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ΚΑΤΣΑΡΓΥΡΗΣ ΒΑΣΙΛΕΙΟΣ. Καθηγητής μαθηματικών Ε::αρβακείου Πειραμ. Λυκείου. ΜΕΝΤΗΣ ΚΩΝΣΤΑΝIIΝΟΣ • Καθηγητής μαθηματικών 100υ Λυκείου Συγγραφική ομάδα: • Καθηγητής Ε.Μ. Πολυτεχνείου . στικέςιδιότητέςτου. Τέλος, ορίζεταιτοσύνολοίR καιοιπράξειςμεταξύτων Οι ορισμοί, οι προτάσεις και τα θεωρήματα διατυπώθηκανμε τηνπιοεύχρηστη μορφή τους, η οποίαδενείναι πάντοτε και η πιο γενική. Από τις αποδείξειςπα στοιχείων του.
Citation preview
: . . ::
. . 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
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
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 )' ( ...) (., .) ; ., .
( '( ( .,,) . . .. )