(,-} )

F(x) = - 2(3 3 +5)3 + 7;

3. f,g IR, gof

) , f,g ) , f,g

4. 20 4:000 . 5 .

. , . :

! ,--------.-------~----- -----..._ -- -_ . . ----.-l

1-

,

1.8 " " . :

*2, t6 [() * [(2),, ,[( ) = [(2)' = 2'

: - IR , , 2 :

f : - IR ,2 ; ; -, , YEf(A) (.).

:

. 138

(.2).

f (.3),

, 2 , :;t:x2 f(x,)= f(X2)

-- -

.2 .3

f: - IR ,

f-I : f(A) - IR,

YEf(A) f. :

f - l(y)=x=f(x)=y

f : - IR :

[- ())= f(f ()= YEf(A)

: = , 3; .

,

f(x) =~

39

f = [3, + 00) , ,2 f(x I ) = f(X2)'

{ - 3 = .JX2 - 3 , =2 .

f - ! : f(A) - IR

f(A) = [ , + 00)., [, + 00),

3

,.0/

f - () = 2 + 3. , f

f-i : [0,+,00) - IR, f - I(x)=x 2+3

f , =, =2+3, X~O.

, , :

) f ,

.)

.

) f: - IR , 2 , ::f.X 2. f

,

= f(xI)- f(X2) -2

, f(xI)- f(xz)::f. , . f(X j)::f. f(X2)' f .

40

) : - IR . [- [ . , 2[()

3. f , , ; , .

)

.',/

,// )

) .

/ j)

42

)

)

'

].. -J ; .

]r. " ,

) f(x)= ~ ~x , i) f(x) = 3";;:.a. ()=2+3 g(x)=3x-5, :

i) (fog)-I, ) r-Iog- I, ;

______ 10 _

'

1. ) f : IR -. IR g : IR -. IR -, gof - .

ii) f : IR -. IR 1 - 1, F(x) = (r(X) + 2f(x) - 3 1- .

2. ; f,g (f + g)(x) . [(f + g)(x) - 2]= 2(f g)(x) - ] ~EA, f = g.

3. f : IR -. IR

f(x I ) - f(X2) ~__ , -21 , X2EIR Xl=FX22 . g(x) = f(x)-~ IR.

2

4 ' 'f() 43 - . = 3) f .ii) f

43

f+ ,

5. f(x)= 2 + ,

< 1 ,,= 1

-

.

Ixl + ; IR.

6. f(x)

7. f f(xy) = f(x) + f(y) x,y EIR* ,

) ( ) = ) f(+ ) = - f(x) . ) .f(x'') = () *.8. (x) = x+~

) f(x)> IR) - .

119. = 10

=6. = .

;

44

xoEIR

2.1 . , ,

.

. , , < , ', >0 e =2,71 ... .

, ;, ; , , .

().

: ,

' , ~ f(x)

', 1.

X -- ~ - "2. 1

,

... f =.1 ---1-,

f ( - -- - ' - .- -

< x~O

2 ' 3f ( + ,() = 1+ ,

45

Ii'1J f(x) = 1 -

:

f(x), ; , 1.

: f e, , , " ,

Iim f(x) = e -

2f () - -- -

Q ,

----' +-._-

:-

, >, f 2,

- , f(x) 6>. f = .

, .2 fI , > , (- > ., ) ,

lim f(x) = f) i:'; - '':0 +

f(x), +, fI f " eI ., f f2 , , ", , ( < ),

lim f(x) = . -

(. f(x), . c -, f2 (. - f f 2. .

46

, :lim_ f(x) = 3 f(x) = 2 - -

, lim f(x) -=1= lim f(x) , lim f(x) - - - + -

, f , , f .

= ~ , f,

.3Jim f(x) = ]

-

lir.n. f(x) = 1-_

2

Jim f(x) = -

[(x)=~ 2 - = 1R \ ( .: ].-------..1.4:I=---=:~::::!::====::... f(x)

[(x)=_X~=_]_(-)

(.3)

, f .

f(x) =1-,

]2

x-=l=O

=,

1R . xEIR* f(-x)=f(x) + :slx l, f ' -=1= =

= - (.4) .

lim f(x) = .\ - 0

; f , f = .

47

.4

:- ; f ' f

-- , ..

Xo =~ (.l) = (.2) ' , .. 2

xo=l (.3).- ; , ,

, ; . .

. . = (.4) .

Xo=~2

(.) '

2.2. ' ; .

f(x) 6>, , , .

; f

' f f (,)(,) (,) (,) .

; ; (,) U (,) , , ( - , ) U ( , + ), XoE IR > 0, (,).

48

:

f (,). ,

lm f(x)= P, EEIR, -

:' , "* , f (x) ,

f(x) - f(x) e , - xol

f( x) - el , f, < - xo l < .

"[ :

> > , " < \ - " l <

If(X)- el

lim f(x), f , , - - + (,) (- , ) 0< Ix -xol

xoEIR :

) lim = ,-

) lim C=C, cEIR-

) , > . > , 0< - <

Ix-xol

2. f(x) = J.!L .

f IR"' . ( - 00,) f(x) = - 1,

lim f(x) = - 1. .... -

(, + 00) f(x) = 1,

li!1] f(x) = 1-

------' -1

lim f(x) =1= lim f(x), (), .\_ 0 .\-0 t

f .

3. f(x) -- 2 + x2-21xl " ' f() 3, 1m =2( - 2) -

f IR \ [2j. f

U(2,2) = (0,2) U (2,4).

(2,2) f(x) f(x) =2+ l-2 =_1_ +2.2( - 2) 2

> . > , (2,2) 0

, , , , , . (3) = 2, U(2,2) f(x) = -} +2. - - 2.

---------- - ----- -----

1. Iim f(x) ->';- :'(0

f(xo) , , > - f

:

= 2)---------------~---~53

)

2

~)------:1

)

2. , , lim f(x).

-

) f(x)=lxl-3 = ) f(x) = - -

=

2 - , :2i) f(x) = l 6- , >2

= 2 ) f(x) = 32 - 6 - 2 = 23. f(x) IxlC2x-l)

g(x) = 2x~-2

:

lim f(x), --

liin f(x), - +

lim g(x), - 2-

lim g(x) - 2 +

4. f [ - ,3] , ; .

) lim f(x) = - -

54

ii) lim f(x) -2

) lim f(x) = -2

) lim f(x) = 3- -

) lim f(x) =3x- J +

) lim f(x) = - -

-1

S. :

(,,, ) < l - ,,/ < ( , ,, ) U ( , )

0 0 lim () = () IR.

-

55

2.3 e . , ; . . .

1 lim g(x)=O , (,) ' f(x) :s:;g(x),

.\ -.\11

lim f(x)=O-

>. lim g(x) = , > , 0< - l <

-

g(x)

2 lim f(x)=f f * O, , ,

-

,\",U(,\,..(i). ; "(.\) ; ', ) ('>0, 1'(.\0

) 1'

) >.

lim f(x) = , =~ > > , , - ' 2 . 0, > , - c

0< Ix-xol

() () (3) , . .

~K :

Iim fI(x) =fIEIR, .. . , lim f,(x) = ( EIR, :

) Iim [atft(x) +.o. +a,.f,.(x)] =a ,f, + . .. + a ,.I',., a t, . .. ,a ,.E IR

) [ fI(x)Hx) ... f,.(x)]= e,e~ ... (\- " 0

, lim f(x) = e *,

) [f(x)] "=I' ", - \:

:

(), Q(x) , :

) Iim () = () , " IR -

) lm QP(X = () , " IR Q(x ,,)* , - ' ( Q( xo)

() = " + - - + 0 . 0 + , + . lim = , *

-"

(), Iim ' = x ~-'

1

. :

) Im (-33+22+2),- -

) Im [(2 2 _1)85 _ 5(32_ 2)92].-1

, , :

i) lim (-3x 3+2x2+2)=-3(-1)3+2(-lf+2=7.- -

ii) lm [(2 2 - 1)85_ 5(3 2 - 2)92] = lm (2 2 - 1)85- 5lm (3 2 _ 2)92 =- - - 1

85 92

[(1- 2)\

3. , , f(x) = 2 _ 2 ,

4 lim f(x) = , EEIR, JJ,.

- -;7/lim !f(x)I=IEI

Ilf(x)I-IEI /::;1 f(x) - lim f(x) - = ,'1(-';:0

!~~ ( f(x) I- e) = , !~~ f(x) = ,

}~I~.\ 1-XJ + 2 -71 ~ }~I~\ (- 3 + 2 -7) "'- - 31= 3

. ,

f(x) = lim f(x) = lim ~IX = lim = 1, f - - -

.

f=O, , (1) 2.2 , :

lim f(x)=O = lim If(X)1 = 0- -

5) lim f(x) = , EE!R , (,)

\ - '\:0

f(x) ~ , lim f(x) ~ -

) lm f(x)=f, lim g(x)=m, e,mE!R , (,) ,\ - ' - \(

f(x) ~ g(X), lim f(x) ~ lim g(X) - -

62

) f < , (2), >

, ( ,) f(x)

2 + 2 > lim ( 2 + 2) = 27 > , -5

lim f(x) = lim J.jX2+2 = \llim ( 2 + 2) = J.j27 = 3. - 5 -5 -5

.Jx2 + 16 - S2. , , f(x) = ., =3.-3

= 3 f. =:- 3

(.JX2+6) 2 _ 52( - 3)(JX2+6 + 5)

2 - 9 +3(x-3)(.Jx 2 + 16 + 5) .J x 2 + 16 + 5

+3lim f(x) = lim --===--- -3 x- 3 .JX2+T6 + 53 +3 3

.J9+T6+5 5

f

h g. f '

h() :;; f() :: g() , (,,, )] . , _

lim h(x) = lim g(x) = , IR !~~ f(x) - - -

(,) [() - h(x) = f(x) - h(x):;; g(x) - h(x) . lim g(x) = h(x) = , lim [g(x) - h(x)] =.

- - -

, 1, lim [f(x) - h(x)] = . - .

64

, f(x) = [f(x) - .h(x)] + h(x) lim f(x) = lm [f(x) - h(x)] + lm h(x) =+ f = f. - - -

" ,

Iim 2 = lm (- 2) = .- -

~ !~~ ( 2+ )=0

lim ( 2-_ ) = -

, xE IR *

- 2 ::;; 2-- ::;; 2

/ / /,,// /

~

..

. 1

. . Ixj~ ~ ::;; 1< ~ ::;; [ ] ,

::;; l xEIR ()

4 '; . [. ~ ] ::;; . [ - ~ ,]

- [ . ~ ]. ( - ) ::;; - - ::;;- .

~ [-+ .--}- ]{- l::;; . .

() . :

xoEIR :

) lim = , -

) lim = , -

65

) , xoEIR

l-I=2I x~Xo 1.lX~Xo 1::52I -; 1::52\ X~Xo \=Ix- a l,

::5 [ - l ::5I - xol lim - xol = ,

-

lim ( - ) = lim = ,- x-~

) .

2.

lim~ =1-

()

< < ~ (.) :

(.)

2.5 . -

, ., ~~ [(2 ] + ~ )] ,

.

u == f(x) == g(u). lim f(x) == uo , f(x) * , * g(u) == f,

]jm g(f(x))= lim "g(u) == f - ' U-Uo

.

1. : Iim

- ",!!",3 --

3

1.

(x-~ ), f(x) == 3 --

3

==x-~ Y==~3 u

lim ( -~ )== - ~ * *~ lim!ll:!:!:! == 1, - "!!,, 3 ' 3 3 u-O u3

limx- .lL.

J

(x-~ )3 .

____ _ - 11m -.:.u:::.=..- == 1 u-O U--

3

67

2 ' 13 -_. .. 1m.-0

23, f(x) = ---'-IJ::....==-.-

13 f(x) = 3 3 ,

, 23 2

Iim3 =lim3~. -0 3 - U = 3lm (~ .u) =31 0=0.- U

"'

1. :) lim (2-3x+4x 2 -x J ) ii)lim(x 2+x-I)I7, i) lim ( 6 + 1 ) 3 ,

- - 1 - - -

2 . , , lm f(x) , : -

< = =

> [ + 1 ,i) f(x) = ~ - - 4, xs 3 = 3> 3I l l ( 2 + 2) ) f(x) = ' x'l= = 2 , =

3. :

) f(x) =-2 ,

2 + .+ 3 ,

< -

Jxlsl> 1 I = = -

i)lim[ 2xJ

_ x2 ( 2 - 2) ]

- 0 Ixl

i) Jim ( _l- + 2 2 - 3). -0

i) lim [( - 2)(3 +_l- )] . - 2

1im f(x) =4, lm g(x) , -2 - 2

68

) g(x) = 3(f(x) 2- 5 i) g(x)= ( f(x) + 2)(f(x)-3) !2f(X) - 11 ii) g() = --'--;_~----J(f() +

s. :--------------~

) lm ,2-13--6

4 - 16ii) Jim-~~-2 x J - 8

1--1-

1--1- 2 ) lim __-,-----

, -

2 1 - 3 + ) m --.,....::...---'---'- - 1 _

.. ( + 3) ,1- 27ll) -

...) ' (2 ) 1m - 2- +-- -2 - 4 4 - ,

xJ+x 2-5x_2 ) lim -------,.--=:..:..=....--=:.... -2 2 - 4

6. :

) lim,- - ]

.JX2+6x+9+3

ii)Jim 2 I -I / + / 2-6 1 -2 + 21

) Jim -221- 2 - -

) Iim 15- 3xl- 13x - 11 - ] _]

) lim l 2 - 2xl -2 - 2

:

3 - -/ ' ) lm -=------"-'-'-- 9 -

-jX+9 - 3i) lim ----::L...:..:-:.....::...----=:- -

ii) lim - .JI-X-

_

.Jj.>....I:..-+--'h'----_ V-"-'---l -_1_1) lm], - 0 h

x J/ 2 - ..J 8) m ---'-----'---'--- -2 - 2 ) lim-2

) IimJ~_\/x , > h-O h

) lim ..JX -.'2- .JX- 3 + 2 -2 -Jx2 - 4

:

j ) lim~., - 0 . + 2

) lim - ..JX+4- 2

} , :::;2. , ~ f(x) = 2 lim f(x) , > 2 - 269

10. lim JQL = fE IR, lim f(x) =

- -

'

. :

j) I i m~,- -

2 ) Iim - , - 3 -25-

x -4.Jx + 4 ) m -----'-____=_-,- 4 (_ 4) 1

6'2+ - ) m -----''-='--'----'------0--

"') . 2 - 1 6111 1m _X- 4 x - .J'X - 2

2. f() () = 2 + - , l m f() f .

r.-O

3. lm f(x), .1(- )

) lm (f( x) - 2x1 +x- ]) = 4 -3

) Iim f(x) + = 2 - 3 - 3

) lm f(x) - 2 = 5,- 3 2 2 - 18

4. f Iim (f( x) + 3 + 4) = 5 Iim f(x) - 2 - - 1

S. lim JQL = 5 lim [g(x)(2x2+ x -IO)] =3 , Iim [r(X).g(X)] - 1 - 2 , -1 ' - 1

6.

') . 3 lm --- ., -

) Iim - - -

" 0) ' ( - 2)111 - 1 .JX2 + 5 - 3

x S: 17 . . [ + 2 ,. f(x) = 1 + + 2 , >1

, , (2 , 2) Iim f(x)

-

70

1 2 2 + + ., S - 18. f(x) = 3x+l , -I0 >.,- )

, 0< - ] <

/ - , > , - -

, -

-:

4 f(x) = 2 Iim f(x) = + 00.( - 2) . -2

, >

4 4 2f(xM

. lim f(x) = - 00, , lim[ - f(x)] = + 00

2. lim f(x) = + 00 - 00, lim If(X)1 = + 00, lim '~ = +003. lim f(x) = + 00 lim g(x ) = + 00 , lim [ f(x) + g(x) ] = + 00

4. lim f(x ) = - 00 lim g(x) = - 00, lim [f(x) + g(x )] = - 00

5. lim f(x) = + 00 lim g(x)=fEIR, lm [f(x) + g(x )] = + 00

6. lim f(x) = - 00 lim g(x)=fEIR, lim [f(x) + g(x)] = - 00

7. lim f(x) = + 00 lim g(x) = + 00, lim [f(x)g(x)] = + 00

8. lm f(x) = - 00 lim g(x) = - 00, lim [f(x)g(x)] = + 00

9. lim f(x) = + 00 lim g(x)= - 00 , Iim [ f(x )g(x )] = - 00

10. lim f(x) = + 00 lim g(x) = fE IR*, lim [r(x)g(x)] = + ~ , ;> - 00 , <

11. lim f(x) = - 00 lim g(x) =fEIR*, lim [ f(x)g(x)] = [ - . > + 00 , f< O

;

12. lim g(x) = + 00 - 00, lim - I- = 0g(x)

;

13. lim f(x) = f, lim g(x) = g(x) > 0(1), limJ1& = + 00, f > Og(x) - 00 , f < O

14. lim f(x) =f, lim g(x) = O g(x)

1.

) ll~ = + 00-

) l+ = - 00 -

1

x=~ lim x=~ =1>0 - 2

) ( ,~ ) > I~ = ~ = -2 -_2

13 lm = lm ~ = + 00,,_~- x -~

2 2

) ( 2 ,) lI!l auv-1!- =0, - -!!' - 22

14, liI!l = liI!l~ = - 00. - !: x-~

2 2

++ 12. f(x) = = .-

f = IR \ 11 lm ( 2 + + 1) = 3 > 0.-

-

> 1 - >0 lim ( - ) = , Iim f(x) = + 00 - -

< - < Im ( - ) = , lim f(x) = - 00.)(- 1- -)-

f = , lm f(x) *- Iim f(x) - '" - -

2.7 JR iR = IRU -, + 00 J. _ IR IR, :

- 00, + 00 aEIR :

- - 00 < , < + 00 , - 00 < + 00 74

2.6. , ( . 2.6) IR iR , :

R

1. +00 + (+00)== +00 - 00 + (-00)=-00

2. +00 + == +00 - + = - , IR

3. (+00)'(+00)=+00 (-00) '(-00) = +00 (+00) '(-00)= -00

. (+ 00 > ( - 00 >04. .( + 00)= '

) Iim f(x) = + 00 - 2

lim f(x) = 00 ( . )

\ - 21

ii) l im f( x) = - 00, lim f(x) =2 \-2 \ -2

(2)= ( . ) / 12

ii ) lim f(x) ~ + 00 \ -2

(2) =2. ( . ) )

)

2

)

2. :

' ) ' Im - , = + 00.\ - 0 ~

.. . ' - 911 ) [1m - --, = . 00

\ - - ( + I)-

3. , ,

76

) ' 4 - 5 1m ,

'

1. , , :

) lim - 9-4 ..[- 2 - 4..jX+ 8

-4i) lm--- - --] -3..[+2

2. , R, , , :

2+ 2+ ii) Iim ----"'---'-~....:.....- 2 f(x)

") . f(x)11 lm-2 = - >-2 ii) lim [f(x)(2x2

- 1)] = + >-2

------ 20U _

'

1. ; . ;) (3,) f(x) ~ g(x) lim g(x) = , lim f(x) =

- 3 - )

) lim (f(x)+g(x), lim (f(x) +g(x)= lim f(x)+ lim g(x)- " -" - " - "

) lm [() = 5, lim f(x) = 5 lim f(x) = - 5 - - " -

) (2- ,6) g(x)~f(x)~h(x) lim g(x)= lm h(x) =f 2 - 2 -2lim f(x) =f

-2

2. ; ; .

) lim...[f(;) = f, lm f(x) = e2 - -

i) f (,) lim f(x) = f(xo)-

77

) lim JQ.L =0, lim f(x)=O - g(x) -

) lim f(x)=f >O, (3,) f(x) >0. - 3 ~~

) lim f(x) = , lim f(x) = - .'" -'

) (2,) h(x):::;; f(x):::;; g(x), lim g(x) = 8 lm h(x) = - 6, -2 - 2

-6:::;; lim f(x):::;;8 -2

3. , , :

i)Jim -l-+l , >- x J

- ) Iim 2 , IR. - (-)

2-5 +4i) lim - ---::-- - - -=- x.J}.:'" 3 +25

78

.Jx2 - 2+ 2 2 - 4. f(x) = , Iim f(x) a EIRIxl - -

f(x) - f5. lm ----'---'-- = , lm f(x) = eX - Xn f(x) + f -

6 [() ! +2 1 + -41-2 ' , ". = 2' , , 11m f(x) = 10. - 5+ 6 x-J

, , .

, ; , .

.

3.1

----8

~----f

f,

9

-- - -1 \

, .

; f, g, h , :

. 1 .2 .3 g, h , f .

f.79

:

- f , lim f(x) = f(xo) -

- g Iim g(x) = e*g(xo) .-

- h .

:

f . f ,

lm f(x) = f(xo )

.

f . ,

Iim f(x) = f(xo)- -. , , 1

lm f(x) = f(xo)- -

:

l'' f . f , > > ,

-l

f :- lim f(x) = lim f(x) :;:. f(xo) ( 2).

\ - - \ --'0 -

-lim. f(x):;:. f(x) ( 3)

\-'0 \ --' ('1

- .

.2 . f :

.

xoEIR lim ( ) = (,, )., - ',

,

IR. Q(x,,) :;:, . l m ~ (,, )\-\" Q(x) Q(x,,)

.

IR lim =; lm = . , ..'(- ' \ - "

~

f(x) = [ 1 - 1.

= 2.

lm f(x)= lm (x2-1) =3 , lim f(x)= lim ( - 3 + 9)= 3 f(2)=-3 '2+9=-2 - -2 - -2+ -2 +

lm f(x) = 3 = f(2), f ; = 2. -2

f :

2. f (x) = ,

,

f,g

lim f(x) = f(Xo) lim g(x) = g(x o) ,- -

lim (f + g)(x) = lm (f(x) + g(x) = lm f(x) + lmg(x) = f(Xo) + g(x o) = (f + g)(x o )- - - -

f + g ' (), (iii) (iv) .

:

,

. .

2 f , (i) fI '(i) kff , f(x o) ~o.

.

1 2 - , f(x) = +2 , 2-1 .

+2

f(x) =~, [ - +,+ 00 ), , 3 +

' [-+,+ (0) 3+ ~o. ; 2 () . . .

f(x)'= [ 1,t - , x ~ o

.

3 ( ) g ;

[(), gof .

.

, [() = ~l , -

= u =-- .-

"---.......--........----- --------- ...

3

-1

-1

)

yt 3 3

+ -1 4 2 3 ( J

)

1.

:

84

2. :

) f(x) =[ 2 2 _1.3+

3. ,

2 2 Ixl :51 34, :5 -) f(x) =

2ii) f(x) = 2 , -1< :52

Ixl>1 -+5 >2

4. :

3-2 1 ) f(x) = -

.JX-I , 2 =1

5. , :

) f(x) = [ 2x1-x -1 *] ii) f(x) =[ -4 =:;4- = x.JX-8 >4

4-

6. :

'----_._- ---- - - ---- _ .- .----- - - -

) f()=~

) f(x) = (3 2 - 1)5

ii) f(x) = )

2-1) [() = --=-='----=-3+ 5

ii) f(x) = (3 J - 2)

) f(x) = 3 2(2 + 7) - 4i;i

---.l

85

2 ) f(x)= ~ , , = ") f() _ , = , =8.

.

lt

2

) f 2.

i) f ' 2.

f_[ 3 ~ -(J+ )+ , x~ 1

4. () - ] 2 2 - ( + ) + , >

. IR

f = (2,15) .

3.4

; ; . ; , , .

( , l())) - - - - - - ~ - - - - -

' 1. iV '() Bo/zano

C f [, ] .

A(a,f(a) ( , f( ) ' C --+--:;r--,f--l----------f.:!-..L---.

. :

(1) f

[,] f() f() 0, [() =0 -3 = -3

2. ~ + 3 =~ .

, - ~ + 3 -.Jx- = .

f(x) =~ + 3 -,,;.

f(9)=~ +3-3=~ >0 f(16)=1 +3-4

- f.- , f . f . ,

f(x) = --, [, 2].- f(x) = [, 2].-- = ~ = ~ = ~ =~ =~

4 4 f .

(, ~) (-1L ~) ( 54ft , 2)4' 4;

---.1L ---.1L 3 6 2 2

1- J3 -[() 2

- + -

, ( , -t) ,( 54 , 2) f(x) < , (-t, ~) f(x) > .~ 6.;. t;j,~6(J1v ;'

Bolzano, ; - .

(2) f

[,] [() * f(), k f () f () ()

(,) , [() =k.

89

[() < [(), [() < k < [() (.4).

l~X) - k)l [,] g()= [() - k 0, g()' g()< ., Bol-zano, (,) ,

g() = f() - k = , [() = k.

()

1(0) -

' "

.4

5 6 f .

,,\\\

'(~)._-~

.5

f()

.-- --~- - 1----,

.6

: f() f . ,

f() . f = [ ,], f() = [f(a), f()], f (.8) . f() := (), [()], f (.9).

90

1(0) \-[\~ ~()j~ "

~ ~,J '() - - - ';- .- .... ""' ,,

l() '1 f (~-~ ~, . -~ ,,'. .

1(0) ) _ _ ~ '" ' '' "

.9

f = (, ) : f() = (Iim f(x) , lim f(x, f

f() = (Iim f(x) , lim f(x, f

:

.. JiU"i; 1' \ 1 \\" ' - .- --

(7) f '

(3) ' : ; [, ], '" " [,] ,

f()sf()sf( ) , [ , ] , f [,] ( ) f(x,,).

. . () = _1- -

( ,2), ; - .

.

f ;

, ; f ().

, f - 1 , f .

91

--'-"------ '-----------

. 1. = 2 - 2 -: (~ ,~ ) 6 4 2. f(x) = 2( - )( - ) + 3( - )( - ) + 5( - )( - ) < < ,

f(x) = ; .

4.

7. f ,

)~ 6 - 6 = 4 .

6. f [, ], f(O) = f(1) = , , ( , ), f(x) = (, ).

) f(x) = 4-92) f(x) = . [, 2]

) ()=+, [,--;-]

) ()=- 2 , [,+]

) f(x)=~, [I,3]

ii) f(x) = 2 2 - 4, [] ,3]

5. f ; [.] f(a) =1= f(), (,), f(xo) = f(a) + f()2

fj!

!

!

i

!!j

: ) f(x) = 3+22--2 ) f(x) = (+) , (-, )i '

1. ; f, , = [0,2] : ) f , 3 .i ) f [0,2), .

ii) f [0,2), 3.! ) f (0,2), 2, ", .. . _

92

5:v) f, ; [0,1)U(l,2], ,vi) f (0,2) ; ; 1 3.vli) f ; [0,1) U (1,2] .

2. :

XIO + l 8+3) ' ' , ' ' ' -----'----'---- +--- = -] -2

( ,2).

ii) ~ +~ = . 4 - 3-

~. = , > * .

4. , IR, - =~ - 2

(O,-f ]. ; f,g [,], f(a) ::;g(a) f() :eo: g() , - [ , ) , f(xo) =-.6~ [, ] ::; f(x)::; f ;

[,]], xoE[O,IJ , f(xo) =x~ , *.'-----_.

______ 30 I _

x ~ +3x-5

1. ()= x ~ 1

' .

>"". . ; ; , lR,= !

2. ' f '() = f(x) + f(y) , IR* f ; = , f ; IR* .

93

- _. -4

L,) ' f f(x+y) = f(x)f(y) X,y ElR f(O)*O

) f ; = , f ; IR .

i) f ; IR [() * , f ; IR .

..........

4i r [,] f(a) *0 , (,) ,

f (a) + [()-

. [ -1 ,1] r 2+( f() = , r (- , ). .

6. f,g [ , ], [( ) = g(p) [() = g ( a ) , [, ) , f(xo) = g(Xo) '

1. ' ; ; 8.00 . . 4.00 . . - \ ! 8.00 . .

4.00 . . .

.

94

8. - ' .( ) .

( ) . .

f [ ., ], ()f( )

, t (, -). , () ,

, -

2 = 1.25

f(2) = (1.25) 2- 2 = - 0.4375

(BAS/C) , f(x) = Nf(x) = FNf Computers. , f Bolzano ' [ , ).

10 REM 20 REM 40 DEF FNf(X) = ... 30 REM .40 DEF FN f(x)=x72-2

100 INPUT ; 110 INPUT ; D120 INPUT ; 200 MESO = ( + )/2210 PRINT "" ;" . ";"d; " ";"FNf (meso)";" ";"MESO "300 FOR J = 1 310 MESO = ( + D)/2320 PRINT ; " ";D; " ";FNf (MESO)330 IF (FNf ()> FNf (MESO) > ) = MESO340 IF (FNf ()> FNf (MESO) < = ) D = MESO350 IF (FNf ()< =0 AND FNf (MESO) >0) D=MESO360 IF (FNf () < = AND FNf (MESO) < = ) = MESO370 J380 MESO = ( + )/2400 PRINT ; " "; ;" ";FNf (MESO) ;" ";MESO500 INPUT ;(/)";$510 IF $="" OR $= '" 100520 ENDOk.

97

2 - 2 = [0,2], , )2 .

! 20a d FNf (meso)

2 -11 2 .251 1.5 - .43751.25 1.5 - .1093751.375 1.5 6.640625-021.375 1.4375 - 2.246094-021.40625 1.4375 2. 172852-021.50625 1.421875 -4.272461-041.414063 1.421875 1.063538-021.414063 1.417969 5.1025-031.414063 1.416016 2.335549-031.414063 1.415039 9.539127-041.414063 1.414551 2.632141-04

,1.414063 1.414307 -8.2016-05i1.414185 1.414307 9.059906-0511.414185 1.414246 4.291536-06il .414185 1.414215 -3.886223-0511.4142 1.414215 -1 .72855-0511.414208 1.4]4215 - 6 .43 7302-0611.414211 1.414215 - 1 . 0728 84-061] .414213 ].414215-2 ;(/)?

! ' 1.414214 .

i

MESO

1.414214

4.1

1 +

~ 1 - ______ 1

1f(x) = 1+- ,

(, + 00) (.)

IR. ; , , .

, . . , ; , ,

.

f , , f(x) . ,

: .1 > > , >

If (x) -

- + Q:I lim f(x) = + 00,

lim f(x) = ( IR, - +

~_l_ .

f, + 00, 1

lim f(x) = - + (

, f ( , + 00):

> , , > f(x) - f\ <

> > , , > , f(x) > .

lim f(x) = - 00, > > , ,- + > , f(x) < - .

; :

) i ' =+- + 00

) lim --\-- =- +OC>

; ' --\-- (, + 00).

) >. > , , :> '> = > '../ = '.JM, , , lim = + 00.

- + 00

) .

.. lim 2 = + 00 , lim 5 = + 00, lim - 3 =0; lim - 2 = .- + -+ -+ -+

() () _ ,

.. lim 3-JX2= lim 2/) = + 00 , lim --;-3-::::==- - + x-+e -+;> R

100

Iim -k- =0.- +

f(x) ( - 00 ,) :

lim f(x) = eE 1R, > > , , - - 00 < _ , f(x) - el <

lim f(x) = + 00, > > , , - - 00 < _ , f(x) > .

lim f(x) = - 00, > > , ,- - < -, f(x) < -.

:

) . _ ( + 00, 1m - - -. - ..00, ) lim =0 - - 00

.. Im 4 = + 00, lim xJ = - 00, lim 6 =0, Im ~ =0.X --Qo -- X - -QO --

?

1. Iim 2 -- 1 =2 - +

( , + (0) .

> . > , >

- -.

, . . 1 2 - 21 . = - , > ...:...:.:---''- - < ,

lim 2 - 1 =2.- +

2. Iim (x J + 3) = - 00 - - 00

> . > , , < - ,

J + 3 < - J < - - 3 < - 3-JM+3101

=3~ , , lim ( 3 + 3) = - 00. - - /

4.2 IR , 2.5 . 90, . . .

, .

lim g(x)=O > !f(x)! ::;g(x), lm f(x)=O.( - + -+

> h(x)::; f(x) ::;g(x) lm g(~) = , lm h(x) = , , )- + - + 00

lm f(x) = - + 00

, 73 .

1

1. lim~ = - + GCI

, xEIR*= ( - 00 , ) (, + (0)

!7 'I::;R 1 ' . ~ -- Im - - = ,

- + OQ - + 00

2. Iim (2 3 - Sx + 7) = + 00

- + CIO

, ::#: 2x3 -5x+7 =2x 3(1-~ +~ )2 2

lim (2 3) = +00, lm (l -~ +~)=l-O+O=I , .- +

3. Jm- + 00

3= 2

lim 3 22+ 2 -7 = lim>L- + '" ----2 - + 5 - + 00

:

3(1+----L. -~)3 3 22(1--1 +~)2 22

32

) lm () = ' !im - + )( - +00

) lim~=~. lim ~-+ Q(x) , -+ (

) *0

() ~ . , ( +-.!!'-" _ 0 _ + ,.

+ 0 0 0 +~ '-\. )=, " g(x)' , (1)~------~----~-------------

g(x)

lm g(x) = + + .. .. + = 1 lm iR - + "- + 00

' lm ', (1), - + c:o

lim ()=!im )( - + 00 - +

) ( , + 00) ( ).

(1), * Q()= g(), lim gI(x)=1+0 + ... +0 =1,

- + 00

~ _ g()Q(x) - g() (2)

lim : iR Iim &f&.( = ., = 1, (2),-+ - + gJ )

, pfx\ ' . Im~= lm--- +'" Q(X) - +___._, '~. Iiml - +00 "

: - - 00 . Bolzano ,'

.

.

() = + - - + .. .+ + ., ;;!: .- .-,

lim ()=' Iim = - 00 Iim ()=' lim = + 00-- - - . - + -+

, , IR , () 0, () Bolzano [.]. ,

-

lim (4 2 + - 3) = 4lim 2 = + , lim '4 2 + - 3 = + )C - oa - -

3. :

) Iim ('42-+ -2)- - 00

i) Im ('4 2 - + 1 - 2),- + OD

i) lim '42 - + 1 = + ( . 2) Iim (2)= - ,

- - --

lim ('4 2- +1 -2)= lim .J4X2_~ - lim (2)=+-(-)= +- - CIO - - 00 - - CD

i) lim '4 2 - + = + lim (2) = + , - + 00 - + m

lim ('42- + 1 - 2) .!.. + co

), . :

f(x)= (.'42_+ 1-2)(.'42-+ 1+2).'42-+ +2

-+.j4X2_X+ 1+2

( -++ )

1- 1+ -

-14

-1+0"';4-0+0+2

/xl ~4-+ +7 +2

lim 'f(x)- + OQ

f (, + ), f(x) = _ ;=== = ==-__14__1_ + _1_ +2 2

105

. , :

i) Iim 5 - 4 =5\- + ~

. 2 _ 111) 11m - - - = + ()

.-" : 3) Iim

- + CI

5 -2I-413

= -()

) lm\- - CXI

=+() ) lim - - CD

3 +42

= - ()

2. :

i) Iim (2' -5+7)\_ + CI

) lm (2x - x'+5-3x J )'( - - OD

) lm ~x+T

) lm ('3'+5 +''-+3) - + (

) Im1(- + 00

) lim J 5 ~ 3'- -00 2 +'+3 ) Iim- -

2' - 3 + 73+6

"') ' '25'+3ll 1m -- - - - -' -.00 3 - 9

) Jim~-x+t ) lm ( 2 + _ 2 - 5 ), - - 00 - 2 ., - + 00 2 - J 2 +4

') . ( 2 + 5 2 + 3 ) 1m -- ---- - 00 + J

3. :

) lim (~- 5)- + 00

) lm (";2- 4 - )Jt- +00

ili) Jim (V!)x2_ X + 3)'1- - 00

) lim- - 00

215-xl +3l-41 +8

) . lim (\&+1 - J,J;:)- + 00

'

, :

') jX-.JX 1m-+ ...;x+T-~ ii) lim- - ~+x-JXf+5+x

ili) lm (vx 2 + x + J +~ -2)- +

2. :

..jiJ+X-xii) Iim

- + 00

2 J )2 - -) Iim

- - c:

L-_ ...." ._.__.__._--' ..._~.._ .__.~~,. .~.~.. .... ..~ - . . _

106

6. .

) l m ) lim () .1-3 - +

) lim () l m ( ) . - !-~

lim (/ 2 + ~- - /~ ~ +x~ ) = .\ -- + 00

!

i

!!!

i

.J!

l

\\ (

\\

\_.

//

//

/

'-

i) () = /4 2 - -2

( I' ) l rn1'- . .

) Iim (.JX2 +x+T + / " - + ) - + 00

( 2 1 ) - ( 2 + 3) ) () = ---'---':--:-----'--"'--'----=-'- 2 2 _ ( + 3)+ 2 .

xX2+l ) r(x) = - "------ - -

Iim r(x) = IR, lim (r(x) - ) , - -

aX,lo9ax

5.1 11 . , *, .

:

* .

: f,g,h, . .. .. ...

: * - IR

(I), (2), . .. , (), ... , 2," " , , .. . !, 20 , ... , .

("2," .. ( ) l* () ,

, = (), *.

:

(.) (,.) , - IJ"~* , = ., ' .

109

10 ; . :

()

, , "* :5

, mEIR , "* ay~m

, m, MEIR , "*

,

- (- 2) , - 2 + l .

, , , .

, [ ,4,9 ,. . . 2 , J ,r -3, -6, -9, ... , -3, ! ( -1,1, -1 ,.. .,( -) ", J .

1. v= -; , * , .

*

< +7ll- 3"

3"= (1 + 2)"~ 1 + 2 ( ),

l"I::::~ < +7 =4,1+2 2 .

2. = __ .l+

*

+ 1 ,, +,-,,=~ +

1----->0'(+1)(+2)

., > ", .

3. 0.-= -" , ' l*, e e

Euler (e = 2,71 ... ), ) , ) .

) *

e"

~ =- ---

(\1+ l)e" =~::::~ =-.L < 1,

vcV+' ve ve e

, . , < , .

i) () ; .

* ' 1 ' " .

5.2

*, . , - + 00. 40 .

, , , .

1 () REIR

lim av=R,- +

, , * , > , - , * , > , > .) () , - 00

lim = - 00 lim = - 00 -+ - 00 ,v- + co

, >, * , > , < - .

+ 00 ->, .

; , V o , l - e ;; - .

112

, *

( * )

, -: .

f (, + 00), * . f(x) * ( f(Y) . , -

f(x) = f(v) = --,--- 2 + 1 2 + 1 40 .

f ( , + (0) :

lim f(X) = CE:R, lim f(v) = C - + 00

.

, lm(~ ) = 1

. ' . f 1, ( ) = -

1-

m(~ ) = 1 . . . . . - ) , , :

'

lim = + 00 l m _1_. =0 '

r.]i) f ( , + 00) ,

f(v) * , > . . .f() = --}-=, f(v) = --}v- 7- ~7 .

13

) . ,

, lf(x) = , xEIR \ ( . l ) + 00 , (f(v) .

2

.1

3

5.3 . , ' :

Iima,, = fEIR, m ,. = mEIR , lim(a,.+ .) = lima,.+ l m ,. = f + m lm( ,. ,. ) = lima,. m ,. = fm

(,.) Iima" f , 1m-- =----=..:..:='- = - , m* ., m ,. m lim ( , = ( .) " = ' '-,

, ; ; , , 1 .

.

. , . . , =( - ) '" , .

' ' ( - 1) ' !lH V * ' .. , =- +2- , .

l* ,. I= 1 ] ' ' Im- -, ' = , Im .a ,.= .

-

11 4

( - ) " ~--:---=--- < _1_, ) + 2 -

2. Im(

- _ .- 2 + - ! * -Jv2 + 1 -= -":"-":"":--":"V-V2+ + .JV2+l +

im(~+)=+ 0 , lim (-J 2 +I- )=lm . . =0 2 + J +

5.4

1. lm ' = , + 00, < > 1

= , . > , >0 , = + .

"= ( + ) " ~ + > lim(ve) = + 00 , lm "= + 00 .

< l l 1, limW = + 00 . '. . lim(-f )" = 0, lim+ =0, lim "= + 00 , lm e"= + 00 .

2. lim'\!a =1 , >

3. lim " ,..- = 1 ( ) "5. lim x,,= , lim 1+- = e"

2,3 4 , 5.

5.5 [ ., 2 ,' '' ' " , ... 1 ,

(.), . . ;

J15

(.), (~,. _I) ( ~,.).

, ;

) ,

.)

, .

,

- (++)" e, -2\' ( + 21) e.

- , = + (- ) " , ., = 2 -+ 2, ~,. 1 = 0- .

- ,.=( 2 ) , ~,.= =O -+ , ., _ = (4 - 1)- =(2 - - ) = - -+ - .2 2

- ,. = (- 1) " , . , = 2 -+ + 00 , a l v - I = -(2-l) -+ - 00.

5.6 .

PEIR, , , ( ;.), X,.:;t:" ,. - " f(x,.) - .

; .

" = + 00 . " = - 00 .

] 16

() ( .), lm = lm , = XoEJR lm [( ) "* lim f(v), f

5 5.4

lim ( +_1_)'=e - .. 00

1. f(x) = , + 00.

, () (2 + ; ), . + 00,

lim f(vn) = !im () = Jim 0=

lm +-}) = lim (2 + ; ) = lm = f + 00 .

117

2. f(x) = ~ " =

J

,

(2~-; ) [- ). 2 +-

2

l f(_I_. ) = l ( 2) = lim = 2

lim fl--1-n

- = (2\' + ; ) = l l J = J + . -

2

() = -' - .

- - / ->'c"'-'''''''''-"",-~"""=.",=="""",,,,,,,,,->i-- .....-., ..._...- _....- . ._---------

. ; ; .; :

' + ( - ) ' ) , = ----'

1' -\') , = -

\ ' +

2.

J \ , + ) , =-=- - - '

) , = \ ' : + " ') ( - ) ',,111 ,. =-- -+1

. 2 + , ) = . _ - ~\ . 5 -, 5') ,. = - --

3 !

3.

11 8

) ( - ) '" = - - -, 2 : +

' "

." \ '

) =--- '-, 7' .

2 : - + 1 ) ,.= ,2 - 3 '

) 1+2 + ... + ,. =----- - 3 :

2 ,, - 1+ 3'' ii ) , = -=---- - '---4 ' .... J ' .

) ,. = .J4v' - +2 -- 2

' +2' + ...+ v' ) , -" -=-----'--'=----'..:( !- )(5 + 2)

) , = ( +-+.- (

. . , ] )'" . -t. IIm( + -.- = e . " - \.J

00 , 1';0 -

( 7 )" ) 0. ,.= J + ) 0." = (1-7 )" ) ( - 2 ) "111 0. ,. = ---+ 5. I im~ = f >O , , -+ + 00 = ,. -+ + 00

,.

'

:: ,. =,3 ...1 lim 0.,,=_1_ .~ 3

. ] - " . . . .2. ,.= ! Im --,-- , E IN ~ , 0

,.

3.

( - ) 'i) 0.,. =

2+ 1i ) " = 2 !~( )

' + 7i ) "

4.

) =( 2 + 1 ) "f ,3 + ]

" ) (4 + 2 )"11 0.,. =3 + 1

5.

) I m ( ~), -

) .Iim (_!- ), -

) I m () - + . _ . .. . _ _ . - - ---. ---- ------J 1 ~

5.7

a ' ', > xEIR, :

'= Jim "' .

, .

: - , xEIR (q,) q,. -

( ,) , ', :

>, xEIR ' = lim ". (.) q,. -

', > xEIR. . :; ; ; ;

.

> *-

f : IR - IR f(x) = ' .

f( x) = '

> < < lim '= + 00 lim '=

lim ' = + 00. > '1- + (:

Iim ' =

\ - _. ,; .\ - .; ~, < < .

. ( , + 00). , . ; 2.

120

~.1 ~.2

f(x) = ' , < "* , ,

f" ' : (0, + 00) - IR, '() = lg" .

f "() = !g , . ;

= ' = = log"y,

' ''~ ' = , lognY . !ognx, ',

> (, + 00) IR. , > J. ; , < < .

Ilim log nx = - 00 - Iim log"x - + 00\ -0 ' lim lognx = + 00 , > .

\ - +

lim !og"x =-oo,

//

//

//

>1

//

//

/

//

//

//

//

//

"", //

/

0

=-~~""". = ..""""""'."...._-- - -------- - -

'

. :

) f (x) = (] + --;- ) ) f(x) = ( ~ 2 + 6)2. :

,

) ( ) ~ e" ) ( ) ~ ( 2 + e) ) () = I,n( - e' )

3. , :

) ( ) = + e ' i ) '( ) = 21 - 34

" ') () 3e ' + ]111 = ----e ' + 2

4. :,

) f(x) = e2, " .. I ) () = l( 2 + - 2),

i) () = )\ .

5. ; :

i 2 - 2 ) ( ) = I.~ X - .

- .

::; 1

I < x < e

x ~ e[

e ' + 2) () =

l( 2 + 1)2,

<

x~ o

6.

) Iim - - CIO

'

) Iim - ..

e" + 2e'+ ] ) lim [( 2 - 3)lnx]\ - 0

. l m f(x) , ; - ... 00

. c ' +2 ' - ) ( ) = , ' 2 '

e "+) ( ) = ' - + 4' , >

' + 4 ' +1

2. :

) , = 2 I(3 ) - l ( 2 + ) , ) , = ( + ) - 2l ,

) ,- = 31 - l ( 2 2 - + ) ) , = I n ( .J~"+I - )

3. :

L i) ,. = --,v+ 1 ") + i1 ,.= n- - -2 2.1

~. :., ., :! \.Ja ,. :=e~ a,. = .e~ a,.=e~ 50U

I - .JX . () = ---.:...~:....::..)+.J ) . ) .

) , , . ) l m () .

'( -

2. \' ( - 2,0) U (, -- (0), ;

,. = (~ l..- )"2 + 3

3. ) ( ,.) , l m ,. = -- 00

ii) ; (.) , ... 1 = Ja ,. ~ -- ] ,. + 2

4. , =

l m ,

- > - :#: /

S. ; () ; x ~ , [ , 1] ) *,

) " ( . J) , $ $ ,. *,

( . . " : ( -- ] )(2 -- ) =) - + 2 - -- ... + =- 6

124

7. AB"r " - .

,2 , " ... ,,, ,2 , . .. . , 2 , " .. . ",, 2 , ...

# ~-------------"

) , 2,, ... 2 ) lim ,.

\'_ + 00

6. ,. - (, R) ', -

) ,. , ', .

) ,.:$ :$ '", = R 2 .

/25

christosRectangle

christosSticky Note

christosText Box

6.1 ; ; ; ; 1 70 ,- ot,

; ; .- . ;

. . ; ; ;

; Ncwton LeibnZ :: ; / , . Newton Leibniz .

; Newton Leibniz. ; ; , ; f ' , 2.000 .

rv ; , , , .

; ; , ; , () ( . 2) ; ; C , () (.3) . ; .(.3 - '.4)

,,

'"1 = 1. =-

,,

~\ , ,(~

. 1 1.2

.3 , 4

( , ; ; , - ] ; ; .

' = () ; C, A(x...f(x,, ; C M(x,f(J( ; ;t: Xo ( . 5

6). , ; ; f

() = f( x) - ( )

(Ex.S 6) C , - ,", .

Im f(x) - f(&). - .... -.,

l2

; , ; ; ; .

,,- - -, ,

(,j

,

/

/ (. ). )(.()

:".. /'~ :\""~'..- '" . ' !(xl - ()- ~ -__ ~./ /

- -::: ~ ~ --------: .. ---=-- ---

.5 .6

" ( . 5 6) C ; , - - , .

Iim f(x) f(xo). - ...- - ; , ; ; ; .

" ; ; , ; ( .6) .

lim f(x) - (,,) '" lm f(x) f(x.,)~ -Jo -" ---'0- - " C A(x".f(x,,) () ; .

.. ; , ; ; ; ; f A(x.,. f(x., ; ; ;.

' .

5 : - 5 , ; 5(1) .

..:

129

( , ], *to ; ; ' ;

. 5(t) - 5(1,)

- t o

10 . 1 - 100 . ;; S , (~) .

:(t,,) = m 5(t) -5(t,,)

-" -.

6.2 . ; , ; -

, , , ' , f(x)-f(x ); - - '" ; f . .

f ' " . ,

\ !X.J(JJ.

' f " . f ".

Im f(x ) - f(x,,).-." -"

; . ; f " [ ' (,,) .

", ; ;; ( , ,,) '\ , ) "" ,... \) ;; ,\ \ ; ;; ( ,'. ) (, .. , ), ; .

130

[ (.,) = lim,-~

f(x) - [(") -

G. W. Leibniz (1677)

..!!.!- dx " ~ " c f' (xo) J .L. Lagrange

(1772) .,

, , f , ,

lim-'

f(x) - [(") "" . , f

lm'- '0' ,

f(x) - f(xo) , , , -

, ., ,

['(",,) ~ lm.-""" -

; ; ......... ...

- * h ; f(x) - f(x,,) (,,) . ; .

- = h - = - , ( () "

" '( ,,) = ("( ) - f{x,,) = - "

(,,) = 1, - (1

f(x" + h) - f(x,,)h

= lim f(xo +) - [

. ) '" + 1 ( ' ( - ) f : Il .

*- -

[ (x) - f( -I ) -(- 1)

= x l+ I - (- I ) I - 1 + 1

= - ,

( - 1) = l m._ -

f(x) -f( - I )- ( - )

= Iim ( - ) = - 2., - - 1

; :;!: .,

'( ) - (,, ) = : + 1 - / -1 - ., -.,

f ' (x,,): Im f(x) - f(x,,) = lim (x + x,,) = 2xQ , x"e lR.' - ' - " . - ""

r x.,e IR.

2. f(x) =.JX . . > .

,*0,

f(x ) -f(O) =-.:JL. ",, _,_, lim - ...; .-o ~

f .

.. , (,+ 00), x;;t: x".

f(x) - f

.. = 1. :::::; >13 - 2, '3. f(x) :::

< ,

() 1(1) =-

, 1 - = x ~ + x + , lim- -

l()- I( I ) -

>I ,

()-() =-

3x - 2 - 1 -

3 ( ) -

( )

f(I)1

f ' (I )=lirn f(x)- f( 1) ;;;3,' - 1 -

f = .

6.3

.

f(x) = Ixl ; ; = ( . ) .

" ;

,

Ixl = lim~ "" _ 1, - .-0 -

!xl - "" Im ~ = 1 - _ + i._0lm. _0 _

! -, f - .1 = . ; ; ' ' .

,

, f , .

.

Iim f(x) =: f(x",) lim ( f (K) - [(x",)~ =: .s_.., s_... *-

[() _ [(,,) = [( - [(,) ( - ,,) , "

f ,

Im f(x) - ( ,,) =: (' (",)s_... -

; . ; ( )

lim ([()- [(,, = ['(,,) ...-~

. ,

f(X)=! xJ , x ~ .:+ , > ! (\' ..= 1.

()

f =: 1. f ; = ,

lim f(x) =: lim f(x) =f(l ) ' l + =: ] ' = l - ( )s- j s_ l _

' f ! 'f(x) = , +( -), x:s > I ' lim f(x)-f(l ) = lim xJ _ 1 = lim ( + + I ) = 3~-I - - - - -

Im f(x) -f(l).- 1 - J = Iim.-1

+ ( - ) - 1

= Im.-1 '

- ) = .

, f = 3 , ; ( ) . - 2.

,.."~.

1. ; ; ,

i) ((x) = x l _ 3x, =- 1 f(x) =~ =- 2,

) f(x) = 2'1 = ) f(x ) =~ = 1

2. f ,

) (('1)=12'12+'1,9. -8,

, , :

i) lm f(x) - ().jX-..[ ) lim,-(f)' - (f() '

.jX- ..j(i

; , ; ; r 8. [() =! + , x:sI~. >1 .

!

135

gO) = g ' ( O) =O , -

'

()- , -\ *,0 . A\' f(xl = :' , ", f .

2. f :' '" -

&( ) = [ [(~ ) " :$ " " . (,,) ( - ,, ) ( ,,) , > "

3. f(XI,",1 ,3:'1 ~ - ~:'I4 -+- ,

; .; f

; .;

) ; ' o -y1

f .

5. f Iim0-0

f(l .. hlh

...5,

f( I ) = O !( f .

6. lim~ .. 4 - .'1.

f ' (O) = 4.

+ f X.yE IR (( ' }') = f(X) + f ) . f x.,E IR- ,

~ ~ f :\,yE IR :: + }')= f(x) -+ f(y) + 5 . f " " IR.

..""r. . [ f(a ) ""O.

[ ] .

136

6.4 ; ; ; ; ; f A(~,f(x.. .

6.1 ; ; (.5 6),

l m.-..-

(:': ) - (,.)'i, - ..

" Iim,-\., '

-) - {.,) - ...

f .:.. ; ; ; ,

= lim [() - [(,) - .

~ [ ' (.).

; ;

y -[(x.) ~[' (x.) (x - x.,) y ~ [(x.,)+ [' (X.) (X-x.,)

; ; ('(.,) ; ; ; r ; ; f :. r . . ' .

; [ ' (.) ~ .

, ;f(x) =Xl + .. = - f ' (- 1) = - 2 ( 6.2 . ), ; ; ; !~O "J\( - I ,2) -2= - 2(+ ) = - 2 .

. 1

A/jo 11m f(x) - (., ) + : - f . . - .. -",

-d ; .; ;; A(x..,f(x,,)

'-. '

.;

.. . x ~Of (x)= _~. .-0 .. " .2

lm.-..

() - 1(0) - "" lm.-(1- - ..fX' = Im ~ .-- -

. = lm - - = + o:>

.- (1- .. - .

l m f(x) -f(O) = + 00 . ; ;.-0 - ; ; f 0(0.0) : 1 (rxJ)

lim

IJ8

g(x) - $(1) = Iim.-1 - = Iim ( + 1)= 2. .- 1- .3

_ - Im g(x) -g(I ) "" lim lim (-=..!..) =- 1, ~-1 + ,- .- 1 ' - . - 1+ A (I, l ) ; g

' - ; g.

f(x) = .JiX (.4) , f

I,'m f( x) - f(O) ' ,rx ' - - - - = lm ~= Im -- ",, + :x>_+ - . _0' - '''[

I,'m f (x) - () " ..::fi... ' - = 1m = 1m - - = - 00, .4. _0 - - .-0- _ _ ..-= 0(0,0) ; f ' ; ,

-~~~ __.'='~' 'T~' _

j>.' '-/

2. , .

,, = 2 ) () =..l...,

) (x) =~ ,, = 3

) f (x) =2x ! , ,, = -

) f(x) = 2" " =

. ; ; ; ; ; ; ; f ; A(xo,f(x,,),

()

,._---~-~-~~

.

l) HV)

--,, ,

, , ,

-.

___ , ,-, , ,, , ,f ---1--- -,

4-,

'\ , 3 4 J . -i . . ; ; ;

; ; f ; \' ~-.

\:$0

iiI f(X)= 1~ + 3

2 ' f .1

~ . r. ; ; ; ; ; { .\.....).

) [( =- ' , : , . =0

) f(x) == ,( . - 31 , = 3 ,,-, \ ) ( _)==, __- , - [ \

-- -/ ,r,j.["-+

- J.,fI4,

T~ 1; : ; A(~ ' - ~ ) ' -1- .

3'. ; ; (()=

(+ ,) , ; -,

. 00 - - .

+ , < ;. f(x) = : ' ; . , ; -~ x ~l ; A( l ,f(l) 4 - -2 = .L ~ ___.J

6.5 f ; , f .

' ; f .

f(x) = 1 + napayroyiOIHll (6.2. . 1). .

f(x) = .JX (, + 0: (6.2, . 2). ' . ( , " () .

' , -

f 1\ ; f . ' 1\ ~~ 1\ f ; . = f(x), ; ; f ; ' 1\ k 1\~ 1\ ~ f(x) ' ( dd

X' )r,jdx dx dx

141

J1. ) =(' /

, " IR, * "

Iim.-.

f(x) - f (x,,) - "

f(x) - (,,)- "

c-c~--- = -"

~ o. (C)' ~ o

2. r(X) '"

. " IR, 4: .,

f( x) - f(x 2 ) '" - ., - .,

=1,

Iim.-.

f(x) - (,,) -"

'" Im 1 = , () ' = .-.

3. f( ,~ '" ", " , . r. > , /

( ') ' = - , IR, *

f(x) - (,) - "

( -" ' - + X -1x,,+ ... + ., - ) = "

Iim'-.

f(,) - (,,), - "

"" Jim {X - ' +X , - lx.,+ ... + x"V- ') "' x.,. - t + x.,. - r+ .. .+ x." -t,,,.-.

( ' ) ' = ' -

4. f(X) "" .jX [ , + 00) (, + 00)

1r::;- ( , + 00)2

6.2, . 2.

S. ; f(x) = g(X) =

() ( ) , = () [} ' ~ -

auv x +h + x2

h( + h) - = -=- -=-_ _

hf (x + h) - f(x) =

h

() , xeIR h :;t: D

2 x +h 2

JL2

J!..2

h

2h2

lim ( +....h... )"" ,h_ O 2

f ' () = lim-

f( x + h) - f (x)h

"" l ' =

( ) ;

g ' (x) = lim-

g(x +h) - g(x)h ""lim

( + h) - =h

_ 2"!!" . ( + h ) . 2 "" 1m~_ o h

h

h

"" - ' = - .

143

6. r(x) =lnx, (, + (0)

(lnx) ' ~_I

, xe(O, + (0) h *0, + h >O :

f(x + h} f(x)h

,

, " Iim ( +--) = e , h-O ~

h

, ,

" ( 1)"1im In(, +--) :lnlm 1+ - -h- O ~ h_O ~h h

=lne;;; 1,

f ' () ;;; lim

f(x +h) - f(x)h

. - 1 -

,

---------

) [(x ) =~ = 4) f(x) = Inx = 2

= - !i) [(x) = x ~

) [() = = ......6

-'

~' :::: ; ' 0 ' 0 ,,,

l-- ~144

r-2 . --~~ ;'~~;~ , ~O , ) ("')=1x~

6.6 . ; ; ; .

; , .

, ilt.

1 ; f, g q.:!~~~~~=3,'~~~'~O,-"X~oE~ f + g .

(f + g) ' ( ) = f ' (.) +g' ( )

, * (f + g)(X)- (f +g)(x. ) f (x) +g(X) - f(xQ ) - g (x.,) _ f(x) - f (xo) + g(X) - g (x.,)

-., - - "

; f,g ."

lim (f + g)( x) -(f + g)( x,,) = lm f (x) - f(x,J + Im g(X)-g(Xo ) = f '(x,,)+ g' (x,,)- - - - - -

(f +g)' () = f' (.) + : ()

, f, g ,

(f(x) + g(X' =f' (x)+ g ' (X)

f], f2, , ,

(f I + f2+ ... + f.)' (x)=fI ' ( ) + f2' (x)+ ... + fx'(x). l

. : ( l + ) ' = ( + (}' = 54+ , xe IR

(lnx + + + 2)' = (lnx) ' + () " + ( )' + (2) ' = _1_ - + 3 2 > .

146

"

2 ; f. g ; " , f g " :

(f . g) ' (,,): [ ' (,,)' g(x,,) + [(,,) . g ' (,,)

, XX~ ,

([ . g)(X) - ([ . g)(x,) = [( ' g(X) - [(,) ' g(x,,) =x- x~

=f( x) ' g(X) - f(x,,) ' g(X) + f (x,,) ' g(X) - f(x,,)' g(x,,) = - "

= f(x) - f(x,) . g(X) + f(x,,) . g(X) - g(x,) - - .,

f,g ; "

Im (f g)(x) - {f . g)(x. } = lm f (x} - ,,) . \im g(x} + f( x.) ' Im g(x) - g(x,,) =.-..., -. ' - '0 - ~ '- ' , -... -.

: [ ' (,,) ' g(x,,) + f( x,,) ' g ' (,,) .

f,g ; , :

(f(x) .g(x)) ' = f '(x) 'g(x) +f(x) 'g '(x)

, (n) ' = () ' l + (l) ' = Inx + , (, + ) .

; ; ; .. , ;; ; :

([( , g(x)' h(x)' = [ ' () . g(x)h(x) + [( ) . g ' (, h(x) + [() . g(x) ' h ' ( )

, ( ! Inx ) ' = ( !) ' lx + x1(lnx) ' ; + lnx() ' =

= 2xlnx + +

f a eIR, (2), ;

[ ( ' )( ) , = ' ' ( ) ,

3 ; f,g ; g(xo) *,0,

t " .' - - " :

g g

;)(-.!.. )'( ) = - g' (Xp )g [g(x,,)] ,

'" ( ~ )') ~ f ' (,) . g(G\~:jIo) 'g'(,) .

\' f .g , g(X);t:O,

( ij (_1_)'_- g' (x)g(x) - --r.wJT( ) (...&L)' = ...r.!& g(x)- f(x) g' (x)

g(x ) [ g(x)] '

,

(_1_ )'__ ( ' + 1)' _ - 4' ' + - (x ' +I) ~ - ('+ I) ' xeIR_1_ ' l- ' 2

(I ~ ) ' = (lnx) ' xl- lnx (x2) ' = _::__-,--_ _

' (XI) 1 '

148

1 -2 ' , -.

, ; . ; f(x) : 1+ \ g(x): - Ix l . ' ; (f + g)(x) : 2, (f g)(x) : 1- 2

( _ ff )(x) (11 : 11:1) : - 1, . 1[1. . :

( ." + ._ I - 1 + ... + a lX + ) ' : . - + (- 1)_ l - 1 + ... + l .. . ( - ) + 2 ' -2) ' : - 31 +4 .

2. QP(X) ()

,

( ' + ) ..

-

=

( 1 + ) '( - ) - ( - ) '( 1 + )( _ }

2x(x -I) -(x 2 + l) ""( _ 1)2

3. () : - , xeR*, l*

( _ ) ' : _ - -

, () ' _ ( , ' - ' - - , (x ~ ') : - -; :~: --- : _ X ~ - 1 - . , 1' (; ; ) (x V ) ' :- ,

(x V ) ' : KXv - 1, x ElR , Ke:Z

; 1 : , () ' : ( l ) ' = O : O - XO ~I

4. f( x) "" :

, () :--,-

'!

. ; , , . '2 2(x) ""(~) () ' - ' () uv + f(x) ""

() ,

t.

) (()

) () +

(2xlx + x)(x 1 + l) -2 _( + 1) 2 -

) (.+ oc ) f '(x)=( ) = ( 2) ' ( + ) _ ( 2 + ) ' x 21nx

2 + 1 ( + I ) l

2xlx+ x ' +x( + 1)1

) R \ { + ; : }. f t(X) =( l + ) ' = + ) "- + }() ' =

= - ( +)( - ) ""2 - + - + 2

0 = + - - . = (- )' +)

2 ( _ 2)

.

2. f(x)= x g(x) + - -. g(x.) :3. g ' ( :.) = '(.,)g(X)

g ., g(x.,) = 3*0, ( '.,)

' >

3. :' ; f(x) == + + 2() = ! + + , "* . A(x o.f(x.,). 8 (Xo.g(x.o) .,...., ,

; V , ' ( ) == 20..+ g '(x., ) == 2.. + , . ; f ' (x..) =g ' (x.,) 2 + = 2.. + 2( - ) == - .,=+ . -*.

,----

'

. :

) f(u) = ]nu + ~12

) f( x ) ~ x ) + lnx + .J5

) f(x ) ~ + l 5

) g(t) =.Jt + , >Oj) f (x) ", Inx + .JX - 7

2. ; ; ; ; ; ; f A(~.f(~)) .

i ) f( x) = x ' + x -6 ~ - - l ) f()=+v", x..=~4

3. ; ; _ Xl + t 4 1t"t ;XOVOl .

4 . :

) f(x) = ) f() = 5 -~ ) 5( 1 ) = +...!....,.1 13 2

\') f(~ J;: ( + ) \'J=( ' - ) ) g(t) "' tlnt+5 t)

) f(x) '" ";;: ;

5. ; ; ) [() = { ' -2); ., = 4

) f() = ", x., =~~ . . 8. _

151

6. ; ; ; ; ; ;

; f(x) >: ..l... 2 - - '- - 3 4 2

i) _+ ) " 2 - 6.7. . .

' , - , +- . xsO 0

8. ; ; ;

') g(X)= - -

'"'

) (() ..;: + Jx, ) ( ) = --,

) P(V) " ..f... , c

,+- -

+4i) f( ) ,. :-3"--'::-:-

) f(X) . - '-

) f(x) =

9. f IR.

11) g(X) = 1+ Kf(K)-.' i ) {X)=~,,

10. f(xl = 2(x + ) B(X)=~+ + IJ-~ ; ; ( ' - - ! + 1l , ' . l1 f ' = g ' ;

11. ; l ," ; ; f ' ,

, ,.) f( x) = x - -

11) f(X) "~'

,

il ) f(x) = 2........L

---..

F m,: - - - ~ - - ' -

12. ; ; ;. F r t '" , m;;.

ml m. f ; k ;; k -,

. /)' df .d,

S:!

13. " 5 + + ..1... , IR- ,

'

1. ; & ; ; ; f{x}= - { - 2,2).

2.

) SI"' I + 2x + J x1... ... +X~- 1 l -

) S: ,", 2x+ 4xJ+ 6xs + ... + 2vx l - I - .

J. " f(x) = ~ -+ -.lL (, .2!.... ) '1'1, 2 ( ' ( ) ..

4. C1, C2 ; ; f(x) a x1 -4x+ 5 g(x) .. x2 + 2x - 4 , C1

A{3,f(3 C l

S. (() =. +. _ - + . . . + + .* pl,Z. .. . . IR, ; .; l

~ =_'_ +_'_ + .. . +_1-f(x) -l - : - .

6. ; . . . ; . ;; f{x) = xJ + : + + '1 0(0 ,0) , (, ) ' .

7. .; ; Y=~ * ,

. ; ,\ : ; \1 .

8. ; . ; 3 2 + + ( = g(X) = 22+ + '10

,

'1 ; 2.

L153

6.7 -

.~ (J , . . .. ..: "

; VQ ; f(x) = ( - 2 + ) ' g(x)= ln(xI + ) . .; ( '( ) g '(X )

. 0 \ . , ; .

1 f g f(x), ; gof

(gof) ( ) = ' ( f(x . ( () l l )

g ' (f( x ~ du p . fI . ) ; ; , f :

i) [()-J' ~ () '" - ' ' ) . , [ ( ) - 2 + 1)' ] ' = 8( -'- 2 + 1 ) "( ' -2 + 1) ' "' 8 ('- 2 + 1) ' ( 3 :- 2)

._) [ ', , ) ]' ~ ',,) [(,011 2.ji) ,

. (f,Z+!)' ~ (, ' . ) - = 2, =-.."==,,2 , : +1 2,l : +1) [ ( ,1J' = ,' l )" ' , )

. . [ (3 : -2 )) ' =-\'(3 :- 2)- (3 : - 2) ' = 6 - ( 3 : - 2),- ) [ , ,' )J' '" - (\ ) " ( ' ( )

. , [ )\' ~ ( x ' + )J' '" 2( -' + ) - [ ( -' + )] ' ==- 2(' + ) ' ( ' + ) ,( ' + ) ' =- - 3 ! [ 2 ( ' + 1)] .

,,, \', "tlltI oto 1100 l, t:- vy.: tvo ~ .

' 54

2 f f '(x.,):;t; O, ., , f - I f(x.,) .

. ( ) (2) ; ; .

, /111. xER,o

('nlxl) ' :-'

, > (lnx) ' ",,-'- .

< (lnlx l) ' "" (In( _ ) ' :; (- ) ,-

( l f() I ) ' ~~ () 1) . . (l I - I ) ' "" (l -XZ) ' - l x :;t; :!: 1 - "" l _1

:-

1 . c' :

(c ') ' ""e'

e' yi ; ; 1nx 1ne' "" x (lne ') ' ""() ' (Ine') ' "" 1 (1)

''" ' ' "(Ine' ) ' =~ , o~. = (e') ' = e'" ,

3. f(x ) "" ', ( > Q :;f: 1) :

( ') ' "" '

, x e IR a' ""e,I"\ (2)

( ' ) ' ""(e,In") ' ""e,In" . () ' "" ' -

4. , x e (O, + 00) (r eR) :

( , + 00) "" etIn" ( )' = (e, lnx) ' ""e,ln' . (. n)' "" ....!.... "" ' - .

; ; .

.4I .

i)f(x) ""VX - 1 . ,) f(x) =e I H

, ,

i) f(x) ""xJ - - 2" (, + (0).

(-'-) ' ( - -'-) ' "!"' - 1 -..!...- 1 _2. _2-f ' (x) = 3 - 2 ""3X J +2 2 "" 3 3 +2 2 =1 1

= 3{! -+ 2x"JX2 , 2, 2 , 2 l

ii)f '(x) ",, (e l'" ] ""(e I+: ) + e +< () = 2x; ~ ' + e l =,

""e l H (2 +)

i) x' ''''e,In,

(' ) ' = (e' I"'}, = e' In'(x lnx) , "" X ' ( l . x + x . ~ ) "" ' (1 + Inx).

iv) (e""' ) ' = ?" . () ' = e ~'_

l . ~ ; f(x~= =

x EIR*

( +) = ( )' + + 1 ( + ) = 2 + + (-;- )-(-;-) ' == 2,

- ,

=

[() - [() f

= 01(,) - [() - f '(O) = Hm, -"

,

+ ! ::5 l l ~~ Ixl = 0 \\ ,~ (, - ) = . /_';-;-';~'9="7'-:,-,-+--+") / -

[ ,() =( 2 ...!... - =

#1 Leibniz Leibniz ..i!...

dx ; . , ; ; y=g(f(x)

u = f(x) = g(U)

.-.Ldx

157

; ; y""g(f(h(X)) = h(x), u = f( v) =g(U) , ,

..l!L =..l!L ..J!!!... ..Jh...dx du dv dx

; ; ; ; ; ; ; .

! ..l!L . ;dx; ; .

( f(x) !g(x)' = f ' (x)! g ' (x)

(, .f(x ' =" f' (x) ( f( x) . g(x) ' = f' (x)' g(x) + f(x) ' g' (x)

(-'-) , _ - g ' (x) (...fuL) '_f' (x) 'g(x) - f( x) 'g ' (x)g(x) ---r.wJ' g(x) - [ g(x) ] ,

(gol) ,() = g ' (f(x) ' " ()

158

(c) '

() ' =

(') ' = .- , VE IJ',l - ([f(x)]'j ' = [ f() '. f' ( )

(') ' = ~ > (.Jf(x) ) ' = ( '() , f(x ) > 02 2.JT\x'() ' = - (f( ) ' = ouvf{x) ' ( ' ()

(} ' = - . (ouvf(x) ' = - f()-f ' (), 1 (Inf(,) , = f; , )' f '(,),(n) =-, f(x) > 0

(I) ' =_1 (In1f(X)!)' = f;X} f ' (X),, 1 ) ' () =-- (f(,) = . f '(x) ! ), - 1 ( )' - [] =--. () = , . f ' (x)

- 'I()

(e ') ' = e' (= (I ") ' = = r"l. f ' ()

( ' ) ' ::=' O"lno (0 '1')) ' = a I"r. - [ ' ()

( ') ' ::=. T x, ~r . IR . (lf('I] ') ' = t [ f )' . ( ' , ) , f(, O

159

_________ _

'

. ; ; r ,8

[ (1) = , g(l) =2, ( ' ( 1) = 4, ( ' (2) = 8 g ' (1) =3, (og gof .

2.

) f(x) '" 3 lx

) .) ~ (3 - 5) ) f( x) = ( - )

( 3, - 1 )' ! ) () = - - -, -2

3. ; .J;; = l : \. \'0

i) ( )= l - 11. ,* 1 ) f (x) :: x j3x - SI. ,.,1..3

4. JII

i) ( x) =~"

) f (x ) = e' +c "2

) ( ) = 2 ' + : ) f(x) = c' lnx

" ) f(x) _ x l'

S. , ,

( x)= x~- ~) () =~ .

= W +..;;) ( ) .. ~2x - 6

) () =~

6. ; , m ffio

m ~ ~l - ~ dm

du

'~.

7. f rn., i) f , f '

) f , f '

8. f rn.,

g(x) ;: f( )

'

g(x) ;: (f() 2 ) g(X)= [ f()] 1

1.

~ f(x): ~

2. f() =' () g(x) = '. () , ~ , 2: 2

) ...E.L = ' - ( + )d,

3. ; ; ; ; ; ,f() : 2 + , (,2)

; + - 2 = .

161

6.8 , f

{' .J!L dx

d'f(",)dx'f "( x.)

; f , . ' " , t f a " .

- ; .. n: ; ; f ..

d' dx1 f(x) . _..,.

f '(x)-f'(",) - ",

( ; ; , f .

f .

, r , r-(x). . , " f.

;: f(x), f N(X) ;

_ , ~ d 1f(x) d dx ; f .

[ " = (f ' ) ' =..!!-(~ )=..!!.'!...dx dx dx: , , /, 3. 4 . . :

[''' d'fdx J d' f f'" dx ... d' fdx'

1. f' (x);:vx - I ,

f(x) ;:x. > 3 f "(x) ;:v(v - l)x - :. f ())(x) =v(v - l)(v -2)x - J

1. 2

2. f(x) = e' f ' () = fU(x) = f(J)(x) = ... = f (V)(x) = e".

3. f(x) = a ' , (> *l ) f ' () = ' , fU(x) = ' ()", f (J)(x) = - . () ! , . . .

4. ; () f()= ( ) g(x)= i) f ' (x) = ouvX, f ~ (x) = - , f (]}(x) = - , . . .) g ' (X)= - , gU(X)= - ,

_

'

1.

i') f(x) _ X+ 2 ~)() (32)'- 2x -l / ..1, = +

2. " f"(x,,) =O.

3. g IR, ; f,

, , ' "(\i) f(x) =~<

prf(x)= g(e' ) u>f(x) = x ' g(x ~)/1~ f,g ; f '(O)= - 1,

d ' g '(0)=6, dx~ (xf(x) + xg(X 0

5. ,

[(0) =4, f ' ( - 1)= 2, [~ (2) =4 rJ)(l) = 6

163

'

. Y/O~ ( lN' ) ; ; f(x) : _I-

2. ; () ""

r>l() = ( ; + ) . VEIN'.

3. f ; ,,;; ,

i) (> =! )('+31 + 2 . .+Sx

>< 1x~ 1

6.9 , ' '11 ; ; 5(t), ; ; ; 5(t) ; . ; ; S [ .

lm 5 (1)- 5 (1. ) ; 5 ' (.)1-... t :aiiA t 1,.d, , ;; f(x) f " . ) " f ' (,,) . , . . - .-. - . . . f ' (x,,) ; ; ; , ; , ; ; ; (

) , . . .

t :- ; ;

t = gt. ; ; , t, u '(t)= (gt) ' = g, g ; .

164

- .

s,

- : _ +

e--.. Q = Q(t) -Q(lo ) S = - to ; ; [ . ,]

~ = Q(t) Q(to) - , -. ,

, . '

; ; Q .

. -

() = lot' ~! , 10 - . . ,

, ; ;

dl - ~ ( ) ' 1 - ~ 1~ = , -2 = - 2 Ioe 2 = - 2 I ~ .,Je' .

lim ~ = lim ( _ _1_ 10_ 1_ )=..=..1 Im _ 1_ = 0, ;,- ~ ... dx ,- ~ ... 2 ~ 2 ,- ~ ", ..[e' ; .

165

2, \' I) t. [\'11';

. \'l 5 cm ".; l) .; ; 2cm " ....("c : ( .; ; 10" , . '

.. - _ .- -~~.:: -:, .; " 0 1 ..

' ( ..) = 411( ( I,,)) : . r ' (.,) ;

( ,,) :. 5cnl " ' ( ,, ) = 2 9!! . .; ; ; ; ""

, ; ;

;+ ) '

,

V'(I,,) = 4 -n - S: .:!. = 200n ~' lscc

3. ll. \' , ; ll 1: ( ' : ' . \' '' (' \'0 S(I) = 20IIvt , SO ; ; . li:tLI '

_ . _ _ 2[, .,- " .; 1. = - 3 - 'i('C ,

u(t) ""S ' (t) "" . 2 , u{ ~l ) "" - :! (_23 ) = - 2 ~ = - ,\''J ; . 1LlJ.J.:L.I.lJ,(h'1.tlJ.J.'.LL

;; ; ;

( )::: ' ( ) -= - 2t ,

' - 7-, , '

sec~_ - - / ,J, ,L ",

r5::: 5(t)

l-4. .; ; {) ( ) .\ ," 6

; .; ; ; :( \ ) = + - : + ., . , . , (x) = ~ . .

:\0 PptOri l ; ; () = [( ) - -

iN>'

( ~ ) ,

() =+ XJ - 20xl + 6oox + 1000 () = n" . ; ; ;

' ) = ' ( ) - ' ( ) = - 3 2 + 2 - : - 3 2 + 2 + - .

- 3 < . - 3x ~ + 2x + - > , - , , < < . . . ; ; ;

: ; ; ;

T'(,) ~ - , '+ 40,- 80 ~ - (, - 20 - '220 )(, -20 + '220 ).

; (20 - .../220. 20 + .../220 )

_____ ___ _

'

. :; ; 1j/OU ; :; ~ ; u1tlo:; u-./3 .

2. ; 1'1 & sec ,

S([) =I J _..1.... :+ 151 + 4, Os ls52

) I l 9 scc;

3. ; r .. 7 - 2 ,

O:S1 5 + . ; .

67

--

4. lX

3m 6

. 6 ; v " 4 m 0,8 m/ sec.

,..., -_ - _'"::>_-- - - -- - _ _ - _"' ,....

,.., _ - - ,.. - - .... 1-r~ _ ": _ ~::;....!..&_ _ ...... ;,.... _ _ _ "V

-_ .......... - ., ...... ----_ ...... ~...,

S. , ;; 3 cm/sec 2 cm/scc voOX

1. " ; ; \- 2 km ' 3 km/h. ; ; _ t ,

-t .

4. ' 360km/ h ; 3 km . ~ ( = 2 km. ; yi = ;

",

"

._ _ _ _ ..JK

,

6.\10 , \

, . , ot ; ; ; ; . . .

- ; ; ; , ; ;

r

169

- ; , : ; ; ; ; ; ; ; ; ; .

f , " , > .

(,, -,,,+) f(x)sf(x,,). , > ,

x (x,, - ,Xo + )r (X) ~( x,,)

.. , . (,,) ; ( ; ; ; ;; ; . .

Q

. 1

(), ; 1.7 . .

:

- ' . , f( ,f() (.l) .

- , ; ; f ( .).

170

; 1:0 . ', >

. ( - , + ) C , , =: [ , ) (,) .

...

_ 3Q

- ~ x~+

Ferma t '(> ; ; , .

f : - IR ; ' , - . - -

f . , , (Q - , + ) C , ( ) ( -, +)

f(x) =S f (xo ) t'(x) -f(Xo)=SO

( -.) . f (x) f(& ) ~O. -

[() [(.) (, + , :$0. -

f , f ' (xe) =: lm f(x) -f(xo)

-", - -

f '(xo)=O .

:

1. , { ) = xJ ' () =: ; (.2).

.2

171

2. ' ' . , f(x) =

=, (.3) .

~.3

..,

3. , ;, ; , . . f(x) = + , ~- = - . ( ' (- ) =- 2#0. ( . 4).

-,

. 4

,

f , f'(x) = !' ''. ; ( f. :

( (,), ;

, ; .

5 ; .

( ) ; ; ( :; l '; &

,,

,

,

.5

, , . . ; , , , , , .

:

; ; ; ; f = [ . ], .

, f, ; ' = [ .] : .

' ; _____

~ K ______ (f ' ( ) = )

~ f '(x)

. 2 < ]) f(x) =

, x ~1

173

i) ,

; f ' (x}=O - 12 ' +24 : = - 12x1(x + 2}= O. , = = - 2.) =l:- J ,

f

(

2.( ' ( ) = _ _1_

.'

. < 1

.> 1

f ' (x) = O 2 =, < I , =

= ( ( 6.4).; ; l = = l .

2. h. . . m/ sec

n h(t) '" . - +gIZ t sec. \' . .

h(t) ; , . ; ;

h ' (t ) = O . - = . () ( ) t = J:!a.. .

g

, ; =...EL .g

Ra/fe

'; ; ,

; . f '(x}=O,

; ; - , ' . .6 () = () .

,, , l)r- - -- -- - -, ,(, )

a , , ,

.6

Rolle

Rolle f

i) [ , ;.... ) (,) i) ) = [(), (, ) , [ '() =.

f [.] , ' ()= [,] . f , . f [, ],

- ; ; "" {,) , [ , ]

" ; [, ], . . , = = ,

f(a) = [(.) < f(, , ) =[() , ( ) ::: (). ; ' f < ( , ) , Fermat , f '(x,) :::O, , .

1. ; ; RoIIe. , ( ' ( ) "" .

175

) f(x) =~ , - 1,1] ) f(x) =x J , [ - 2, ] ,

i) ) = J, { - 2,2J ) f(X) = { - + 2,3 - l s xS O

O

; + = (, ~ ).

t ; Rolle . ; .

r

i) ; [ ,] ) 1lJ.i 6 -

; ; ; ; f , ; .

" ; Rolle , ().

. , l - ] :S - ] .

::: , ; *. . . < . f(x) ::: ; (.)

(.) . , ; , (,) . - ( - ),

[ - = l (- ) l u s [ - ] .

2. x+l :sc :sxc + I.

> 0. + :Sc' ::sxe' +

x::se' - I::sxe' I:s e' - l ::se' (1)

f(t) :::( ; [, ], ( ,) ,

e'- eO ' e' - l-=::-~'- = ' ( ) = .' (2)

e' ;

'

. , f(x) '"

,, = - .

1.

-

1=i) f(x) = - -

ll) f(x)= 2Ix-I I+ 3

3. ; ;

) f(x) - 3' -l3~ ) f(x) =..::l (3 - 2)3

4. - 3 + =0 (- ,).

5. )' - xJ = 5 (1,2).

6. '+ + = , > .

7. ; ; ; ; ; ; - 1,1].

i) f(x) _ lx 2 _xl ) f(X) =1 4-X),3

I xl :t: -

. ( \ x,ye lR := OOu:

e' - e'11) e'<

, '

1. ( 1. [.] () " ( ) , ( t 1. (,).

2. ; ; RoIle f(x) '" [, ] , '" - (,).

3. 3 + 2 s= + (0,1).

4. f [0,1) () < 1 ( ' ()* XE[O, JJ. ;x"e(O,J) . ( ") '"' ,, .

5. ~ +!!L:.L + .. . + .J!L +~=O, *, + 2., ' +,_ I ' -1 + ... + lx+~,",O (0,1).

6. ( [0,4], ((0) = 2:S ( '(x) :SS (,4), 9:s:( 4):s21.

7. ' t l . , .

8. :

) 2 -~ < In2

6.11 t t ; ; . ; XU .

1 f ' ('() "" . f .

.. . ;t-x".. . . >", f ; ; ;

... ] . ; (" ,) ,

f(,) - ffx.) = '()(, - .) . f ' () ==

f(,) - (.) = f(,) = f(x.), f .

' , .

; f.g ; '

( '(x) =g '(x). c , f(x) = g(x) + c .

2. . ~K . .. f(x)= ( - ~ : :~~ f '(x)=O f IR .

. f:(O, + () - R ( l (:;) ,, _I- f(e) = 3

(, + ) ( (1nx)' = _1_ . Eot ( ' () :: (1nx)' ,

f(x)= lnx + . f(e)=3, Ine + c = 3 l + c = 3 c = 2. f(x)=lnx +2.

2. : - R '=. I(X) = tt",

' ( ' = f. F(x) = (~~) , xe R, , F' (.) ~ f ' (')e' - f(.) (o') ' =

(e ')2f(')e' - f(' )e'

" =0,

F(x)= c 2..:c f(x) = ", c . ' f(x)=ce', f ' (x) : (ce' )'= ce'= f(x) xeR. f ' = f.

'"'

; ;'( ) = : (. 2) :- (- 00,0) , ;

' ( ) = 2 . ' " .

- (, + 00), ' () = 2 , f " .

,

.2

,

2 f, [ . ] :

) f ' (x O ( ,) , f [ .] ) f '(x) < O ( , ) , f [ . ] .

": (, , , < X~ . f ; '; ; I ; (X "X:J. ("; ) ,

f( , , ) - f(,,): '() (" - ,),\) f ' ( 1 - , >, f(x:) - f(x,O f(x ,) -::: f(x:),

f [ . ] .

) f ; [a, pJ .

: . (2) ;

( ,) ( ,) ( .) . , . , I) = ' ;

( , + 00) ( , + 00) f ' (x) :::: I ~ > 0. , 2"r:x (2) , f ; [ , + 00 ).

2. ; ; (2) . , f(x)= ' .; ( - , ; f ' (x) > 0 ( - 1, 1), ( ' (0) = 3 0)= 0.

183

. r.;tri / :

) f(x) :::: In- I- ,

) (() : , (- ~ .-f- ) ) Hx) :~

) (. + 00) '() ""( _ 1m ) ' :::: _ _ 1- 0,

( - ~ . ; ).

) r ",

f ( - 00, 1J [2, + 00) ; [1,2J.

; f ' ; f ;

- - +

f ' - + ~ -f "---~ / ~ f ( - 00, - ] [ , + 00)

[ - 1,1] .

3. xeR e' ~ 1 + .

f(x) = c' - 1- , IR xe IR f ' () = e ' - . f '(x) =e ' - I f ' (x) = e' - I > 0, f ; [, + 00). {, + 00)

f(x ) ~ f(O) e ' - I - x~ I - I - O e ' ~ I + x; xeIR e '~ l + x.

_

'

. ; f,g f '(x) = g(X) g ' ( ) =: - f(x) IR , (f() + (g () .

2. f( ) = ...;;+: , g g( - 3) =: 1 g' (x) =: f' (x) xe lR. .

185

3. f

) ( '() =__ (, + 00) f(4) = 2-1, ) ( ' ( ) = + xe lR f (+ ) = 0

) f ' (x)= e' ( + ) xe lR f(O) .. - 34. ;

) f(x) =-

,,) f(xl = ,,) f( x ) = "jX ~ -4X

5. ; :

l fIx) =c' -X

j,, ) f(x) = x'

.. ) f() 10,11 =- -,

,'> f(x) = In(inx) -

"ii) f(x) = -,-,

6. ; ; (( ) = 2 ! + : + fix + 5 ; IR .

'

. ; , ;

::; ; ; .

2. ;

, ,,,,

-'-8 m,h,

7/7/6/777

J . f(xl = ' - 4 + 2. f{x) = O .

186

. ' ; ; . ,g I \VJ g\v/ > g "' .... '

) f(x). - ) ,

( - ,) ,, = - .

- -; ( - ,). - (. + 0: , = .

187

)

f ( , ) ; x~ (, ) .i) f ( ,., ) ; (..,) ,

f(x,,) ; f ( .) .) ; ( ,,,) ; (",) ,

n x,,) ; f ( , ) .

) ,,) , '1 (,.,) ( "" )

( '1) > (,,) ()' : , ,,). a , 6 ( , ,, ) , ( , ,,) ( ) < ( ), ; ()

f( "",) < f( y) < '( ) , (, ,,). ; ."

..) < f (y) ~ lim () :::: (,,).'--..,

,,) < ,, ). . ( ,,, ) ; ; . ; ( ,,) . )

: .

, ; .; ; ; ; (2) 6 .1 :

(1 )' ( ...) (., .) ; ., .

( '( ( .,,) . . .. )