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*** . .
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+i=+i
+i=0
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=0,1,2,3 z z = = )
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12
,
12
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z24z+8=0
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zw
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=i1821+i2003: .i. 2i. -1 .2. 0)
=(1+i)2004(1i)2004: .1. i. 0.-i. -1) z 3, = z : .3z.2z.1z.9z.9z ) : .2z zz = . z z = .22z z = .4z 2 zz= = . z w z w + )
z 2 5i 7 + + = : . . . .. ) z, z 5i 2 = - : .z=7i. z=3i. z=5+2i .z=25i . z=2i) 2z 1 z 2 , = : . z 4 = . z=1 . z=-1.z=i . z.)
z 1 i z 3 5i = +
: .x+y+2=0 . 2xy+3=0 . xy+3=0 .2x+y4=0 . xy+4=012 )
z 3 7i 5 + =
: .(x3)2+(y+7)2=25. 3x7y5=0 .2 2x y13 7+ = .2 2x y17 3 = . (x+3)2+(y7)2=251.4 )
,
C
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.
13
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>0,
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.,
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:(C):(xx0)2+(yy0)2=2
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=2.
:2 4z 2 z 4 zz 4 z (1)z= = = = 15)
w=x+yi
z=+i.
:(1) 4 1 zz (2)z z 4== = (2)4i 1 zw z z 4i z 4i z iz i i( i) ( )iz z 4 = + = + == + = + = + + + = + +
:w=+(+)ix+yi=+(+)ix y x2(3) y y x2+ = = + = =
z=+i
C:x2+y2=4,
:2 2(3)2 2x y y x 4 42 2+ + = + = (x2+y2+2xy)+(y2+x22xy)=162(x2+y2)=16x2+y2=8
w=x+yi
z=+i
:(2)4i 1w z z 4i z iz (1 i)zz z = + = + == + = +
:wz1 i=+
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,wR 2 2, = :C1:x2+y2=816 ().
z=+i
w=x+yi.
:2z 2 z 4 = = 24i 4iz 4iz 4i( i)w z z z ( i)z zz 4z+= + = + = + = + + =
=(+i)+i(+i)=()+(+)i:x yx 2()y y x2+ == = + =
2+2=4,
z=+i
x2+y2=4.,():2 22 2 2 2x y y x4 x 2xy y y 2yx x 162 2+ + = + + + + =
x2+y2=8
zw
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z
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z,
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w=+i
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,
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z0
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z 0 z 1. = = : z 0 z 0, = = z 1, = :z6=1(z3)21=0(z31)(z3+1)=0(z1)(z2+z+1)(z+1)(z2z+1)=0(z=1
z=-1
z2+z+1=0
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,
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R>0
. [10]
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) 1w .w= [13]) ,w,z-C.[12]
,:=x+ix,=y+iy,=+ix+y+=x+y+=0:)1 1 1 , , = = = [8]25) ++=0[9]) 2+2+2=0[8]
z
w
:2 2z, w 0, w z z w = = .:) z w 1 = = [5])1 1z wz w= = [4]) w=-
z[4]) w3=-1z3=1[4]. )
z
w.[4])
z
:z2004=1[4]
26
3.1 )
f
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00
x0,
>0 . f
(x)1 . 00
x
R) i) f
x=,
:[ ](1)h 0 h 01limf ( h) f () lim f ()f (h) f ()2 + = = f () 0h 0 h 0f () limf (h) 2f () limf (h) 2 = = h 0limf (h) f (0) (3) = f0.
f
(0)=0,
f
(x)=0
x
R.
f
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f
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R.
f
x=,
-:h 0limf ( h) f () (4)+ = :(1) (3)h 0 h 0 h 01 1limf ( h) lim f ()f (h) f () limf (h)2 2 + == = ==
1 1f ()f (0) f () 2 f ()2 2= = = ,
f
R.6.3
f:RR
:f2(x)+2f
(x)+2x0
x
R
f
x0=0.:f2(x)+2f
(x)+2x0f2(x)+2f
(x)+112x[f
(x)+1]22x[ ] 220 f (x) 1 x 0 f (x) 1 x + + x f (x) 1 x 1 x f (x) 1 x + + :( ) ( )x 0 x 0lim 1 1 lim 1 x 1 x = + = x 0limf (x) 1.=
x=0
:f2(0)+2f
(0)+10[f
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(0)=-1
[f
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x0=0.6.4
f:RR
:f3(x)+3f2(x)f
(-x)=-2x3
x
R.: ) f, ) f
(x)=x
.. ) ( )2x 1x - 2A = lim .f(x) - 1
) 2x +2xB = lim .f (x)
.) :f3(x)+3f2(x)f
(-
x)=-
2x3(1)(1)
x
x
:f3(-
x)+3f2(-
x)f
(x)=2x3(2)(1)(2):f3(x)+3f2(x)f
(-
x)+f3(-
x)+3f2(-
x)f
(x)=0[f
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(-
x)]3=0f
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x)=0f
(-
x)=-
f
(x)
x
R.
f.)
f
(-
x)=-
f
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f
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x
R.B. ) :( )2 2 2x 1 x 1 x 1x 2 x 2 1A lim lim lim (x 2)(x 1) (x 1)f (x) 1 = = = =
48 - :2x 1 x 11lim lim(x 2) 1 0(x 1) = + = 0
g()>1. h()=f
()g()=f
()=[f
()1]0
f
()0.(1)
x2:2 22 22 2 3x f (x) 3xx x (3)x x x + :2222x 0 x 0 3x 3xlim lim 3 9x x = = =
(3)x 0f (x)lim 9.x=
f
(0)=9.
.fx0.,()Cf( )0 0A x , f (x ) :
=f
(x0)
():yf
(x0)=f
(x0)(xx0)()xx,:==f
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(x0)=g(x0)(1) f
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(x0,y0)
CfCg.(2)CfCgx0.64 . CfCg,,x1x2,:yf
(x1)=f
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(x1)=g(x2)f
(x1)x1f
(x1)=g(x2)x2g(x2),CfCg:2 11 22 1g(x ) f (x )f (x ) g(x )x x= =
.Cf(,)
:
(x0,y0)
. ()Cf:():yf
(x0)=f
(x0)(xx0) ()
(,)
f
(x0)=f
(x0)(-x0). x0,.
8.2 CfCg
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g(x)=x3x+3
.) .) Cf
Cg.)
(,)
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Cg.
=f
()
=g(),:f
()=g()(1)
>0
Cf
Cg
,:f
()=g()(2)65(1):ln++2=3+3(3):1f (x) (lnx x 2) 1x= + + = +
g(x)=(x3x+3)=3x2-1(2):2 3 211 3 1 3 2 1 0 ( 1)(3 3 1) 0 1 + = = + + = =
32+3+10
R
(0,) 21 1 2f (x) + f =x x x
x>0,) f
(x)=lnx,
x>0.) :xf (xy) f 2f (x) (1)y + =
(1)
x=1
y=x
:1 1f (x) f 2f (1) f (x) f 0 (2)x x + = + =
) (1)x(y):x 1f (xy) y f 2f (x)y y + =
x=1
y=x
:21 1 1 1 2f (x) x f 2f (1) f (x) f (3)x x x x x + = + =
f
(1)=1
x0
(
x>0).84 ROLLE...... ) (2):21 1 1 1f (x) f 0 f (x) f 0 (4)x x x x + = =
(3)(4):2 12f (x) f (x) f (x) lnx c, x 0x x= = = + >
f
(1)=0,
0+c=0c=0.
:f
(x)=lnx
x>0
f
(x)=0
x
(,)
f
=[,],
f-..;
f
(x)=0
x
(,)
(,).
-:12c , x (, )f (x)c , x (, ) =
f,:12c , x (, )f (x) c, x c , x (, ) = =
f,:- +1 2x x lim f (x) lim f (x) f () c c c = = = =
f
(x)=c,
x
(,),
f(,).
85
. ) Rolle-.[3]) f
f
(x)=f
(x)
x
R.
f;[3]. ) f.f
f
(x)=0,
x,f-.) fg
f
(x)=g(x)
-x,c,f
(x)=g(x)+c
x
.[14]. ()-().) f [,]
f
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(),
f- Rolle. ) f
[,]
(, ),
Cf
(,f
())
(,f
()). )
f:R
f
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x
A, f. )
f
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x
,
f
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f
(x)=f
(x)
x
R,
f
(x)=cex. [5]86 ROLLE......
f:RR
:f
(x+y)=f
(x)+f
(y)+3xy(x+y)x,y
R. ) f
x0=0,
fR.[6]) f
x=,
.[7]. f
x0=0
f
(0)=0.) fR.)
Cf
Cf
,.[12]
f:(0,+)R
:f
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(1)=-1f
(x)=e2f(x)
x>0:) f,[5]) f
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(x)]2,[5]) :g(x)=-
f
(x)e-f(x),x>0,[5]87) f
(x)=-
ef(x)
x>0,[5]) f
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lnx,
x>0.[5]
f
[,]
(,)
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f . ) f
[,],
- f
[,]
f- .
92 -
. ,-,.:(x)B(x)A(x)B(x)(x)
x
,
.)
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.)
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,().1....932:A(x) B(x)A(x) B(x) (x) B(x) (x)
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ee,
...-(
f
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-,
>e).. , 0 g(x) = xe - (1 + e )ln(1 + e ) :) g,) g(x)0,) f
(0,+).) :x x x x x x x x xx1g(x) [xe (1 e )ln(1 e )] (e xe ) e ln(1 e ) (1 e ) e1 e= + + = + + + =+
x x x x x x x xe xe e ln(1 e ) e xe e ln(1 e ) = + + = + = xx x x x x xxee [x ln(1 e )] e [lne ln(1 e )] e ln 01 e= + = + = 0. f
(1)[f2(1)+f
(1)+]=-10,
f
(1)0,
:2 22 ln < -
,>0
>.
x,
=x
:2 2 2 2x x2xln x 2xln x 0 (1) < + < x2xln ,-.
xln x.x(1):2x 2ln x 0 x + < :2x f (x) 2ln x , x x= + :2 2 2 2 22 2 2 2 1 2 x 2x (x )f (x) 2 1 1x x x x x x+ = = = = f
[,+)
f
(x)
:f
() + < < < .98 - 2 2xg(x) 2xln x ,= +
x
((1)),:
g()=0x 1 xg(x) 2ln 2x 2x 2ln 2 2x x = + = + g()=0 1 2( x)g(x) 2 2x x= =
g(x),
g
[,+).
:x>g(x)0.[7]) :x3x2+
lnx
x>0
=1.[9]106 -
:f
(x)=2x33x212x+8) f.[4]) f.[4]) f.[5]) f.[5]) :2x3+8=3x(x+4)[7]
107
13.1 A.)
f
;) f ;)
(x0,f
(x0))
Cf;) f
(x0,f
(x0))
-
Cf
x0
,
f
(x0);) ;)fx0
f
(x0)=0,
(x0,f
(x0))-
Cf
;)
f
(x)=x46x2+3.. ) DeLHospital0.0 ) xx 0e 1lim ;x
) DeLHospital . ) 22x2x lnx 1lim .x lnx 2++ ++ +
)
x=x0
Cf;)2xf (x) ;x 1=
108 DELHOSPITAL)
y=
Cf+-;) 22x x 2f (x) .x x 3+ += +
)
y=x+
Cf; ;) 2xf (x) 3x 5x 1= ++. .)
y=x+
Cf,;)Cf,2x 3x 3f (x) .x 1+ +=+
13.2 ) =x0
Cf, ,) f ) =
Cf+(-), ().) 222x 1f(x)x 2x 1+= + ) Cf+,
0.
:==)2x 2f (x) 2x 1x 3+= + ++ 109) DeLHospital: i)x 1x 1e xlimx 1=ii)2xxxlime+= iii)xlnxlimx+= iv)x 0xlimln(x 1)=+
13.3 ) f ,f. ) 1f (x)x=
(-,0)
(0,+). ) ff
,f. )
f
(x)>0
x, f. ) f,
f
(x)0f
(x)>f
(0)f
(x)>0117f
(-,0]
[0,+)
-R.f
: x0,f
(-,0]
[0, +),
R.f
f
(0)=0.)
f
fR.
f
(x)
f,:f
(0)=0f
(-,0]
[0,+).
f
(-,0]
[0,+);
.f
.)
f
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f
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x0=0.. )
f
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x0,
f
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f
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f
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x=0.) :ex+e-xx2+2ex+e-xx220f
(x)0f0,:f
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x
R .118 DELHOSPITAL x=x0 y=y0 y=x+,00x xlim f (x)= 0x xlim f (x)+= 0xlim f (x) y+=0xlim f (x) y= [ ]xlimf (x) (x ) 0+ + = [ ]xlimf (x) (x ) 0 + = x0- Df
-.+-+-:0x xlim f (x)
0x xlim f (x)+
:xlim f (x)+
xlim f (x)
:xf (x) limx= [ ]x limf (x) x=
14.4
f
R
:f(0)=f
(0)=0f
(0)=2) x 0f (x)A = lim .x
) 2x 0f(x)B = lim .x
) 2x 0 x + f(x) = lim .f (x)ln(x + 1)
) :x 0 x 0f (x) f (0) f (x)f (0) 2 lim 2 lim 2 (1)x 0 x = = =
=2.119)
f
x0=0,
,:x 0limf (x) f (0) 0= = 00 DeLHospital:2x 0 x 0 x 0f (x) f (x) 1 f (x) 1B lim lim lim 2 12x 2 x 2 x = = = = =: 00x 0 x 01 f (x) 1 f (x) 1 1B = lim == lim = f (0) = 2 = 12 x 2 1 2 2
f
x0=0,
f
(x)
x0=0
x 0lim f (x) = f (0) = 2.) 2x 0f ( x )lim ,xx2.:( )222x 0 x 0f (x) xxxf (x) ln(x1)x x x f (x) lim limf (x)ln(x 1) +++= =+
:00x 0 x 0 x 01x 1 ln(x 1) 1lim lim lim 1x 1 x 1 + +== = =+
:1 1 12 1+= =
14.5 f
x0=0,
f
(0)=1
f(0)=0,:) x2x 0x - xeA = lim ,x
120 DELHOSPITAL ) x 0f(x)B = lim ,x
) xx 0xf(x)lim = -1.x - xe
) DeLHospital:0 0x x x x x x0 02x 0 x 0 x 0x xe x e xe x e e xeA lim lim lim 12x 2 x = == == = )
f
(0)=1
f
(0)=0,
:x 0 x 0f (x) f (0) f (x)f (0) 1 lim 1 lim 1x 0 x = = =
:00x 0 x 0 x 0f (x) f (x)B lim 1 lim limf (x) f (0) 1x (x) = = == = = = f
,:x 0limf (x) f (0)= ) :2x 0x x x xx 0 x 0 x 02 2 2x 0xf (x) f (x)f (x)xx xx xe x xe x xex x xlimxf (x) B 1lim lim lim 1A 1 x xelim = = = = = =
14.6 32x - 4f(x) = .x
)
f
.)
f
.)
f.)
Cf.)
Cf
f.121)
R
-
x3x24=0.)
f.) f
=R*.
:3 4 3 3 3 32 4 3 3x 4 3x 2x(x 4) 3x 2x 8 x 8f (x)x x x x + += = = =
f
(x)=0x3+8=0x3=-
8x=-
2,f
(-,-2]
(0,+)
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x0.
,f
(-,0)
(0,+).
Cf-.) ()().) :332 2x 0 x 0 x 0x 4 1limf (x) lim lim (x 4)x x = = =
:32x 0 x 01lim(x 4) 4 0 limx = < = +,
x=0
Cf.122 DELHOSPITAL :33x xf (x) x 4lim lim 1x x = = [ ]32 2x x xx 4 4limf (x) x lim x lim 0x x = = =
,
y=x
Cf+-.) -f-,.)
x3x24=0Cf.
x=0.
x0
:33 22x 4x 4 x f (x) x = = = Cf
y=,
R.
,yy:
0.
>-3
,
>0,
y=
Cf.) Cfyy.( )*f . = +x + x 0lim f(x) = - , lim f(x) = + f
=(0,+),
f()=(-,+).
,f,f(),R.12314.7 --
f:RR
.:1g(x) =f(x) ) g.) : + x xlim g(x) lim g(x) ) Cg.)
g.)
g.)
g.)
g.) :f
(x)=0(x=-
2
x=0
x=2)g:=R{-
2,0,2}) x xlim f (x) lim f (x) += = + :x xlim g(x) 0 lim g(x) 0 += = ) x 2 x 0 x 2lim f (x) 0, limf (x) 0 limf (x) 0. = = = f-
2,
0
2
:
x 2 x 2lim g(x) lim g(x) + = = +
x 0 x 0lim g(x) lim g(x) + = + =
x 2 x 2lim g(x) lim g(x) + = = +124 DELHOSPITAL ,
x=-
2,
x=0
x=2
Cg.x xlim g(x) 0 lim g(x) += =
y=0
-Cg(-)(+).) f
(-,-2],
(-,-2)
1g(x)f (x)= .:
x1B e dx, x 1x 1 = > +
2 21 1 dx, x 0,2 x x = +
B. ) .) I xxdx; =
) A x2xdx. =
) ;) lnxA dx B xdx,x= =
x
(0,).)22x 1A dx, x 3.x 5x 6+= > +
12915.2 .) f (x)g(x) dx =.) I lnxdx, =
=. ) ( )A f g(x) g(x)dx, =
=
,
u=
du=.) 32 x 1I 3x e dx,=
=.) 2 99A x(x 4) dx, = =+c.)22x 3A dx.x 3x 2+=+ +:22x 3 A cx 3x 2+=+ =++ +
B. ) : f (x) dx = f (x) dx =
)
f
(x)0
x
[,],
f (x) dx
) cdx =
) : f (x) dx , =
R [ ]f (x) g(x) dx , + =
,
R.) f
[,],
f
(x)0
[,]
f
[,],
f (x) dx
)f
,,
.
:
f (x) dx =+
130 . ) f
[,]
: i)( )xdf (t)dtdx=ii)( )tdf (x)dxdt=
iii)( )df (y)dydx=
) g,( )g(x)f (t)dt =
) 32xt 1xf (x) e dt,+=
f
(x)=) Gf
[,],
: f (x)dx =
) : i)0 xdx = ii)120(3x 2x 1) dx + =
iii)2 1x02xe dx =iv)e1 lnxdx =
15.3 .) : . f (x) dx f (x) c = +.
F
(x)=f
(x),
Ff .f (x) f (x)e f (x) dx e c = + .(x)dx ln (x) c(x)= +
.
f (x) dxf
(x)) 2x 1f (x) ,x+=
x>0,
: .21 1f (x) dx cx x= + + .1f (x) dx lnx cx= +
.1f (x) dx ln( x) cx= + +.2 31 2f (x) dxx x=
.
131) 2f (x) ,x 1=+
x>-1,
f (x) dx: .2(x+1)+c.
2ln(x+1) . 2ln(-1x)+c .2ln(x+1)+c . 2 x 1 c + + )
f
(x)=xx,
f
(0)=f
(0)=1,
: .f
(x)=x+x+1. f
(x)=xx+2 . f
(x)=2x3x+1.f
(x)=xx+2 . f
(x)=-
xx+2) 13f (x) x = (1,2),
: .33 5F(x) x x4 4= + . F(x) x 1 = +. F(x)=x2+1 . F(x) x x 1 = +. F(x) x x = + . ) 220A (3x 2x 1) dx, = +: .3. 5. -7 .-4. 6) 32 12 21 2x 7x x dx 2 dx,x 5 x 5+= ++ + : .12= . =3 .32= .=5. =2) I 2 f (x)f (x) dx =: .f
()f
(). f
()+f
() . f2()+f2() .f2()f2() . f
()f
()f
()f
())
Cf
(0,0)
(1,1),
- 10I f (x) dx =: .2. 1. 3.5 . 0132 ) x21f (x) 1 t dt, = : . f (x) x x = . f
(x)=2x . f
(x)=-
2x . f (x) x = . f
(x)=2x)fRt6 40 xf (x) dx t t , = +: .f
(1)=7. f
(1)=5. f
(1)=3.f
(1)=8. f
(1)=10) 51I x 2 dx,= : .5. -3 . 9.7. 10) x 12txg(x) e dt+=: . . . .. ) 2 h22 h 01L lim 5 t dth+= +: .2. 3. 5 .1 . 7) [ ]0f (x) f (x) xdx 2 + =
f
()=1,
f
(0): .1. 2. 3 .5 . 4. ) - : .32f (x) dx
.1 32 1f (x) dx f (x) dx+
.1 32 1f (x) dx f (x) dx
.0 32 0f (x) dx f (x) dx+
.
133) CfCg
x=
x= : . [ ] f (x) g(x) dx
. [ ] [ ] f (x) g(x) dx f (x) g(x) dx
. [ ] [ ] f (x) g(x) dx f (x) g(x) dx +
. g(x)dx f (x)dx +
. [ ] [ ] f (x) g(x) dx f (x) g(x) dx
) f
(x)=x2
y=-
x+2
: .3. 5. 7 .92. 6)
f
(x)=x3
g(x)=x
x=-
2
x=1
: .114.73.85 .137.112 )
f
(x)=x
g(x)=x
x=0
x=2 : . 3. 5. 2 2 . 3 5. 4 2 ) : .13.57.2 3ln3 2+ .3 5ln2 3+ .
3ln22ln3134 15.4 .)
F
f,
G
f
F+c,
c
R. )
F
(x)=f
(x),
x
,
f (x) dx F(x) c. = + ) f (x) dx f (x). = ) 1dx ln(x 2) c,x 2= + ++(x>0). ) xdx x c. = + )x x dx c. = + ) 1dx 2 x c.x= + . ) ( )xf (t)dt f (x), =fR. ) 2x40f (x) 1 t dt, = +8f (x) 2x1 x . = + ) 14 3 20(5x 4x 3x 2x 1) dx 5. + + + + = ) 0 xdx 0. = ) 2 20 0f (x) dx f (x) dx. =
135
..: 11f(x)f (x)dx f (x) c, 1 1+= + +
21f (x)f (x)dx f(x) c2= +
f (x)dx ln f (x) cf (x)= +
1f (x) 1dx f (x) c, 1 1 f (x) += + +
f (x) f (x)e f (x)dx e c = +
f (x)dx 2 f (x) cf (x)= +
. I A(x)dx =(x)f
(x),.. 1I f (x)dx=
x=f
(y),
dx=f
(y)dy.
f
()=
f
()=,
:( ) [ ] 1 I f f (y) f (y)dy yf (y)dy yf (y) f (y)dy= = =
,Cf1fC
y=x,
Cf.
136 16.1
f:RR*:x1 1 2+ =f(x) f (x) e
x
R
f
(0)=1,
:) :( )xI = f(x) + f (x) e dx )
f
(x)=ex,
x
R
.) .:( )x x xI f (x) f (x) e dx f (x)e dx f (x)e dx = + = + =
x x x xf (x)e dx f (x)e f (x)e dx f (x)e c = + = +
) :x x1 1 2 f (x) f (x) 2f (x) f (x) f (x)f (x) e e++ = = ( )()xf (x) f (x) e 2f (x)f (x) + = ( )x 2 x 2f (x)e f(x) f (x)e f(x) c (1) = = +
(1)
x=0
:f
(0)=f2(0)+cc=0:f (x) 0x 2 xf (x)e f(x) f (x) e , x= = 16.2 fg
[,],
FfGg[,]. f(x) dx = g(x) dx, :) F()G()=F()G(),)
(,)
,
f
()=g().137) FGfg
[,],
: f (x) dx F() F() =
g(x) dx G() G() =
: f (x) dx g(x) dx F() F() G() G() = =
F()G()=F()G())
f
()=g()
h(x)=f
(x)g(x).
h
H=FG,
:( ) H(x) F(x) G(x) f (x) g(x) = =
x
[,] H
[,]
[,]. H
[,],
(,).:H()=F()G()H()=F()G()
H()=H(),
.-Rolle
(,)
,
H()=0.
:H()=0f
()g()=0f
()=g()16.3 2xt t4f(x) = e dt ,
x
R
.)
f
(x).)
f
.) 20I = 3f(x) dx. ) :( )22 2x xx x 2 f (x) e x 2xe , x = = 138 ) :f
(x)=0x=0f
(-,0]
-
[0,+).) :[ ]() 2 2 2200 0 0I 3f (x) dx 3 (x)f (x) dx 3 xf (x) 3 xf (x)dx = = = ==
2 2 2 2x x x x2 20 06f (2) 0 3 2x e dx 0 2 3x e dx = =
x
[0,2],
x x. = :( )3 3 3 2 2 22 x x 3 x0 0 0I 2 3x e dx 2 e (x )dx 2 e dx = = = =
[ ]32x 8 802 e 2(e 1) 2(1 e ) = = = 3 22 x0A = x e dx ( )3x = y.
I f (x)dx =-,: x=+-y dx=-
dy: I f ( y)( dy) f ( x)dx J = + = + =
2=+J,
1I (I J).2= + -
+J
.139:-I f (x)dx =
x=-y.
16.4 x-xxI = dx.e + 1:)xx-xe xI = dxe + 1 ) =)
x=-
y,
dx=-
dy.: x y- xx ( y)( y)I dx ( dy)e 1 e 1 = = =+ +
x y x x- - -yy xx xe xdy dx dxe 1 e 1 1 e = = =+ + +
) :x x x- xx xe xI dx I dxe 1 e 1= =+ +
.:x x x x x- -xx xe x xx xe x2I dx dxe 1 e 1 e 1 += + = = + + +
x x- -xx(1 e )dx xx dxe 1+= = =+
[ ] -- -x( x)dx xx xdx = = + =
[ ]-( ) () x 2 = + =
2=2=.140 ()..fh(x)I g(t)f (t)dt, =g(t),: - ,:( ) ( ) I g h(x) f h(x) f (x) = ,,, .. : I f (x t)dt I f (xt)dt.. = =
xt=u
xt=u,
... xx tI e f (t)dt.=-: xx tI e e f (t)dt,=ex( t). e-x, e-xI..f
(),x..f
()--f.14116.5
f:RR
:yy xxx, y f(t)dt = e (y - 1) - e (x - 1) )
f(x)=xex.)
Cf,
xx
x=1.) :x xy x x yy yf (t) dt e (y 1) e (x 1) f (t) dt e (x 1) e (y 1) (1) = =
f,xyf ( t ) dtx.y(1)x.:( )xx yyf (t) dt e (x 1) e (y 1) =
f
(x)=ex(x1)+exf
(x)=xexex+exf
(x)=xex
f
(x)=xex(1).) ,
x=1,
-
f
(x)=0,
x=0.
f
(x)0
x
[0,1]
1 1 1x x0 0 0E f (x) dx xe dx x(e )dx = = = =
1 1 1x x x0 0 0xe e dx e e e e 1 1 = = = +=
1..().
142
. ) f;) f
;[4]. :) 0dx dx = =
)1dx x dx , 1x= =
) xdx xdx = =
)2 21 1dx dx x x= =
)x x dx e dx = =
[5]. fFf,:)
G(x)=F(x)+c,
c
R,
f,[3])
G
f
G(x)=F(x)+c,
c
R.[6].()-().)
f
. 143) F
f
- . )
F
G
: f:AR,
G(x)=F(x)+c. ) f (x)dx f (x) c. = + ) Ff, F(x)dx f (x) c. = + ) f (x)g(x)dx f (x)g(x) f (x)g(x)dx. = ) ( )f g(x) g(x)dx f (u)du, = :u=g(x)du=g(x)dx [7]
f,g:RR
f
(0)=1
g(0)=1.
fggf.
h(x)=f
(x)+g(x)
(x)=f
(x)g(x),
:) h(x)=h(x)(x)=-
(x),[8])
H(x)=e-xh(x)
(x)=ex(x)
,[8]) f
(x)=ex
g(x)=ex,
x
R.[9]
f:RR
:(x2+1)f
(x)2x
x
R)
f
(0).[5]144 ) 22xf (x) .x 1=+
[5])
f.[9]) Cf,xx
x=1.[6]
:1 2xx 2 41 xf (x) g(x) f (t)dt x 0(1 x ) 1 x= = + +
)
g(x)=0
x
R*.[13]) 1I f (x)dx, =
R*.[12]
145
.:) f (x) dx f (x) dx f (x) dx. = =
)
f
(x)0
x
[,],
f (x) dx
) f (x) dx = [ ] f (x) g(x) dx + =
)
f
,,
,
: f (x) dx =+
) f
[,],
f
(x)0
x
[,]
f ,)f,
xF(x) f ( t )dt =, :
F(x)=
x
, ( )g(x)f (t)dt , =g-
f
g.) : f (x)g(x) dx =
( ) f g(x) g(x)dx , =fg-
,
u=g(x),
du=
u1=g()
u2=g().146 ) fg
x=
x=
(
0
-
[,],
[,]
,:f () = 152 17.30 fg
[0,1]
:f
(0)=g(1),f
(1)=g(0)f
(0)f
(1))
(0,1)-,
f
()=g().) ;17.31
f:RR -
xf
(x)2x2+x
x
R,
f
(0).17.32 :2xf (x)4 x=
)
f.) :x 2 x 2M limf (x) N lim f (x) = = ) f.17.33 f:h 0limf (x h) f (x)+ = ( )( )2 2f (x) x f (x) x 0 + =
x
R.
f.17.34 f
[1,4]
:f
(1)+f
(2)=f
(3)+f
(4),f
(1)f
(2)f
(3)f
(4):)
(1,2)
,:f (1) f (2)f ()2+= ) f.17.35
f:RR -
[f
(x)x2][f
(x)+x2]=2x2+1
x
R.)
f
(x)=0.) f,:f
(2000)=4
106+117.36
f:RR
f
(xy)+f
(x)+f
(y)+3=x+y+xy
x,y
R.)
f
(1)=0.)
f.) :2xx2xlimf(x) 1++
17.37
f:RR
f2(x)=1+2ex[1f
(x)]
x
R.
f
(2004)
) f,:2=-1) ,f-
x0=1.17.40 :22x , x 0f (x)x0, x 0= =
:) f
x0=0,) Cfxx.17.41 f:x+2f
(x)x2+x+2
x
R.
:) f
x0=0,) f
x0=0.17.42 fx0=2
:h 0f (2 h) 3lim 5h+ = :) f
(2),) f
x0=2.17.43 f-
x0=0
f
(0)=1,
:x 0f (3x) f (2x) f (x) 3f (0)A limx+ + = 17.44
f,g:RR -
x0=,
:)2 2x f (x) x f ()A limx =
)x g()f (x) g(x)f ()B limx =
17.45 f-
f
()0,
f .17.46
y = f
(x)
x
[,].
(,)
Cf
(,f
())
-,
(,f
())
(,f
()),
: 2+= 17.47 :f
(x)=x2+x+2) Cf
:y=3x+5.154 ) Cf
(2,7).17.48
f
(x)=x3.
Cf,:) ::y=3x+5) :7 , 93 17.49
f
(x)=lnx
(-1,-2)
(2,1).) .) -Cf.17.50 -
f,g:RR
g(x)g(x)0
-
x
R.
:f (x)h(x) ,g(x)=
x
RCh
(,h())
-,:f ()h()g()= 17.51 :f
(x)=x2+x+21g(x) 1x= ,Cf
Cg
:x=1,
-.17.52 :2xx 1f (x)e+= ) f
(x)f
(x).) f
f
.) f Cxx.17.53
f:RR
.-:) f
,) f
f
(0)=0.17.54
f,g:RR
g(x)=f
(x2x+2)
x
R,fR.Cf(2,3):y=4x+6Cg
(1,g(1)).17.55 :2x x 1, x 1f (x)x , x 1 + 0.15717.75 :2xf (x)x 1=
:) f,) f.17.76
f:RR
,:2f
(x)=[1+f
(x)]x
x
R
f
(0)=f
(0)=1
f
(1)=3:) f
,)
f.17.77
f,g:RR-
[f
(x) g(x)][f
(x) g(x)] = 0,
x
R.
f
(0)=1
g(0)=0,
:) [f
(x)g(x)]2=1,) f
(x)=g(x)+1
x
R
f
(x)=g(x)1
x
R.17.78 :21f (x) 2(x 35) 750x = + ) :23200f (x) 2 , x 0x= ) f-.) f.17.79 :f
(x)=x+(1)x2+1) f-.)
f
(x)=0.)
f
(x)0.)
0e.)
exxe
x>0.)
+1 > ( + 1)
e.)
xx
x>0,
-
=e.17.81 :f:RR
f
(x)>0
:(f
(x)ex+1)(f
(x)+ex1)=0
x
R.
f
(x).17.82 :f
(x)=(2x28x)lnxx2+8x+2158 ) f-.)
f.17.83
f:RR
:2f3(x)+3f2(x)+6f
(x)==2x3+3x2+6x+5
x
R.
f-R,f-.17.84
,>0,
x+x2
x
R.
=1.17.85 :f:RR
f 3(x) + 3f
(x) = x3 + 3x
x
R.) f-.) f
(x).17.86 :f
(x)=3(21)x4+4x3+12x2+12x+2,
R,f
(-1)f.17.87 :f:RR:f
(1+x)f
(1x)2x
x
R:) :g(x)=f
(1+x)f
(1x)2x,) f
(1)=1.17.88 :f
(x)=-
x3+12x4) f.) -
f
(x)=0.) f.) f.17.89
N*
2,
x0
:2(x+1)>2+2x+(1)x2
17.90 -270cm2.-2,5cm3cm.) x-,(x)x.) ,.17.91 159,.-,-.
-30.) --.) -.) -.17.92 :f
(x)=x39x2+12x(lnx1)+22) f
(x).) f.) ,,
Cf.17.93 :4 3 2x x xf (x) x 12 6 2= + + +
,,
R.
,f.17.94 :f
(x)=x3(++3)x2+(+4)x+1,f
(1)f(2,f
(2))
Cf.17.95 :f
(x)=6x2(lnx1)2x3+3x2+2002) :g(x)=lnx+1-x) f.17.96
f:RR
.:[f
(x)]3+3f
(x)=x33x+2
x
R,
:)
f,)
Cf,)
f.17.97 :2x x 3f (x)x + +=+
,,-
Cf
:x=2y=3x+5160 17.98 :2( 1)x 2x 3f (x)3x 2+ +=
():2xy+=0.
,()
Cf
+.17.99 f
(x) +
():y=3x2,
:22xf (x) xf (x) 3x x 3A lim2f (x)(x 1) 6x 5++ + +=+ +
17.100
f:RR + ():y=3x+4, R ,:2xf (x) 6xlim 1xf (x) 3x 5x 2++= + +
17.101 fR
R.
,-DeLHospital,:00x f (x) f ()f () limx = ==
x x [f (x) f ()]lim limf (x)(x ) = =
:x limf (x) f ()= f
x=.-f
.-;17.102
f:RR
:exf
(x)+x=1+(x+1)f
(x))
ex=x+1.)
f
(0).)
f
(x).17.103 fx0=0,
f
(0)=0
f
(0)=1.
-:)x2x 0x xeA limx= )xx 0xf (x)B limx xe=
17.104
f:RR
-:ef(x)+f
(x)=xx+1
x
R.)
f
(0)=0.) f
(x)f
(x)f
(x).) :4x 0f (x)A limx= 17.105
f:RR
-:f3(x)+3f
(x)=3ex3x-3
x
R.) x 0limf (x).
161) f-.) :x 0f (x)A limx= )
f
(x)=0.17.106 :3 222x 3x 6x 1f (x) 6lnxx+ += ) f
(x).)
f
-.) :6x2lnx=2x3+3x26x+1) f
(x).17.107 :2x x 2f (x)x 1+ +=
)
f.) f.17.108 :22 3xf (x) xx= + ) f.) -:x3x23x+2=0
R.17.109 :22x 1f (x)x 1=+
) :3322x 3xf (x)(x 1)+=+
) f-.) :22 2 21 xf (x) 3(x 1) x 1=+ +
) f
Cf.) f.)
Cf.) f.17.110 :f
(x)=x+1+ln(x2+1)e-x
)
f
(x).) f-.) :ex[x+1+ln(x2+1)]=1) xA lim f (x).+= 17.111 :f
(x)=ex1ln(1+x))
f.162 ) f -.)
f.) x 0limf (x)-f.) :1+ln(1+x)=ex
)
x>-1
-:ex1+ln(1+x)17.112 f:f
(x)=6x+4
x
R.
Cf
(1,5)
::x+7y+10=0
f.17.113 -fxx 2x 1++
x0.f,
Cf
-
(0,2).
17.114 :)2 A xdx, x ,2 2 =
)2B xdx, x (0, ) =
17.115
f:RR*,
-R.-Ff,CF
:y=2004.17.116 :*f:+
f
(x)=2f
(x)
f
x0=0
2.-
f.17.117 f:RR
:f
(0)=2(1)f
(x)f
(x)=ex(ex+1)
x
R.) :f2(x)=(ex+1)2
) f.17.118 :)2A x lnxdx =
) B ln(x 2)dx = +
)lnx dx, x 02 x= >
16317.119 f,R.-
g(x)=(f
(x)+f
(x))ex,
x
R.
:( )xg(x)dx f (x) f (x) e dx = + =
x xf (x)e dx f (x)e dx = + =
x x xf (x)e dx f (x)e f (x)(e )dx = + =
x x x xf (x)e dx f (x)e f (x)e dx f (x)e = + =
:xg(x)dx f (x)e (1) =
g(x)dx-g,:xg(x)dx f (x)e c = +
(1)-c;17.120 :)2xA dx, x (0, )1 2x=
) B x ln(2 x)dx = +
17.121 :)2lnxA dxx=
)2x B dx, x ,1 2x 2 2 = +
17.122 I ln xdx, =:+-1=xlnx17.123 :) A x ln(1 x)dx, x (0, ) = +
) B (x 1) xdx = +
)x e dx =
17.124 :)122x 3I dx, x 1x 3x 2+= > + +
)3 222x 4x 8x 6I dx, x 2x 3x 2+ + += < + +
17.125 :)2x x2x x3e 4eA dx, x 1e 3e 2= > +
)x xB e 1 edx = +
17.126
f:RR
-xxxe-2x
x
R.
f
(0)=-1,
f.17.127 :xg(x) (x t)f (t)dt =
f
f
(x)>0
x
R.) g(x).) g.17.128 2x4 31tf (t)dt x 2x 3 = + -
x
R,
f
(1).164 17.129 :f:(0,+)Rx1xf (x) lnx 1 f (t)dt = + +
x>0.17.130 :)420xA dx x=
)30B x x 1 = +
17.131 :)2e1lnxA dxx=
)440B xdx =
17.132 :4 3 22312x x 16x 9A dxx 9x =
17.133 2 xt1f (x) 2 e dt, =:10I f (x)dx =
17.134 : *40I xdx, = ) : 21I I 1=
>2.)
5.17.135 :214 8C: y x x9 3= + 225 10C : y x x9 3= + )
xx.) -
xx.) -.-C1C2..17.136 f.) : 0 0xf (x)dx f (x)dx2=
) :20xxI dx3 x=+
17.137 f-,
f
(0)=0
f
(0)=1,
:x0x 0xf (t)dtA limx x=
16517.138 fR:x 11 xtf (t)dt f (x) f (t)dt =
x
R
Cf-
(2,1),
f.17.139 :f
(x)=-
x2+4x
(3,f
(3)).
Cf.:) .) -,
Cf
xx.17.140 :x2+y2=8:21y x2= 17.141 :*f: (0, )+ + :22004f (x) x f (x) f (1)e= = ) f:1xf (x) 2004e= ) --:2f (x)g(x)x=
xx
x=1
x=2.
166
18.1.1 :z2+z+=0,
,
R1,:) 1, =) 2, ) 2z z 0 + + = 1.18.1.2 :f,g:RR :) f
(xy)+x+y=xy+f
(x)+f
(y)
-
x,y
R, ) g(xy)+g(x)+g(y)=xy+x+y+3
x,y
R.18.1.3
f:RR
:f2(x)+2x2xf
(x)+1
x
R.)
f
(1).) fx0=1.18.1.4 fx0=0
:23x 01xf (x) x lim 7x= ) :2x 01limx 0x =
) x 0limf (x).
)
Cf
.)
f
(0)=0.18.1.5
f:RR
-
x1=2.
:3f (x 1), x 1g(x)f (2x), x 1 + 1) f -.)
f.) :2x 2x x 2 2 x 3x 2 + ;) -
x=-
x.18.2.9 :)21 x xA dx, x ( , )1 x+ += +
)2(x x)x B dx, x ,2 2 x+ =
)2 x dx, x 0,2x 2x 2 = +
)22x dx1 x=+
18.2.10 :f:RR:)xx t x0x e f (t)dt xe f (x)+ =
x
R)x3 216xf (x) 6 f (t)dt 4x 3x 11 = + +
18.3.1 z,:5z z = )
z . ) :5z z = ) -.18.3.2
f,g:RR :170 (f
g)(x)=x21
x
R.) f g ,
f
g
.) fg
11
-
fg
11.) fg.18.3.3
f:RR -
x0=0
x
R
:f3(x)xf2(x)x2f
(-
x)=x2x)
f
(0).) x 0f (x)lim .x )
f
(0)=1.18.3.4 :f
(x)=x33x2+2x+1-Cff,:) ::x+11y+17=0) ::x+y2004=0)
(3,7).18.3.5
f:RR
:f
(x2)x23x+2f
(x3)+2x-4
x
R.)
f
(x)=x2+x.) ()-1M , 0 .2 -:i) ()Cf-
R.ii) Cf-.iii)-,R,:1y2= 18.3.6 :)1 x 1 1lnx 1 x x+< 0)1 e e < <
,>0
0.) 2x 2 x
x>0,-
=e.18.6.7
f:RR
-
f
(0)=f
(0)=0
f
(0)=2.) x 0f (x)lim .x
) x 0f (x)lim 2.x= ) :2x 0 x f (x)lim 1f (x)ln(x 1)+=+
18.6.8 f--
:y=2x+3.) xf (x)A lim .x= 176 ) xB lim [f (x) 2x].= ) ,:2 22 3xx 3 f (x) 2x x 3lim 4x f (x) 2x 2x 1+ + + += +
18.6.9
f:RR
.-:x0g(x) (x t)f (t)dt, x = )
g(x).)
g
.) Cg.18.6.10
f:RR
-
f
(0)=0
f
(0)=2.) x 0f (x)A lim .x= ) :( )x20dg(x) x f (t)dtdx=
) :x20x 0 x 0x f (t)dtx f (x)B lim limx x x x = =
) :x220 x 01lim x f (t)dt 12x 2x 2 =+
18.7.1
f:RR
(f
f
)(x)=2x
x
R.)
f
(1)=1.) f11.-
f-1;) f-R.) :f
(x2+x)=f
(2x+2)18.7.2
00,
N*.
-:) :xxg(x) =
(0,+),) =e.18.7.3 :f
(x)=3x4+4x3+44
,
R.) f -.)
f.) :34+43+440
,,
R.17718.7.4 -
y=f
(x),f
[0,5]
(0,5).
) f.) f.)
Cf.18.7.5
f:RR
-,
f
(0)=2
x0=0
-0.:f (x)x, x 0g(x)0, x 0= =
) x 0f (x)A lim .x= ) 2x 0f (x)lim 1.x= ) g-
x0=0.) g.) g(x).18.7.6 F-
f
(x)
:2F(x) x 1f (x)e 2xe+=
x
R.
F(0)=1,
:)
F(x)=x2+1,) f,) CfCF,) CfCF.18.7.7 fR.:) :[ ]20f 3 f (x) f (x) xdx 22 = + =
f
(0)=1,) :f
()=1[ ]04f (x) f (x) 2xdx 2 + =
f
(0)=2.18.7.8 :) A xln(x 1)dx, x 1 = + >
)x2xeB dx, x 1(1 x)= > +
18.7.9 :)30x 2A dxx 1+=+
)3x 10B e dx+=
)2xxe dx1 e=+
178 )83x 1 dxx x 1+=+
18.7.10 :2lnxf (x)x= ) Cf
xx.) :x 0 xlimf (x) lim f (x) +
) -Cf,xx:x=1x=e
18.8.1 zw:3 7z w wz 1 = = ) zw.) zw.18.8.2 :22x 2x 1f (x)x 1+ +=+
) f-.)
f.)
f.)
Cf.18.8.3
f:[0,+)R
-
f
(0)=0
f
(x)>0
x>0.)
x>0,
-,
f
(x)=xf
().)
f
.) :f (x)g(x)x= .18.8.4 :) 2x+3x+4x=9x ) xx=2x+4,x>018.8.5 :f:RR :3 x 21f (x) f (x) 2 e x x 12+ + = +
x
R.) f
(0).)
exx1=0.) f.) f-.18.8.6 :( )2 2f (x) xln x x 1 x 1 1 = + + + +) f
=R.) :179( )221f (x) ln x x 1 f (x)x 1= + + =+
) f-.) f.) :( )2 2xlnx x 1 x 1 1 + + = + 18.8.7 :x21e x, x 02f (x)ln(x 1) x, x 0x+ = + >
) f,:=1) f,.18.8.8
f:RR
-:2 21f (x ) f(x)4
x
R.)
f
(0)
f
(1).) f.) f.18.8.9
f:RR
-
f
(0)=1:( )x x y0 0 02x tf(t)dt f(t)dt dy xx 2 + = +
x
R.) :x0xf (x) f (t)dt x xx = +
)
x0
f-.)
f
(x)=x
x0.) f.18.8.10
f:RR*
f
(x)=2xf 2(x)
x
R.
f
(0)=-1:) :21f (x)x 1= +
) :10I f (x)dx =
x=y.18.8.11 -:)22f (x) ln xx= + )2 4g(x) lnx 2 xxx= +
180
18.9.1
=3+4i
z z z 50. + = ) z-().) z0,.z0;)
w
C
w 2, z w 3. 18.9.2
f:RR
:f
(0)=f
(0)=f
(0)=2yf (x y) f (x) yf x2 + = +
x,y
R.
:)y y yf (x y) f x f x2 2 2 + = + + +
x,y
R, ) f
(x)=2+2x
x
R, ) f
(x)=x2+2x+2.18.9.3 :f
(x)=x4+2x3x+1) f.)
Cf.) Cf
.18.9.4 :)42x 0xlim 12x 2x 2=+
)2x 0xx xlim 1xe 1 x2=
)2 xx 0 x elim 1x x =
)2x 01xx lim 1x= 18.9.5 :2x x 2f (x)x 1+ +=
) f-.) f .) f.)
Cf.)
f.)f.18.9.6 :)
ex-1=xlnx+1)
ex-11xlnx18.9.7 F-
f:RR
:f
(0)=1f
(x)F(-x)=1
x
R181:) F(x)F(-x)=1
x
R, ) F(x)F(-x)=1
x
R, )
g(x)=e-xF(x)
-R, )
f
f
(x)=ex.18.9.8 fR:8 64 3f (3x)dx f (4x)dx =
:)2412 f (x)dx 0, =
) :x12g(x) f (t)dt =
...-
[12,24],)
(12,24)
,:f
()=018.9.9
(2,-3)
-
x2=4y.:)
,) -.18.9.10 :t 4xtxe 5tf (x) dt, x1 e+= +)
f.)
f
(x)=x5+x.18.9.11 :f (x) 1 x = +
x
[0,].) f-.)
f.) -
Cf,xx
x=0
x=.18.9.12 :f
(x)=x33x2+5x62+23
) f.)
Cf.) -R,
Cf-.
182
19.1
z=x+yi,
x,y
R
.)
+i
:z + 8iw =z +6 )
x
y,
Im(w)=0.)
x
y,
Re(w)=0.) ().) .).)
z-
6(x,y)(-
6,0).
:[ ][ ]2 2x (y 8)i (x 6) yi z 8i x yi 8i x (y 8)iwz 6 x yi 6 (x 6) yi (x 6) y+ + + + + + + += = = = =+ + + + + + +
2 22 2x 6x y 8y ( xy xy 6y 8x 48)i(x 6) y+ + + + + + + += =+ +
2 22 2 2 2x y 6x 8y 8x 6y 48i(x 6) y (x 6) y+ + + + += ++ + + +
) :Im(w)=08x+6y+48=04x+3y+24=0,
(-
6,0).183) :2 22 22 2x y 6x 8yRe(w) 0 0 x y 6x 8y 0 (1)(x 6) y+ + += = + + + =+ +
) :x2+y2+6x+8y=0(x2+6x)+(y2+8y)=0(x2+2
3x+9)+(y2+2
4y+16)=9+16(x+3)2+(y+4)2=52
(1)
(-3,-
4)
=5.
(-
6,0)
-
z,
Re(w)=0.) (1)
(0,0).
C.)
C:(x+3)2+(y+4)2=52,
(-
3,-
4)
=5,
z 3 4i 5. + + = 19.2 z - 1 = z - 3i ,
z
C .) z
x3y+4=0.)
(1,0)
(0,3);)
z z .)
z=x+yi.
:z 1 z 3i (x 1) yi x (y 3)i = + = + 2 2 2 2(x 1) y x (y 3) + = + x22x+1+y2=x2+y26y+92x6y+8=0x3y+4=0().184 ) z 1 z 3i =
(1,0)
(0,3).
.) z,z 1 z 3i , = -.
.
:1 3= =
:y=-3x
y=-3x
x3y+4=0
:2x 3(3x) 4 0 10x 4 0 x5 + = + = = :2 6y 3x 35 5 = = =
:2z (1 3i)5= + 19.3
z
3z + z = 0. ) z. ) 3z + z = 0. ) z1z2,31z 32z . ) 2004 20041 2A = z + z . ) 3 3z z 0 z z . + = = :( )( )3 23z z z z z z 1 0 z 0 z 1 = = = = = 185) z 0. = :3 3z z 0 z 0 z 0 + = = = z 1. = :3 3 2 2z z 0 z 1 0 (z 1)(z z 1) 0 (z 1 z z 1 0) + = + = + + = = + =
=14=-30.
>0,.186 19.5 z7 3z z = 1. ) z = 1. ) z zz
.) 7 3z z = 1. ) 7 3z z 1. = z z , = :7 3 7 3 107 3zz 1 z z 1 z z 1 z 1 z 1 = = = = = z 0
z
C.) :2 1z 1 z 1 zz 1 zz= = = = ) 1z ,z= :37 3 7 4 2 21z z 1 z 1 z 1 (z 1)(z 1) 0z = = = + =
(z=1
z=-1
z=i
z=-
i)..19.6
z z - 2 - 2i = 3 2. ) z
.) : i) ii) ) :z 2 2i 3 2 = 187 z (2 2i) 32, + = z
(2,2)
32. = ) C
(2,2)
32 = :C:(x2)2+(y2)2=18(1)
C
.i) -z.:
:y=x(2)(2)2 2(x 2) (y 2) 18 + =
(x2)2+(x2)2=18
(x2)2=9x2=3(x=5
x=-1)
x=5
y=5.
x=-1
y=-1.
z=5+5i.ii)
z=-1i.
19.7 fR:(
f
f
)(x)f
(x)=x
x
R)
f
(x).)
f
(0).) f-1f
11.
:f
(f
(x))=x+f
(x)(1)
x1,x2
R
f
(x1)=f
(x2).188 :( ) ( )(1)1 2 1 1 2 2 1 2f f (x ) f f (x ) x f (x ) x f (x ) x x = + = + = f
11,
f-1:f
(R)R.) (1)
x=0
:( ) 11f f (0) f (0) f (0) 0= = 19.8
f:R(0,+)
g:R(-,0).
,
R
f
()=
g()=,
:) 0
x
R,
=f
()>0.
g(x)x
xP(x)..) g(x)0f
(x)>f
(0)
f
(x)f
(0)
x
R.
f
(0)=0
.212 ) :ex+x2=x+1ex+x2x1=0f
(x)=0
f
(0)=0
f
(x)>0
x0.
x=0
f
(x)=0.)
f
(0)=0
,:f
(x)0ex+x2x10ex1x(1x)
x
R.19.30
f(x)=xx,
00.)
+1>(+1)
>e.)
2x=x2
x>0.) f
=(0,+).
:2lnx 1 lnxf (x)x x = =
f
(x)=0lnx=1x=ef
-f
(0,e]
[e,+).)
>e
f
[e,+).
ln 1f () f (e) eln e< < < lne0.
:e x e xlnx 1f (x) f (e) elnx x lnx lne x ex e
exxe
x>0.)
e + > + +
ln+1>ln(+1)+1>(+1)
)
x>0
:x 2 x 2ln2 lnx2 x ln2 lnx xln2 2lnx f (2) f (x)2 x= = = = =
(0,e]
f,
11.
f
(x)=f
(2)x=2215
[e,+)
f,
11.
:f
(2)=f
(4)4
(e,+):f
(x)=f
(2)f
(x)=f
(4)x=4
x=2
x=4.19.33
f(x)=ex-1lnx1.)
f
.)
ex-1=lnx+1.)
x+elnx
x>0,
=e.) f
=(0,+).
:x 1 x 11f (x) (e lnx 1) ex = =
x
x 121f (x) e 0x= + >
x
f
(1)=0f
,: x0f
[1,+).
f
(1)=0
f.) :ex-1=lnx+1ex-1lnx1=0f
(x)=0: f
(1)=0 xf
(1)f
(x)>0 f
(0,1].216 x>1f
(x)>f
(1)f
(x)>0 f
[1,+).
f
(x)>0
x1,
x=1
f
(x)=0.)
g(x)=xelnx,
x>0.
g(x)0=g(1)
x>0.
-Fermat
g(1)=0.
:xeg(x) lnx= eg(1) 0 ln e 0 ln 0 e= = = = eh(x) lnxx=
h(e)=0.
=e
:exe+elnxex-11+lnxf
(x)0,0f.19.34 :)
xx=ee
=(0,+)
x=e,)
x+elnx
x>0,
=e.) :x e x ee ex e lnx lne xlnx e lnx lnx 0x x= = = = = :eg(x) lnx ,x=
x>0:eg(e) lne 1 1 0,e= = =
x=e
g(x)=0,
.21721 eg(x) 0x x= + >
x>0.g(x),
x=e
-.)
f
(x)=xelnx,
x
=(0,+).
-
f
(x)0
x>0.
f
(1)==0,
:f
(x)f
(1)(1)
x>0.
(1)
x0=1
(-)f
(x).,
x0=1
-
=(0,+)
f
x0=1
(-).Fermat
f
(1)=0.:xef (x) lnx=
f
(1)=0lne=0ln=eln=e=ee(2),
xx=ee
(0,+)
x=e.
(2)
=e.19.35 :x + x f(x) = ln - lnx - ln2 x + x +
>0.)
f
.)
00f2(x)+1>0
x
R f.) (1)
x=0
:f3(0)+3f
(0)=0f
(0)[f2(0)+3]=0f
(0)=0
x=0
f
(x)=0.
f-(),-.) f
=R,
:
x0f
(-,0)
(0,+).f.) f
=R,
f,.19.39 :f(x)=2+ex-1g(x)=2+e1-x
)
f
g.) CfCgfg.)
x=
CfCg,-
Cf
Cg
.223) fg
=R.
x
R
: f
(x)=ex-1>0,
f. g(x)=-e1-x0,
g(x)=xexex+1.)
g(x).)
f
=(0,+).224 )
x
R
: g(x)=(xexex+1)=ex+xexex=xex
g(x)=0x=0g
(-,0]
[0,+).: xg(0)g(x)>0 x>0g(x)>g(0)g(x)>0
g(x)>0
x0.)
x>0
:x x x2 2e 1 xe e 1 g(x)f (x) 0x x x += = = >
g(x)>0
x>0.
f.19.41
f:R*R,
:1f (x) =x
x0f(1)=f(-1)=2)
f.)
Cf
.)
f.) xx
Cf
.
Cf
,.) 1f (x)x=
x0.
( )1ln xx= R*-,:ln x , x 0f (x)ln x , x 0 +
225: f (1) 2 ln 1 2 2 = + = =
f
(1)=2ln1+=2=2 f (x) ln x 2, = +
x
=R*.) :f ( x) ln x 2 ln x 2 f (x) = + = + =
x0.
f,Cfyy.) f
=R*.
:( )x 0 x 0limf (x) lim ln x 2 = + = ( )x xlim f (x) limln x 2+ += + = +f
f
()=R.)
y=
Cf-
(,f
())
(,f
()).
:f
()=f
()==-yy.Cf,:f
()f
()=-1f
()f
(-)=-121 11 1 ( 1 1) ( ) = = = =
(1,2)
(-1,2).19.42
f,g:RR ,
:f(x)g(x)=x4x
R
y=3x7
Cf
+
.226 ) : i) x +g(x)limxii)2x +g(x) + 5x + 2xlimxf(x) - 3x + 1
)
y=2x3
Cg
+
.) i)
y=3x7
Cf+,: [ ]x xf (x)lim 3 limf (x) 3x 7 (1)x+ += =
f
(x)g(x)=x4g(x)=f
(x)x+4(1)x x xg(x) f (x) x 4 f (x) 4lim lim lim 1 3 1 2x x x x+ + + + = = + == =
ii)2 2x xg(x) 5x 2x f (x) x 4 5x 2xlim limxf (x) 3x 1 xf (x) 3x 1+ ++ + + + += = + +
( )2x xf (x) 2x4x x x1x4f (x) 4x 4 2x 3 4 0 0lim lim 17 0 xf (x) 3x 1f (x) 3x+ ++ + ++ + + + + += = = = + + +
)
y=2x3
Cg+,-:[ ]x xg(x)lim 2 limg(x) 2x 3x+ += = xg(x)lim 2,x+= ()(i).:[ ] [ ] [ ]x x xlimg(x) 2x limf (x) x 4 2x limf (x) 3x 4+ + + = + = + =[ ](1)xlimf (x) 3x 4 7 4 3+= + == + = ,,
y=2x3
Cg+.19.43
f(x)=2+ln(x1)
g(x)=2ln(x1),
x>1,
Cf
Cg
.227.:)
Cf
Cg,)
f
g..
Cf
Cg
..) fg
=(1,+).
CfCg
f
(x)=g(x).
:f
(x)=g(x)2+ln(x1)=2-ln(x1)2ln(x1)=0x1=1x=2CfCg
(2,2).) :1f (x) 0,x 1= >f.x 1 xlimf (x) lim f (x) . += = + f,
f
()=R.: [ ]1 g(x) 2 ln(x 1) 0x 1= = 0.
-,-:(-2,-1),(0,1)(2,+))
x
Df
:3 32 2( x) 4( x) x 4xf ( x) f (x)( x) 1 x 1 = = =
f.,Cf.) :3 2 2 32 2 2x 4x (3x 4)(x 1) 2x(x 4x)f (x)x 1 (x 1) = = =
4 2 2 4 2 4 22 2 2 23x 3x 4x 4 2x 8x x x 4(x 1) (x 1) + + + += =
f
(x)>0
x
Df,
f:1=(-,-1),2=(-1,1)3=(1,+))f1,23.:3 32 2x x x xx 4x xlim f(x) lim lim lim xx 1 x = = = =
32x 1 x 11lim f (x) lim (x 4x)x 1 = = +
:f
(1)=(-,+)=R230 3 32 2x 1 x 1 x 1 x 11 1lim f (x) lim (x 4x) lim f (x) lim (x 4x)x 1 x 1+ + = = = = +
:f
(2)=(-,+)=R332 2x x x 1 x 11 xlim f (x) lim (x 4x) lim f (x) limx 1 x+ ++ + = = = = +
:f
(3)=R,f:f
(Df)=f
(1)
f
(2)
f
(3)=R) :x3x24x+=0x34x=x2x34x=(x21)
x=1
x=-1
.,
x1
:32x 4x f (x) x 1= =
f
(1),
f
(2)
f
(3)
R
f1,23f
(x)=
1,23.:1
(-,-1),2
(-1,1)3
(1,+)19.45 fR
f2(x)xf(x)+x23=0
x
R .)
x=
f,
=-1
=1.)
Cf
.)
x=
ffR,
f
()=0.
:f2(x)xf
(x)+x2-3=0(1)231:2f
(x)f
(x)f
(x)xf
(x)+2x=0(2)
x=,
:2f
()f
()f
()f
()+2=0-f
()+2=0f
()=2
x=
(1):f2()f
()+23=0(2)2(2)+23=0323=02=1(=-1=1)) f
x=.
f-,
f
()=0.
(2):2[f
(x)]2+2f
(x)f
(x)f
(x)f
(x)xf
(x)+2=02[f
(x)]2+2f
(x)f
(x)2f
(x)xf
(x)+2=0
x=
:2[f
()]2+2f
()f
()2f
()f
()+2=0[f
()]2f
()+1=0,
=14=-3-1.)
f(x)
.)
x1+ln(x+1)
x>-1,
=e.)
x+1>0x>-1.
=(-1,+).
Cfyy
f
(0)=11ln1=0,
(0,0).)
x
:x x1f (x) [e 1 ln(x 1)] ex 1= + = +
f
(0)=11=0.x21f (x) e 0(x 1)= + >+
x
.f
,x=0-
f
(x)=0.f
(x): x0 f
2=[0,+)
f
2=[0,+).) f
f
(0)=0
()f
(x).) Cfxx,
f
(x)=0.234 : f
(0)=0,
x=0
. Hf
(x)
(-1,0],
:xf
(0)f
(x)>0 Hf
(x)
[0,+),
:x>0f
(x)>f
(0)f
(x)>0
f
(x)>0
x0,
x=0
f
(x)=0.)
f
(0)=0
f
(x).
x>-1:f
(x)f
(0)ex1ln(x+1)01+ln(x+1)ex
)
1=(-1,0]
:x 1 x 0lim f (x) limf (x) f (0) 0 = + = =
f1
:f
(1)=[0,+)
f
(x)0
x
=(-1,+),
f:f
()=[0,+)f
(2),
2=[0,+)
,
[0,),
R
{+}.
f
(1)=[0,+)
:f
()=f
(1)
f
(2)=[0,+)) :g(x)=x1ln(x+1),x>-1: g(x)0=g(0)
x>-1
()g(0)()g(x).235
x0=0
=(-1,+)
g x0=0.Fermat
g(0)=0.
:x1g(x) lnx 1= +
g(0)=0ln1=0ln=1=e19.48
f:RR
f
(0)=f
(0)=0
f
(x)0f
(x)0g(x)>g(0)g(x)>0 x0f
(x)>f
(0)f
(x)>0)
x>0
xf
(x) > > > > < 0.
h(x)=f
(x)x,
x0.
:22 21 f(x)h(x) f (x) 1 1 01 f(x) 1 f(x)= = = 0h(x)0
f
(x)>0.
g
[0,+),
:x>0g(x) > >
xf
(x)0.) Cf
y=x+1.
:23x 1 1f (x) x 1 x 1 x 1 x3 x= + + + = + = :[ ]11 11 1213 333 1 1E f (x) (x 1) dx dx 3lnxx x x |= + = = + = | \ .
1 13ln1 3ln 3 3ln3 21 3 = + + =
19.55 3 2xxf(x) = e .x + 2
)
f.)
f
.) : x + x -A = lim f(x) B = lim f(x) ) ,, x -2lim f(x). ) x 0lim f(x) .) :x+20x0x-2
x0:Df=R{0,-2}252 ) :3 3 3 3 3 2 2 2 2x x x x x2 2 2x 2x(x 2) x x 3 x 4x 3f (x) e e e e ex 2 x 2 x 2 (x 2) x (x 2) + + = = + = = + + + + +
3 3 2 2x x2 2x 4x 3(x 2) x x 6e e(x 2) (x 2)+ + + = =+ +
x
Df.
f
(x)=0x2+x6=0(x=-3
x=2)f:(-,-3][2,+):[-3,-2),(-2,0)(0,2]) 3 2xx xxlim lim e 1,x 2+ += + =+
=+. 3 2xx xxlim lim e 1,x 2 = =+
=-.)
x0.
:32xx 2 x 21lim f (x) lim x ex 2+ + = = + +
x 2lim f (x).) :x 03lim ,x= 3xx 0lim e 0.= x 0lim f (x) 0 0 0.= = x 03lim ,x+= + 3xx 0lim e .+= + :253( )33 3x3x x22xx 0 x 0 x 0 x 02 33x1 2 1xx xee 3 elim x e lim lim lim2+ + + + = == = ==
( )( )3x32xx 0 x 023x1xe3 9lim lim e2 2+ + = = =+
x 0 x 0lim f (x) lim f (x), + x 0limf (x).19.56 ,,:
P(x)=x3+x2+x+.
,,-.P(x).;:.-,.:P(x)=3x2+2x+,
P(x)0
x
R.
:
=1.
,:P(x)=x3+x2+x+
=2,
:=4212=42122=-820,
P(x)-.P(x).f
=R,
x xlim P(x) lim P(x) += = +(
>0),P(x)
P(A)=R.254
0
P(A),
P(x).-,P(x).19.57
f:RR
:f(x+y)=f(x)ef(y)-y+f(y)x,y
R) f(0).)
R*,
f()=0,
f(x)=0
x
R .) f.) .) :f
(x+y)=f
(x)ef(y)-y+f
(y)(1)
x=y=0
:f
(0)=f
(0)ef(0)+f
(0)f
(0)ef(0)=0f
(0)=0)
f
()=0,
0.
(1)
x=
:f
(+y)=f
()ef(y)-y+f
(y)f
(+y)=f
(y)
y=x
:f
(+x)=f
(x)(2)(1)
y=
:f
(x+)=f
(x)ef()-+f
()f
(x+)=f
(x)e-(3)(2)(3):( )f (x) f (x)e f (x) 0 f (x) 01 e= = =
x
R,
0
1e-0.)
f
(x)=0
(1).()f
x=0,
f-R.255
f
(x)0
x0.
(1)
y=-
x
:f (0) 0f ( x) x f ( x) xf (0) f (x)e f ( x) f (x)e f ( x) (4)= + += + = (4)x
x:f
(-
x)ef(x)-x=-f
(x)(5)(4)(5):f
(x)f
(-x)ef(-x)+f(x)=f
(x)f
(-x)ef(-x)+f(x)=1f
(-x)+f
(x)=0f
(-x)=-f
(x)(6)
x0,
f
(x)f
(-
x)0
x0.
f
(0)=0,
(6)
x=0.
f.) (1):f
(x)=0
f
(x)0
x0.
f,(5):( )f (x) x f (x) xf (x)e f (x) f (x) 1 e 0 x = =
x0
:1ef(x)-x=0ef(x)-x=1f
(x)x=0f
(x)=x
f
(0)=0,
f
(x)=x
x=0.
f
(x)=x
-(1).,,
f
(x)=0
f
(x)=x.
19.58 F
f:RR ,
:F2(x)F(x)F(x)
x
R ,
0:) F(0)=F(),)
f(x)=0
R .256 ) :F2(x)F(x)F(x)(1)(1)
x=0
:F2(0)F(0)F()(1)
x=
:F2()F()F(0):F2(0)+F2()2F(0)F()[F(0)F()]20F(0)F()=0F(0)=F())
00,
(0,+).. f
x0=1,
f
(1)=2:) f
(0,+),) f,) I = f(x)dx. .) :f
(xy)=f
(x)+f
(y)+xyxy(1)
x=y=1:f
(1)=f
(1)+f
(1)+111f
(1)=1) f
x0=1,
:x 1limf (x) f (1) 1 (2)= =
x0>0.
:[ ]00 0 0 0x x h 1 h 1lim f (x) limf (x h) lim f (x ) f (h) x h x h = = + + = (2)0 0 0 0 0 0 0h 1 h 1f (x ) limf (h) lim(x h x h) f (x ) 1 (x x 1) f (x ) = + + == + + = 258 f
(0,+).
0xhx= :
x=x0hh1,00x x h 1lim f (x) limf (x h). = ) f
>0,
:x limf (x) f ()= [ ]h 1 h 1limf (h) f () lim f () f (h) h h f () = + + =
g(h)=f
()+f
(h)+hh,:h 1limg(h) f ()=
f
(h)=g(h)f
()h++h [ ]h 1 h 1limf (h) lim g(h) f () h h f () f () 1 1 = + + = + + = :h 1limf (h) 1 f (1)= = f
x0=1,
-f
(0,+).. ) 0xh,x=
x=x0h
h1
xx0.
-
x0>0.:00 0 0x x h 10 0 0f (x) f (x ) f (x h) f (x )lim limx x x h x = =
0 0 0 0h 10f (x ) f (h) x h x h f (x ) 1limx h 1+ + = =
( )0 0h 10f (h) 1 (1 x h x h)1limx h 1 + + = =
2590h 10x (h 1) (h 1) 1 f (h) f (1)limx h 1 h 1 = + =
[ ]0 00 0 01 1 1f (1) x 1 (2 x 1) 1x x x= + = + = +1f (x) 1x= +
x>0.) :1f (x) 1 f (x) (lnx x) f (x) lnx x cx= + = + = + +
x>0.
:f
(1)=1ln1+1+c=1c=0
f
(x)=lnx+x.) :I f (x)dx (lnx x)dx (x)lnxdx xdx = = + = + =
2 21 x xxlnx x dx xlnx x cx 2 2= + = + +
19.61 f
[0,],
0 f(x)dx = 2 Ff.)
F(0)F().)
(0,)
,
f()=.) Ff,:0 f (x)dx F() F(0) =
:0 f (x)dx 2 =
F()F(0)=2,
F(0)F()=-2.260 )
f
()=
g(x)=f
(x)x,
x
[0,].g
G(x)=F(x)+x,
:G(x)=(F(x)+x)=F(x)x=f
(x)-xG
[0,]
(0,).
: G(0)=F(0)+1 G()=F()1:G(0)=G()F(0)+1=F()1F(0)F()=-2().RolleG
[0,]
(0,)
,:G()=0f
()=0f
()=19.62 f
f(1)=f
(1)=0.) : 1 120 01f(x)dx = xf (x)dx2
) 36f (x) = ,x + 1: 10I = f(x)dx ) .:[ ]1 1 1100 0 0f (x)dx xf (x)dx xf (x) xf (x)dx = = =
12 2 21 10 00x x xf (1) f (x)dx 0 f (x) f (x)dx2 2 2 |= = + = |\ .
1 12 20 01 1 1f (1) x f (x)dx x f (x)dx2 2 2= + =
261) ():2 21 1 1 123 30 0 0 01 1 6x 3xI f (x)dx x f (x)dx dx dx2 2 x 1 x 1= = = = =+ +
31 1330 0(x 1)dx ln(x 1) ln2x 1+ = = + = +
:21303xI dxx 1=+
x3+1=y,
3x2dx=dy.
:2 21 11I dy ln y ln2y = = =
19.63
f:RR
xt - f(t)0f(x) = e dt
x
R .)
f
(x)ef(x)=ex
x
R .)
f.) :xt f (t )0f (x) e dt (1)=
(1)x,-f.(1):f
(x)=ex-f(x)f
(x)ef(x)=ex(2)) (2):( )f (x) x f (x) x f (x) xf (x)e e e (e ) e e c (3) = = = + (1)
x=0
f
(0)=0,
(3)
x=0
:ef(0)=1+cc=0
ef(x)=exf
(x)=x
x
R.262 19.64
f(x)=(x1)lnx,
x>0.) f.) f,.) f.) Cf.) :I = f(x)dx )Cf,xx
x=e.) f
=(0,+).
:x 1f (x) lnxx= +
x>021 1f (x) 0x x= + >
x>0f
.: x0,f
(0,1]
-
[1,+).)
xf
(1).
x>1
f
(x)>f
(1).
f
(x)f
(1)=0
x>0.
f
(1)=0
f.) xlim f (x) ,+= + f0
=(0,+),:f
()=[0,+)263) :x 0limf (x) ,= +
x=0
Cf.xlim f (x) ,+= + Cf.) :2 2 2x x x 1I (x 1)lnxdx x lnxdx xlnx x dx2 2 2 x = = = =
2 2 2x x x xx lnx 1dx x lnx x c2 2 2 4 = = + +
)
f
(1)=0
f
(x)>0
x1.:
x
(0,1)
f
(x)>f
(1)f
(x)>0,
f-
(0,1],
x
(1,+)
f
(x)>f
(1)f
(x)>0,
f-
[1,+).
f
(x)>0
x1.
f
(x)=0
x=1.
-
[1,e]
f
(x)0,
:e2 2e11x xE f (x)dx xlnx x2 4 |= = + = |\ .
2 2 2e e 1 e 3e e 12 4 4 4 | | | |= + + = | |\ .\ .
19.65
y=f(x)
R: xt x02 e f(t)dt = (e + 1)f(x) - 4
x
R)
f
.264 ) f(0).) f:xxef (x) = f(x)1+ e
x
R)
f.) :( )xtx01f (x) 2 e f (t)dt 4 (1)e 1= ++
.f.)
x=0
((1))
f
(0)=2.) :2exf
(x)=exf
(x)+(ex+1)f
(x)xx xxee f (x) (e 1)f (x) f (x) f (x)1 e = + =+
x
R.) :xx xxef (x) f (x) f (x)(1 e ) e f (x)1 e= + = +
f
(x)(1+ex)(1+ex)f
(x)=0x xx 2 xf (x)(1 e ) (1 e )f (x) f (x)0 0(1 e ) 1 e+ + = = + +
xxf (x)c f (x) c(1 e )1 e = = ++
:f
(0)=22c=2c=1
f
(x)=1+ex
x
R.26519.66
f:RR
: g(x) = f(x - t)dt ,x , < ) g.)
g()=g().)
(,)
,
f()=f().) :g(x) f (x t)dt =
xt=y.
dt=-
dy
: x x 0 x x x x 0g(x) f (x t)dt f (y)dy f (y)dy f (y)dy f (y)dy = = = = + =
x x 0 0f (y)dy f (y)dy =
( )x x 0 0g(x) f (y)dy f (y)dy f (x ) f (x ) = =
x
R)
g()=g(),
: 0 0 0 0f (y)dy f (y)dy f (y)dy f (y)dy =
0 0 0 0f (y)dy f (y)dy f (t)dt f (t)dt 0 (1) = + =
t=-
y,
dt=-
dy.: 0 0 0f (t)dt f ( y)( dy) f ( y)dy = =
f,
f
(-y)=f
(y)
y
R.
: 0 0 0 0f (t)dt f ( y)dy f (y)dy f (t)dt = = =
266 : 0 0f (t)dt f (t)dt 0 + =
(1),
g()=g().) : g()=g(), g
[,], g
(,).Rolle,
(,)
,
g()=0.:f
()f
()=0f
()=f
()19.67 e21lnxA = dx.(1 + lnx)
) 2x-121x - 1A = e dx.x
) .)
1+lnx=y.
lnx=y1
x=ey-1.
dx=ey-1dy
:e 2 2y1 x12 2 21 1 1lnx y 1 x 1A dx e dy e dx(1 lnx) y x = = =+
) ():2 2x 1 x 121 1x 1 1A e dx (x 1)e dxx x = = =
2x 12 2x 1 x 1 x 11 11(x 1)e 1 e 1e (x 1)e dx xe dxx x 2 x = + + = + =
[ ]2 2x 1 x 111e e e ee dx e e 1 12 2 2 2 = + = + = + =
26719.68 :2x-322x-31(x - 1)eI = dx(x - 1)e + 2 - x
)
x=3y
: 22x-31(2 - x)I = dx(x - 1)e + 2 - x
) .)
x=3y,
:dx=-
dy:2(3y) 3 32y1 22(3y) 3 32y2 1(3 y 1)e (2 y)eI ( dy) dy(3 y 1)e 2 (3 y) (2 y)e y 1 = = = + +
3 2x2 23 2x 2x 31 1(2 x)e (2 x)dx dx(2 x)e x 1 (x 1)e 2 x = = + +
e2x-3.) -.:2x 32 22x 3 2x 31 1(x 1)e 2 xI I dx dx(x 1)e 2 x (x 1)e 2 x + = + = + +
2x 32 22x 31 1(x 1)e 2 xdx 1dx 1(x 1)e 2 x + = = = +
12I 1 I .2= = 19.69
f:(0,+)R
: x1- f(t)dtf(x) = e
x>0268 .:)
f
(x)=-
f2(x)
x>0)1f(x) =x
x>0. Cf-.:) =,) ,..) :x1 f (t )dtf (x) e (1)=
x>0.
f,x1 f (t)dt-.,(1),f,:( )x x1 1f (t )dt f (t )dt (1) x21f (x) e f (t)dt e f (x) f (x)f (x) f(x) = = == =
:f
(x)=-
f2(x)(2)
x
R.) (1)
f
(x)>0
x>0.
(2):2 f (x) 1 1 11 ( x) (x) x c (3)f (x) f (x) f (x) f(x) = = = = +
(1)
x=1
f
(1)=e0=1
(3)
x=1:11 c 1 1 c c 0f (1) = + = + = :1 1x f (x) , x 0f (x) x= = > 269. ) :21f (x)x=
x>0.
(,f
())
Cf.:yf
()=f
()(x)21 1y (x ) =
y=0
x=0
:21 1(x ) x x 2, = = =
(2,0),21 1 2y (0 ) y , = = 2B0, . :A B A B20x x y y 2 0 1, , ,2 2 2 2 ++ + + = =
.) ,:1 1 2E OA OB 2 2 ..2 2 = = = ,-.19.70
f(x)=x2(e-x2)
g(x)=x2(x2ex).)
f(x)g(x)
x
R .)
f(x)=g(x).) CfCg
x=1.) fgR.:f
(x)g(x)x2(e-x-2)x2(x2ex)x2(e-x2x2+ex)0e-x+exx220(1)270
h(x)=ex+e-xx22.
: h(x)=exe-x2x,
h(0)=0 h(x)=ex+e-x2,
h(0)=0 h(x)=exe-x
h(x)=0ex=e-xx=0h(x)h
(-,0]
[0,+).
: xh(0)=0 x>0h(x)>h(0)=0
h(x)>0
x0.
hR,h.:
xh(0)=0.h
(-,0]
[0,+).h(0)h,:h(x)h(0)ex+e-xx220(1),
f
(x)g(x)
x
R.) h
(-,0]
[0,+).
: xh(0)h(x)>0 x>0h(x)>h(0)h(x)>0
h(x)>0,
h(x)0
x0.
h(0)=0,
x=0
h.:f
(x)=g(x)=0x2h(x)=0x=0271) CfCg
(0,-1).
f
(x)g(x)
x
[0,1],
:[ ]( ) ( ) [ ]1 12 x 2 2 x0 0E f (x) g(x) dx x e 2 x x e dx= = =
( ) ( ) ( )15 31 1 12 x x 4 2 2 x x0 0 00x 2xx e e dx x 2x dx x e e dx5 3 = + + = + + =
( ) [ ] ( )1 12 x x x x001 2x e e 2x e e dx5 3 = + + + =
( ) [ ] ( )1 1x x x x001 13e 2 xe e 2 e e dxe 15 = + + + =
[ ]1x x01 1 13e 2 e 2 e ee e 15 = + + + =
1 2 2 13 5 13e 2e 2e ee e e 15 e 15= + = 19.71 fgR:f
(x)g(x)=(x2+2x1)ex
x
R)
h(x)=f(x)g(x),
-
(0,-1).)
Cf
Cg.) :f
(x)g(x)=(x2+2x1)ex(1):h(x)=(x2+2x1)ex
h(0)=-1,
Ch
(0,-1).
-:2 x 2 xh(x) (x 2x 1)e dx (x 2x 1)(e )dx = + = + =
272 x 2 x x 2 xe (x 2x 1) (2x 2)e dx e (x 2x 1) (2x 2)(e )dx = + + = + + =
x 2 x xe (x 2x 1) (2x 2)e 2e dx = + + + =
2 x 2 x(x 2x 1 2x 2 2)e c (x 1)e c = + + + = + :h(0)=-1-1+c=-1c=0
h(x)=(x21)ex.) :f
(x)=g(x)f
(x)g(x)=0h(x)=0(x21)ex=0(x=-1
x=1)
h(0)=-1 0x
)
f
.)
Cf.)
f.) f.)
f.)Cf,
xx
x=1
x=e.) lnx 1f (x)x+=
=(0,+).
:2 21x x 1 (lnx 1)lnxf (x) , x 0x x += = >
f
(x)=0lnx=0x=1f
(0,1]
[1,+).
f
(1)=1
()-f.) :x 0 x 01limf (x) lim (lnx 1)x = + =
x 0 x 01lim lim(lnx 1) .x+ = + + =
x=0
Cf.x x x xlnx 1 (lnx 1) 1lim f (x) lim lim lim 0x (x) x+ + + ++ += == = =
y=0
Cf+.277)
1=(0,1]
2=[1,+),
:
f
1
2,x 0 x 1limf (x) lim f (x) f (1) 1, = = =
f
(1)=(-,1],x x 1lim f (x) f (1) 1 lim f (x) 0,++ = = =
f
(2)=(0,1].f:f
()=f
(1)
f
(2)=(-,1]
(0,1]=(-,1]) :2 4 3lnx x 2xlnx 1 2lnx f (x)x x x = = =
x>0121f (x) 0 1 2lnx 0 lnx x e x e2= = = = = f (0, e :) e,+
( )( ) M e, f e , :3 eM e,2e
Cf.) f.
f.278 )
f
(x)>0
x
[1,e],
:e e e1 1 11 lnxE f (x)dx dx (1 lnx)(1 lnx)dxx+= = = + + =
e2 2 211 1 1 1 3(1 lnx) (1 1) (1 0) 2 ..2 2 2 2 2 = + = + + = =
e1 f(x)dx
1+lnx=y.
1dx = y dyx: 22211y 4 1 3E = y dy = = - = ..2 2 2 2
19.75 : x12xt + ln tf(x) = dt ,t + 1
x0) f(x) = ln x
x0.) Cf,xx
y=1.) x12xt ln tf (x) dt,t 1+=+
x0.
:11 x xx12 2 2 21 1 1xt ln t t ln t t ln t t ln tf (x) dt dt dt dtt 1 t 1 t 1 t 1+ + + += + = + + + +
( )2 2 2 2 21 1x x1xlnx ln x 1 x ln x 1 xln xf (x)x 1 x x 1 x(1 x )1++ + = = + = + + + +
22x xln x 1 xln x 1x x(1 x )+ + = =+
x0.279:ln x , x 0f (x)ln x , x 0 +
:121t ln tf (1) dt 0 f (1) 0t 1+= = =+
=0
=0.
f (x) ln x =
x0.) f,:f
(-x)=f
(x)
x
R*.
Cfyy.Cf
x>0,
f
(x)=lnx,Cf.:f (x) 1 ln x 1 x e x e = = = =
=21,
1-Cf,
y=1.
:[ ]e ee111 1E e 1 lnxdx e (x)lnxdx e xlnx x = = = =
=e[(ee)(01)]=e1
=2e2.19.76 f(x) = 2x + x ,
x0.)
f.)
f.) f-1.) 2 -10I = x f (x) dx. 280 )
x>0
:1f (x) (2 x) 02 2x x= + >+
2+x>0.
f
=[0,+),
f
[0,+).)
f
(0)=0
xlim f (x) .+= + :xf (x) 2x x x2x = + = + :x 1 1 x 1x x x x x x1lim 0,x+= :xxlim 0x+= :xxlim2 2 0x+ + = >
:x xxlimx2 lim f (x)x+ + | |+ = + = + | \ .
f
f
()=[0,+).) f,
11.
f-1
f,
[0,+).)
x=f
(y).
:
dx=f
(y)dy
0=f
(y)y=0 2 f (y) 2y y 2 y 2 = + = = 281:( )2 2 21 10 0 0I xf (x)dx f (y)f f (y) f (y)dy f (y)yf (y)dy = = = =
22 22 2 20 001 1 1 y f(y) dy yf(y) f(y)dy2 2 2 |= = = | \ .
2 22 2001 1 1 12f(2) (2y y)dy 2 4 y y2 2 2 2 = + = =
2 2 2 2 21 14 (4 1) (0 1) 4 2 22 2= + = = 19.77 31 - xf(x) = .(1 + x)
)
f
.)
f
.)
Cf.) Cf.) f
=R{-1}.:3 23 6 4 1 x (1 x) (1 x) 3(1 x) (1 x) 3(1 x)f (x)(1 x) (1 x) (1 x) + + + = = = = + + +
4 41 x 3 3x 2x 4(1 x) (1 x) + = =+ +
x
.
f
(x)=0x=2f(-,-1)
(-1,2]
[2,+).
1f (2)27= .282 )
x-1
:4 34 8 5 x 2 (x 1) 4(x 2)(x 1) x 1 4(x 2)f (x) 2 2 2(x 1) (x 1) (x 1) + + + = = = = + + +
5 59 3x 6(3 x)2(x 1) (x 1) = =+ +
f
(x): f
(-,-1)
[3,+),
(-1,3], 1M3,32 Cf.) :x 1 x 1lim f (x) lim f (x) + = = +
x=-1
Cf.:x xlim f (x) 0 lim f (x) += =
y=0
Cf..)
Cf
:1 130 01 xE f (x)dx dx(1 x)= =+
1+x=y
:dx=dy28322 23 3 2 21 112 y 2 1 1 1E dy dyy y y y y | |= = = + = |\ .
1 1 1(1 1)4 2 4 = + + =
,-.
19.78
R=1.-.-,:) ,) ,,.) ,(-
(0,0)).:C1:(x1)2+y2=1
(1,0)
R=1.:C2:x2+y2=2
(0,2)284 C1C2:2 22 2 2(x 1) y 1x y + =+ =
2x ,2= :2 2 22 2 2 (4 ) 4 y x y4 2 = = = 2 2 4 , (0, )2 2
:222 4 224 2y (x 0) y x0 = =
y=0
:2M2x2 4 =
22(AM) , (0, 2)2 4 =
) :22AM x()2 4 = =
( )( )( )2 2 22 0 0 0 2 2 2 4 limx() lim lim2 4 2 4 2 4 + = = = +
( )( )2 222 0 0 2 4 lim lim 2 4 44 (4 ) + = = + =
4.28519.79 (2917.)-6.6.66,.66,,.f(t)g(t),h(t)-,
t
[6,12].
:) h(6)h(12)
290 ) :22 2100 x 100K(x) 20 1 20x x = + =
K(x)=0x=10 (x) -
x=10.) :(10)=20(51+10+10)=20
71=1420) :100 10010 x 10= = .19.83 1600..-425.2.-20.40,-.) (x): 16K(x) = 40 x + + 160 x
x.) ,;) .291) 1600,-32002,:3200
2=6400 :3200 32100x x=
100. :32 64020 x x = 40x,x 40.:640 16K(x) 6400 40x K(x) 40x 160x x = + + = + +
) :22 216 x 16K(x) 40 1 40x x = =
(x)=0x216=0x=4 -
x=4.
-,4.) :16K(4) 404 160 40 168 6720 4 = + + = =
292 :32 328 x 4= = 19.84 2000,5000.-500,-1000..:) 2000,) ,,) .) 2000x..,.:=
=1000
=500
:1000=500=2
=2.
x2000,
=x,:xx 2 2= = :x(x) 5000 , x 02= > 293:x(x) 0 5000 0 x 100002> > <
x
(0,10000). 500 1000 ; x =
:500x x1000 2= = ) (x):xE(x) (2000 x)(x) (2000 x)50002 = + = + :x 1E(x) 5000 (2000 x) 4000 x2 2= + =
(x)=0x=4000,
4000+2000=6000.) :4000E(4000) (2000 4000)50002 = + =
=6000
3000=1819.85 (t)-,t.2000201800.:294 ) (t),t,) 40.) (t)(t)--,:(t)=(t)
(t)>0,
:[ ](t)(t) (t) ln(t) (t)(t)= = = 1t c t1ln(t) t c (t) e (t) ce+ = + = = 1cc e . = :
(0)=2000c=2000,
(t)=2000et(1),20 209 1 9(20) 1800 2000e 1800 e ln .10 20 10= = = = 1 9ln t20 10(t) 2000e , t 0. = (()):(t)=(t)(t)(t)=0(t)e-t(t)e-t=0( )t t t(t)e 0 (t)e c (t) ce = = = .:t1 9ln t2020 109(t) 2000e 200010 = =
(t)=(t)
.295) 40:( )21 99 9ln 402ln ln20 1010 10(40) 2000e 2000e 2000 e = = = =29 812000 2000 2081 1620 10 100 = = = =
19.86 6..149m3.-
te4-tm3/h,
t-6..,25m3/h.612,:) Q(t),)
Q(t)
t,) ,) .(
e454.)) ,:Q(t)=-
(te4-t+25)m3/h) :4 t 4 t 4 tQ(t) (te 25)dt t( e )dt 25dt t(e )dt 25t = + = = =
4 t 4 t 4 t 4 tte e dt 25t te e 25t c, t 0 = = + +
(t=0
6..).:Q(0)=149e4+c=149c=149e4
:Q(t)=(t+1)e4-t25t+149e4
Q(t)=(t+1)e4-t25t+95,t0296 )
f
(t)=Q(t).
: f
(t)=Q(t)=-(e4-tte4-t) f
(t)=0e4-t(1t)=0t=1-,
t=1,
-
6+1=7..) t
Q(t)=0.
-:Q(4)=5100+95=0
Q(t)0
x
R.
f.) f:( )x xx x xlim f (x) lim (x e 1) lim e 0 = + = = xx xlim f (x) lim (x e 1)+ += + = +f
f
()=R.(f-f
().,,.)) f,,
11.
:f-1:f
(R)R
f
(R)=R
f-1:RR.)
f-1(x)=x
f
(x)=x.
:f
(x)=xx+ex1=xex=1x=0) .:2 2x x 2 2 2 x x 2e e x 2 x x e 1 (x 2) e 1+ + + + + + 2f (x ) f (x 2) (1) + f,(1):x2x+2x2x20x
[-1,2])
f
(0)=0,
f
f
(x)x
x
[0,+),
f-1(x)0
x0.
:e10E f (x)dx=
298
x=f
(y),
dx=f
(y)dy,
:( ) [ ]1 1 11100 0 0E f f (y) f (y)dy yf (y)dy yf (y) f (y)dy= = = =
12y0y 1 3f (1) e y e e 1 (0 1 0)2 2 2 | |= + = + + + = |\ .
y=x:[ ]121 1x x0 00xE e f (x) dx (e x e 1)dx ex e x2 = = + = + =
1 1 3e e 1 (1) 22 2 2 = + = =
)
x=f
(y),
dx=f
(y)dy.:1 1 1 12 2 200 0 01 1 1I f (y)yf (y)dy y f(y) dy yf(y) f(y)dy2 2 2 = = = =
12 2 2 201 1 1 1 10 10 2e f(x)dx e e 2e 2e (3e 5)2 2 2 2 3 3 3 = = + = =
299
20.1
f:CC
( )f (z) f z z z + = +
z
C.:)( )f z z , = )
f
(z)=z
z
C.20.2 ,,.:) z z z, = ) ,,: + + = + + .20.3 u,zw:22z z 1wz z 1+ += +
z
C:x2+y2=1.) u u u. = ) 1z .z= ) w.
.20.4 :z=x+yi
x,y
R:2 2z 1 z 3 2i 6 + =
(x,y).) C,.) C,,-.20.5 ,-,.: 1 w + + = =
:) w w, = ) w,) ,.20.6 ,,300
(, ),
.) .) : z + +=+ +
,,.) z--.20.7 z1:z z1z z+ = ) z.) z
z3=1
z3=-1.) :2004 2004 20041 2 8A z z z = + ++
z1,z2,,z8
z().20.8 ,,-.,:++1,:)1 1 1 , , = = = )
(++)(++)=,)
(+)(+)(+)=0,)
.20.9 ,,,:()4+()4+()4=0:x=,y=,=x y = = :) 1, = = ) x y , = = ) ,.20.10
f: R R
(x+y)f
(x)=f
(x)+f
(y).
-:) f
(0)=0, ) f
(1)=0,) f.20.11
f:RR:f
(x)x+11+f
(x+y)f
(x)+f
(y)
x,y
R.
:) f
(0)=1) f
(x)+f
(-x)2
x
R) f
(x)=x+1,
x
R30120.12
f:RR
:f
(xy)+f
(x)f
(x)f
()1
x,y,
R.
:) f
(0)=1
f
(1)=1) f
(x)=1
x
R20.13
f:RR
:f
(x+y)+f
(x-y)=f
(3x)
x
R.
:) f
(0)=0,) f
(2x)=2f
(x)
x
R,) f
(3x)=f
(x)
x
R,) f
(x)=0
x
R.20.14
f:RR
:f
(0)=0:( )2f xf (x) f (y) f(x) y + = +
x,y
R.
:) ( ) f f (x) x =
x
R,) f2(x)=x2
x
R,) f
(x):f
(x)=xf
(x)=-
x20.15 :f:(0,+)R :xyln(xy)yf
(x)+xf
(y)f
(xy)
x,y>0.
:)
f
(1)=0,)21 1f f (x)x x =
x>0,)
f
(x)=xlnx,x>0.20.16
f:RR
:f
(1)=1,f
(x+y)=f
(x)+f
(y)21 1f f (x)x x =
x,y
R.
:)
f
(0)=0,)
f,)
f
(xy)=f
(x)f
(y),)21 1 f (x)f1 x (1 x) =
21 1 2x f (x)f1 x (1 x) + =
x0
x1,)
f
(x)=x
x
R.20.17
f: R R
:1f (x) xfx =
x0,
f
(x+y)=f
(x)+f
(y)1
y-
x.) :2f (x) xf 1x =
x0.) :2 2f f (x) 1, x 0x x =
302 ) f
(0).)
f.20.18 -
f:RR
-:( ) ( ) f f (x) x f 1 f (x) 1 x = + =
x
R.20.19 :f:RR :f2(x)2f
(x)x=4x+2x+1
x
R
:f 32 =
) f,.) f
(0).) :g(x)=f
(x)x,x
R,.)
f.) :xf (x)A limx+= 20.20 -
f
(-x)=3x2
x
R.) f
(0).) Cf-
(0,f
(0)).) f.20.21 .,-150m2..2m3m.) .) ,;) -;20.22 :f
(x)=x3+x2+x+3
x0=1
-,
4.)
=0.)
.20.23 :xf (x) ln (x)ln(x)2 =
x 0, .2 = 303) f.) f.20.24 :xx exe
x>0,
:) :xx exf (x) e =
,) =e.20.25 :f
(x)=x48x3+22x224x+
R) f -.) f.) :xA lim f (x), B f (1), f (2)= = = x f (3) E lim f (x)+= =
R.)
f
(x)=0,
R.20.26
f:RR*:f
(x+y)=f
(x)ef(y)-1
x,y
R:) f
(0)=1,) f
(x)=ef(x)-1
x
R,)
y=ey-1,)
f
.20.27 :xx1f (x) , 0 1= + < ) f -.)
f.)
f.) :xx1 1 + = + ) f-.20.28
f,g:RR
f
(x)>0
x
R,xlim f (x)= xlim f (x) .+= + )
f.) -
f
(x)=0.)
(f
g)(x) x (g
f
)(x)
x
R,
g=f-1.20.29 :2f (x) 5 144 (20 x) 3x 108 = + + ) f -.) f.304 ) :25 144 (20 x) 3(36 x) + = 20.30
f:RR -
f
(1)=1
f
(x)0.) f -.) :22lnx 1ln2x ln(2x 2)2 1 x = ++
20.32 :f:RR:21f (x)1 x=+
x
R.
f
(0)=0,
-:)
f-1,) f
(x)=x
x , ,2 2 ) f-1(x)=x,
x .2< 20.33 f-
,
,
)
(e+2)+(e-+2)>2
3
N*
R*.20.34
f:(0,+)R
-
f
(1)=0,
-:( ) f f (x) f (x) 0 x 0 + = > :) f
(1)=1,) (f
f
)(x)=x
x>0,) xf
(x)+f
(x)=0
x>0,) f
(x)=lnx
x>0.20.35 :32(1 x)f (x)x= ) f-.305) f ,,-.)
Cf.)
f.) f.)-
x3+(3)x2+3x1=0.20.36 --:K(x)=6
10-6x26
10-3x+4/x.100.) 500.) 200-.20.37
f,g:RR
-xf (x) f (t)g(t) dt =
x
R.) f
().) :h(x)=f
(x)e-G(x)
(
R),
G-g.) f.20.38
f:RR
:10f (x) x f (xt) dt 2 x = + ) f-.)
f
(x)=f
(x)
-
x
R.) :g(x)=e-xf
(x),x
R.)
f.20.39 :)101A dx, x 0x 10x= >+
)61B dx, x 0x x= >+
20.40
f:RR
-
[0,]:f
(0)=-1,f
()=1f
(x)=f
(x)
x
R.) :g:RR,g(x)=f
(x)+f
(x)) 0 f (x) dx 0. =
) :20 (x x)f (x) dx 0 =
) :0 xf (x) dx =
306 20.41
f:(0,+)R
-:x1f (t)f (x) 2004(x 1) dt x 0t= + >
.) :xf
(x)f
(x)=2004x) :f (x)g(x) , x 0x= > )
f
(x)=2004xlnx.. -Cf,xx
x=e.20.42 :120ln(x 1)I dxx 1+=+
:)120ln2 1I dx,2 x 1=+1 yxy 1=+
)ln2I8= 20.43 :f:(0,+)R:x1 f (t )dtf (x) e=
x>0.:) f,) f
(x)=-
f2(x)
x>0,)1f (x)x=
x>0..
Cf
.:) =.) ,-.20.44 :0122ln(x 2)I dxx 1+=
) 1 2yx ,y 2+= +:0122ln3 1I dx2 x 1=
) 2ln 3I .4= 20.45
f:RR
-
f
(0)=f
(0)=2.
10g(x) f (xt) dt, =
x
R.:)2x 0f(x) 4lim 8,x= ) [ ]1g(x) f (x) g(x)x=
x0,)
g
x0=0,)
g(0)=1.20.46 :2xf (x) e , g(x) lnx = = 3071 e0 1I f (x) dx g(x) dx = +
=e.20.47 :f:RR* :14 3 30f (x) 1 2x t f (xt) dt =
x
R.
:)x3 30f (x) 1 2 t f (t) dt =
x
R, ) i) :421g(x) x ,xf (x)= ,ii)241f(x) ,x 1=+
x
R, )41f (x)x 1=+
x
R, ) [ ]xlimxf (x)x 0.+= 20.48 f g f g
g(x)+g(-x)=1
x
R.) :0 0f (x)g(x) dx f (x)g( x) dx=
) : 0f (x)g(x) dx f (x) dx=
) :22x-2xA dx1 e=+
20.49
f:(0,+)R
-
f
(1)=3.
Ff:1 3f (x)Fx x =
x>0,
:) 1h(x) F(x)Fx =
(0,+),)1F(x)F 1x =
x>0,) :3F(x)(x)x=
(0,+),) f
(x)=3x2
x>0.20.50 :f
(x)=2(x)+2(x)) :20 0f (x) dx 2 f (x) dx =
) :20A f (x) dx =
) :20xf (x) dx2=
308 20.51 :f
(x)=x33x2+3x) f.)
Cf
::y=x) Cf
1fC .
20.52 :120ln(x 1)A dxln(2 x x )+=+
)
x=1y,
-:120ln(2 x)A dxln(2 x x )=+
) 1A .2=
30920
20.53
f:RR
:( ) f x f (x y) f (2x) y + + = +
x,y
R.:) f
(0)=0) (f
f
)(x)=x
x
R) f
11) fR) f
(x)=x
x
R20.54 :3 23 22x 6x 5x 2, 2 x 1f (x) x, 1 x 12x 6x 5x 2, 1 x 2 + + + = < < +
)
f
(x)
f
(x).) f
.) Cf.20.55 :22x 3xf (x)x 2+=+
) f-.) -
f.
.) f .) Cf.) Cf
xx.20.56 :23 xf (x) x e = ) f-.)
Cf,xx
x=1.20.57 :2 xt0f (x) e dt =
) f -.) f.)
Cf,.20.58 fg-
[,],
-
(,)
,: g() f (t)dt f () g(t)dt =
310 20.59 :xx t x1 e lnt dt e lnx + +
x>0,
:=e20.60 f[,]
: f (x)dx 0 =
(,)
,:f (x)dx f ()( ) =
20.61 ,: 0 2 = = + = :) 2=) ==
311
.fx0.fx0-,
f
(x0)=0.[14]. :) fx0,f .) fx0,) fx0x0 Cf,)
f
x0,) f
[,],
(,)
,
x0
(,)
,[5]. ()-().) f
x=
f
(x)=0
x
R{}, f. ) fR,, [,]
Rolle. 312 ) f,- f. )
f
(x)>0
x
,
f . ) FGf,
F=G+c. ) f f
() ,. [6]
f:A=(-,)R
:f2(x)2f
(x)+2x=0
x
(-,)f
(-,0)
(0,),
:)
f
x0=0,[8]) f
(x),[9]) f
(x),f.[8]
f:RR
f
(0)=0
:f
(xy)=f
(x)f
(y)+x
y
x,y
R.
:) f
(0)=1[10]) f
(x)=x[15]313
f,g:(0,+)R
f
(1)=g(1)=0,
f
(x)=-
eg(x)g(x)=-
ef(x)
x>0.
:) fg,[5]) f=g,[8])
h(x)=e-f(x)x
(0,+),[5]) f
(x)=-
lnx,
x>0.[7]
. )
z=+i
,
R
;)
f:AR
11;)
f:AR
,;[3]. 1 2 1 2zz z z =
z1,z2
C.[7]. () -().) z , = 2z ,z=
>0
z
C. )
,
C
2+2=0==0. 314 ) f
(,),
f m. )
f:AR
11,
f
(x)=y
y
R. )
f
(x)=0
x
,
f. [5].
(x)=-
x
x
R.[10]
f:RR
:f
(0)=f
(0)=1f2(x)=f
(x)f
(x)
x
R :) f
(x)=f
(x)
x
R,[15]) f
(x)=ex,
x
R.[10]
F
f:RR
f
(0)=1
f
(x)F(-
x)=1
x
R.)
g(x)=F(x)F(-
x)
.[8])
F(x)F(-
x)=1
x
R.[8]) f.[9]
:t 4 2xtxe 5t 3tf (x) dt, x1 e+ += +315) f
.[15]) :x 4 2x-e 5x 3xI dx, 1 e+ += +[10]
. )
f:AR;[1])
f
(x)0
x
[,]
f
[,],
f (x) dx ;
[1]) f
,x
,
xF(x) f (t) dt; =
[4]. f
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