olokliromeni sillogi askiseon
TRANSCRIPT
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1.
. -
1. ,
+i.
: , 95, 5, 6.
2. x, y
,
+i
. : , 95, 7.
3. i,
4
.
: , 95, 8.
4. +i,
i, .
:
12
2
i3
++++.
5.
+i
, x , y
.
:
i)x3y2()
36y2x3(z ++++++++++++==== , * .
6. z
, zz ====
zz ====
. ( ).
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Re(z)=0 Im(z)=0.
: , 96, 8.
7. z
z , z=x+yi ,
.
: , 96, 5.
1. ,
.
: , 100, 1.
2. z
z,
z=x+yi x y.
: z
9zi41z3z ====++++====++++ .
3.
, , zzz 2 ====
.
: , 1
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, zzz 2 ====
,
. : , 102, 10.
1. z ,
z=x+yi
x y.
. (
).
:
) z z ==== >0,
z (0, 0) .
) z zz 0 ==== >0,
z K(x0, y0) . (
K(x0, y0) z0).
) z 21 zzzz ==== ,
z
z1
z2.
) z 2zzzz 21 ====++++ ,
z z1
z2, 2 2= 21 zz .
) z 2zzzz 21 ==== ,
z z1
z2 2= 21 zz .
: , 101, 6, B 2,
B 5, B 6, B 8, B 9.
2. z
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z, z
.
: , 101, 8.
1.
.
: ) z1=8+6i 5z2 ====
21 zz ++++ .
) 3i5z ++++
12z .
2.
,
:
) z (), :
)OM(),(dz min ======== . ( ).
) z (, ), :
z min ==== z xm ++++==== .
) z 1
y
x2
2
2
2
====++++ , :
z min ==== z xm ==== .
) z 1
y
x2
2
2
2
==== , :
z min ==== .
: , 101, 7, 102,
8.
3.
,
:
) z1 z2
(1),(2) : ),(dzz 21min21 ==== .
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) z1 z2 (, ), :
2zzxm21
==== .
) z1 () z2
(, ) : ),(dzz min21 ==== ),(dzz
xm21 ++++==== .
) z1 (,1)
z2 (, 2) : 21min21 zz ====
21xm21zz ++++++++==== .
) z1 z2 1
y
x2
2
2
2
====++++ ,
: min21
zz 2zzxm21
==== .
:
1.
.
2.
(2)
(3).
)z(f ==== z=x+yi,
.
: 2iz
iz2w
++++
==== z
(0,0)
=1.
) w.
) wz .
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.
1. 0z,z 21 1z1 ==== 2z
z
z
z4
1
2
2
1 ====++++ .
:
) 2z2 ==== .
) 0, z1 z2
.
2.
z1 z2,
21 z5
i34z
++++==== .
) .
) (1, 2)
=2
.
3. 2zzz 321 ============ 4zzz 321 ====++++++++
8zzzzzz 133221 ====++++++++ .
4. z1, z2 z3 z1
2+z22+z3
2=z1z2+z2z3+z3z1
.
5. z1 z2 : z1z2=1+i.
z1 (0,1)
=1.
) z2
.) z2 .
) 0
w=(z1z2) .
6. z 2z310z ====
.31z ====
7. 5i42z ====++++ .
) z .
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) z.
8. z
z2w
==== 0z . z
(0,0) 1.
) w
.
) w .
9. z=x+yi, x, y
6z
2zw
==== , 6z .
) w +i, , .
) Re(w)=Im(w) (x,y) z.
10. zz3 ==== .
) z0
|z0|.
) .
11. z
z1zln ==== .
12. z=x+yi x, y (x,y)
.
w=4z
z
++++ z -4.
) w +i.
) (x,y) :
i) O w
ii) O w .13. , ,
. 1 ============ ++=0,
:
) ======== .
) , .
14. z= i3
24
4
++++
.
) z.
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) z .
) o z .
) .
15. , , x2+y2=1.
w=
))()(( .
16. z :
(4+3i)z(43i)z =50i.
) z.
) z 5.
) z
.
17. z
5i86z ==== .
)
z.
) 5 z 15.
) z .
) :
i) z .
ii) z .
) z1, z2
, 21 zz .
18. z w 6z)22i( ====++++
)i33(w)i1(w ==== :
)
z.
)
w.
) w .
) wz
.19. z 2iz ==== .
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) z
.
) z1 z2
.)
22iz3ziA ++++++++====
.
) 7i53z3 ++++ .
20. z1 z2
.zz
zzw
21
21
++++====
) 0w
2008 .
) 21 zz ==== .
21. z )zRe(2)z
4zRe( ====++++ (1).
) z.
)
i43zc ++++==== .
) c.
) z1, z2, z3 (1)
321133221 zzz2zzzzzz ++++++++====++++++++ .
22. z Re4
1)
z
1( ==== .
) z
2z =2.
) z lm(z)=1, Re(z)=2+ 3
Re(z)=2- 3 .) 4
1 ().
23. z : 3zz220102010 ====++++ .
) 1z2010 ==== .
) 1z ==== .
) i2z .
24. z=+3i, .) z.
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) i5z ++++ .
25. z, w6zi
i2z3w
++++
++++==== 2z ==== .
) w.
) wz .
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11
2.
. -
1. .
.
: , 146, 11.
2. ) ,
.
,
.
) ,
.
x.
.
x.
: , 148, 6.
x1,
x2 Af x1
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1-1
1. f 1-1 :
: x1, x2 Af
f(x1)=f(x2) x1=x2. : f
. ( :
, 1-1 .
: ).
: , 156, 2.
2. f 1-1,
x1 x2 Af 21 xx
f(x1)=f(x2).
: , 156, 2, viii)
3. f,
1-1
f(A). ( y=f(x) x
). f-1
f(A) Af. , f-1, y
x. : , 156, 2.
4. :
) f-1 f(A)
Af.
) :
i) f(x)=y x)y(f 1 ==== .
ii) y))y(f(f
1 ====
)A(fy
.iii) x))x(f(f 1 ==== fAx .
) f Af
f-1 f(A)
.
: 21 yy
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2121
11
21
11 yy))y(f(f))y(f(f)y(f)y(f >>>>>>>>>>>> , .
)y(f)y(f 21
11
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x
1)x(f 1 ==== .
y=x).
) : f
Af :
)A(fAx,x)x(f)x(f)x(f 1 ======== .
: )x(f)x(f 1==== f
Af. x)x(f Bx .
x)x(f >>>> x)x(f > . f
Af 1f
f(A).
)x(fx)x(fx)x(f))x(f(fx)x(f 111 >>>>>>>>>>>>>>>> ,
. x)x(f
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:
1. f2(x)=g2(x) f(x)=g(x) f(x)=-g(x)
x f. x)x(f ====
g(x)=x. f2
(x)=g2
(x) x f(x)=g(x) x [0, ++++ ].
f
====
BAx)x(g
x)x(g)x(f .
2. f(x)g(x)=0 f(x)=0 g(x)=0 x
.
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.
1. f(x)=x4-(3-4)x3+(3-3)x2-(3-2)x+2-3
. Cf
.
2. :f : 2x2 2)x1(f)x(f ====
x .
) 2
1)x(f)x1(f ==== .
) 1x2)x(f ==== x .
3. :f :
323 x2)x(f)x(f3)x(f ====++++ x .
) f.
) x)x(f ==== x .
4. :f : )y(f)x(f)yx(f2 ====++++
y,x .
) f(0)=0 f.
) f f(0).
) 0)(f ==== 0)x(f ====
x .
) 2)0(f ==== 0)x(f >>>> x .
5. f : 2x1)x(f ==== .
) Af.
) Cf .
) h=fof.
6. f :
) 1x)x(xf)x(f 2 ++++==== x .
) *:f )x
1x2(2)
x
1(f2)x(f ++++====++++ *x .
7. f, g : g(0)=0
y6x6)1x(3)y(g)x(g)y1(f)x(f2 2 ++++====++++++++
y,x .
) x6)1x(3)x1(f)x(f2 2 ++++====++++ x .
) x2x)x(f2
++++==== x .
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) gf ==== x .
8. :f :
yxe)y(f)x(f)yx(f ++++++++ y,x .
) 1)0(f ==== .
) )x(f
1)x(f ==== x .
) xe)x(f ==== x .
9. f, g :
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15. :f
(3, 2) (5, 9).
) f .
) )x(f1
.) 9))xx(f2(f 21 ====++++++++ , x .
16. 1xx)x(f 3 ++++++++==== .
) f .
) f.
) )x(f)x(f 1 ==== .
17. xx
exln)x(f ++++==== .
) f .
) x)x(f 1 ==== .
18. :f x)x)(fof( ==== x .
) 2008)x(f ==== .
) :g 1-1 x(fx ee)x(g ++++==== ,
x)x(f ==== x .
19. :f x)x(f)x)(fof( ====++++ x .
) f 1-1.
) f(A)= .
) x)x)x(f(f ====++++ x .
) 0t)t(f ====++++ .
20. :f x)2)x(f(f ==== x .
) f 1-1.
) f(A)= .
) )x(f 1 f.
) f f
.
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3.
. -
x0
1. (0/0))
Horner (x-x0)
.
: , 174, 3.
)
. (
).
++++ .
++++ .
++++ .
33 2
33323 ++++++++ .
3
2332
++++++++ . 3 23
23 ++++++++ .
: , 175, 4.
) ,
x
)x( 0
x
1x :
1x
)x(lim0x
====
0x
1xlim0x
====
.
: , 175, 6, 7.
2. ,
.
,
.
: , 175, 5, 9.
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3. . x x0
:
) (0/0) .
: , 175, 1 vi).) (0/0). :
i) x0 ,
.
.
: , 176, 2 i).
ii) x0 ,
x0
(x-x0).
: , 176, 2 ii).
4. :
f(x) x0 f(x) x0.
g(x), f(x)
.
: , 176, 4.5. f(x) x0
, f ,
f(x),
f(x) . ( :
).
: )x(flim0x
x)x(xf
x .6. : :
) )x(flim0xx
)x(h)x(f)x(g
x0. )x(glim0xx
)x(hlim0xx
====
)x(glim0xx
)x(hlim0xx
====
)x(glim0xx
= )x(hlim)x(flim00 xxxx
==== .
: , 175, 8.
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) )x(flim0xx
)x(g)x(f )x()x(f (
A(x) ) x0, 0)x(glim0xx
====
0)x(Alim0xx
====
. )x(g)x(f)x(g
0)x(glim))x(g(lim00 xxxx
========
.
0)x(flim0xx
====
. ( ).
: )x(flim0x
x)x(f 0.
) f(x)
x x0. 1))x(K(
)x(g)x(f
0)x(glim0xx
====
. ( .) 0)x(flim0xx
====
.
: )x(flim0x
)x
1(x)x(f ==== .
7. ))x(g(flim0xx
g(x)=u,
0xx
u)x(glim0
==== ( )
)u(flim0uu
. g(x) 0u x0 ))x(g(flim0xx
=
= )u(flim0uu
.
: , 175, 7 ii), iii).
8. (/0) :
)
. .
.
: , 181, 2.
)
,
.
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: , 182, 3.
9. :
)
: 011
1
x...xx)x(f ++++++++++++++++====
. 0
xxxlim)x(flim
++++++++
==== .
.
: , 187, 1 ii).
)
:01
11
0111
x...xxx...xx)x(f
++++++++++++++++++++++++++++++++====
.
0 0
xx x
xlim)x(flim
++++++++
==== .
.
: , 187, 1 ii).)
x
.
. ( : xx2 ==== ++++x xx2 ====
x ).
: , 187, 2, 3.)
,
( )
.
: , 187, 2.
)
x.
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: , 187, 4.
)
: ++++====++++
x
xelim , 0elim x
x====
,
++++====++++ xlnlimx ====++++ xlnlim0x .
.
: xx
xx
x 2e
elim
++++
++++++++
.
)
,
x .
1))x(K(
.
x
.
: )1xx2x(lim 3x
++++++++++++
.
:1. 1
x
xlim
0x====
, 1
x
xlim
0x====
, 1
x
xlim
0x====
, 1
x
xlim
0x====
,
0x
1xlim
0x====
.
2. 0)x
1(xlim
0x====
.
3. 0x
xlimx
====++++
0x
xlimx
====
.
4. 1)x1(xlim
x====
++++ 1)
x1(xlim
x====
.
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.
1. 1)x(flim1x
====
:
)
1)x(f
3)x(f)x(f2lim
2
2
1x
++++
)
1)x(f
23)x(flim
1x
++++
1)x(f
1x0 ==== .
2. :f 2x
)x(flim
0x====
.
) 0)x(flim0x
====
.
) ( ) :x)x(f
x)x(f2lim
0x ++++
++++
.
3.
:g,f 2))x(g)x(xf(lim1x====++++
4))x(xg)x(f(lim
1x====
. )x(flim
1x
)x(glim1x
.
4. , 2x)x()x( ++++ x .
+=0.
5. , : xx
x)x4(lim
240x====
++++
++++
.
6. , , 2x
xxlim
2
2x====
++++++++
.
) 4+2+=0.
) , , .
7. 44xx2
1xxlim
3
x====
++++++++
++++++++++++
. *N .
8. , :
0)x1x4x9(lim2
x ====++++++++++++++++++++ .
9. , ),1( ++++ . : xx
xx
x
32lim
++++
++++++++
.
10. :f 2)x(f
)1x(flimx
====++++
++++.
) )x(f
)2x(flimx
++++++++
.
) .
11. :f x)x(f)x(f2 ====++++ x .
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) f 1-1.
) )x(flim0x
.
)
2
1
x
)x(flimx
====++++
.
12. :f 2x
)x(flimx
====++++
3)x)x(f(limx
====++++
. :
1xx)x(f
xxx)x(xflim
2
x====
++++
++++++++
.
13. f :
xx3)x(xfxx ++++ x x0=0.
)x(flim0x
.
14. f : 22x)x(f)2x( ++++
x x0=2. )x(flim2x
,
.
15. )14x7x1x23xx9(lim 222x
++++++++++++++++++++++++
.
16. f,g : 4)x(f)11x(lim0x
====++++
124x
)x(glim
0x====
++++ x0=0.
) )x(glim0x
.
) )]x(g)x(f[lim0x
.
17. f,g : )x(xf2)x(g)x(f 22 ====++++
x0=0. )x(flim0x
)x(glim0x
.
18. 1x
)x(lnflimx
====++++
)x(flimx ++++
,
)x(flimx ++++
.
19. (x) 3 :
1)3x(
)x(Plim
23x====
5x2
)x(Plim
2
5x
.
20. ++++ ),3(:f 18xe6e2x(f2x(f4
====
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x ),3( ++++ .
) f
f-1.
) 3ee)x(flim x2
x21
x
++++.
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4.
.
1.
f
f x0 f(x0).
f ,
f ( , ,
...) xx0 0xx f
.
: , 197, 2, 4.
2.
Af,
f xx0 0xx ,
)x(f)x(flim)x(flim)x(flim 0xxxxxx 000
============ ++++
.
.
: , 199, 1, 2.
3. f x0
f(x0) )x(flim0xx
.
4. ),(x0
)x(g)x(f 00 ==== , Bolzano [, ]
)x(g)x(f)x(h ==== . 0)(h)(h
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)x(g)x(f)x(h ==== .
: f [-, ], >0
)x(f ],[x .
],[x0 00 x)x(f ==== .6. Cf, Cg
x0 ),( x0 ],[
4. 5.
: , 200, 6.
7. f(x)=g(x)
(, ), (, )
(, ) (, ) [, ] [, ] Bolzano h(x)=f(x)-g(x).
: x4=11-2x
(-2, 2).
8. : f
0)x(f x Af, f
Af.
,
( ).
: , 199, 9, 7.
9. ,
. :
a) f [, ]
f(A)=[f(), f()].) f [, ]
f(A)=[f(), f()].
) f (, )
f(A)=( )x(flimx ++++
, )x(flimx
).
) f (, )
f(A)=( )x(flimx
, )x(flimx ++++
).
) f [, )
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f(A)=[f(), )x(flimx
).
) f [, )
f(A)=( )x(flimx
, f(].
) f (, ]
f(A)=( )x(flimx ++++
, f(].
) f (, ]
f(A)=[f(), )x(flimx ++++
).
) ++++ .
: , 199, 10.
10. :) : Bolzano
f(A)
.
) : f
.
) : ) )
.
: , 200 5, 6.
11. x0(, ) f(x0)
[f(),f()] [f(),f()],
( Bolzano).
: 3)x(4
x)x(f
3
++++====
37 ]2,2[x .
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30
.
1. :f
>>>>
==== ,
0xx
)x7(
0xe)x(f
x
f x0=0.
) .
) )x(flimx
.
) )x(flimx ++++
.
2. :f
====
====
0x2
1
0xx
x1
)x(f2
.
) f .
) )x(flimx ++++
.
3. :f
>>>>
++++++++====
,,
xe)4x2(
x5x)x(f
x
22
f x0=.
) , .
) f.
4. :f yx)y(f)x(f x, y
.
) f .
) g(x)=f(x)-2x, x
.
) f(2x)-f(x2)>4x-x2.
5. :g,f :
)x(g2)x(xf21)x(g)x(f 22 ++++====++++++++ x .
) f, g x0=0.
) 2x x
)x(flim
++++.
) g ,
g(x)=-2x .
6. :f :
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1x)x(fx ++++
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13. :f )x(fe2)x(f x2 ==== x
.
) xe)x(f)x(g ====
.) g(0)=1 f.
14. , 0 :
0)525)(( .
15. f, g [, ]
0)x(g ],[x . ),(x0 ,
x
1
x
1
)x(g
)x(f
000
0
++++
==== .
16. f [-1, 3].
1, 2 )3,1( : 2f(1)+f(2)=f(0)+f(1)+f(2).
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33 -
5.
.
1. f(x0)
x x0.
,
,
.
: , 220, 2, 4.2. ,
.
: , 227, 2, 238
4.
3. f
x0 ,
.
.
: , 218, .
4. f
x0 ,
.
: , 228, 1.
5. f f(x0),
f
, x x0 f(x0). f
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34 -
,
f(x0) f x0
.
:) :f :
43 x2)x(f)x(f ====++++ , x . f(0).
) f x0=0 x
3223 xxx)x(fx)x(f ====++++ . f(0).
6. f x0
.
K(x), f(x0), f(x)
.
f x0.
: , 221, 8, 240,
7.
7. f
f(x0), f(x0)
f x0 .
f ( )
f(x0) ( f x0)
f x0,
f(x0).
:
) , 221, 5, 6.) f x0=0 f(0)=0 x)x(f
x , f(0).
8. f ,
, x=x0+h x=x0h
.
:
) f x0=0 x, y
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35 -
f(x+y)=f(x)+f(y)+5xy,
f f(x).
) f x0=1 x, y
*
f(xy)=f(x)+f(y), f * f(x).
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36 -
.
1. :f
x0 x0>0,
>>>>
====
000
0000
xx)x(f)xx(f
xx)xx(9)x(f)xx2(f)x(g
x0. f(x0) g(x0).
2.
>>>>++++
====
xx
xx)x(f
3
x0=, , .
3. :g,f 222
x)x(g)x(f ====++++ x .
) f, g x0=0.
) f, g x0=0
1)]0('g[)]0('f[ 22 ====++++ .
4. :f
>>>>++++
====
1xxln
1xae)x(f
x
.
) , .
) f 1-1.
5. :f x0=0 x
: xx)x(fx)x(xf)x(f 2223 ==== .
) f(0).
) f(0).
6. :f :)yx(xy3)y(f)x(f)yx(f ++++++++++++====++++ x, y f(1)=4.
) f(0)=1.
) f(x)=3x2+1 x .
7. f x0=0 7x
x
1x)x(f
lim3
2
0x====
.
) )x(flim0x
.
) f(0)=0.
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37 -
) f(0)=0.
8. f x0= f()=2.
) h
)(f)h2(flim
0h
++++
.
) h
)(f)h(flim
2
0h
.
) h2h
)h(f)h2(flim
2
0h
++++
.
9. Cf :f
(0, 0) 4x
)x(xf4)x(flim
2
2
0x====
++++
.
) 0)x
x2)x(f(lim 20x ====
++++ .
) f(0)=-2.
10. :f
f(x3+x+1)=7x3-x x .
) f(3).
) f(3)=5.
11. :f x1=2
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38
6.
.
1. f
x0 f(x0), f(x), f(x0)
y-f(x0)=f(x0)(x-x0).
: , 220, B 3.
2.
(x1,y1), (x0,f(x0))
y-f(x0)=f(x0)(x-x0). x0.
: , 239, 10.
3. (x0, f(x0)),
x0.
)
xx =f(x0)( x0).
: , 238, 5.
) y=x+
f(x0)=( x0).
: , 239, 11.
)
f(x0)=( x0).
)
f(x0)=-1 f(x0)=-1/( x0).
: , 239, 9.
) A(x1, y1) , B(x2, y2)
, f(x0)12
12
xx
yy
==== ( x0).
4. Cf
(x0,f(x0)), Cg ,
f(x0)=g(x0) f(x0)=g(x0). .
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39
: , 240, 3.
5. Cf
(x1,f(x1))
Cg, Cf, g(x)
g(x0)=.
Cg.
: , 240, 4.
.
1. y x x=x0 y=f(x),
f(x0).
y=y0, y0=f(x0) x0
f(x0).
: , 243, 1, 2.
2. y x
y0, x
x, y x0, y0,
y0.
: , 245, 8.
:
f(x),
f(x).
-
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40
.
1. f(x)=x
3
-3x
2
+2x+1.) Cf :
x+11y+17=0.
) Cf
: x+y-2009=0.
) Cf
(3,7).
2. f(x)=xlnx-x.
) Cf
(1,f(1)).
) )
.
3. 1x
1x)1(x)x(f
2
++++
++++++++++++++++====
x
2x)x(g
++++====
x=1.
) +=3.
) Cf, Cg ,
=-11 , .
4. x
32x)x(f
22 ++++++++==== 16x)x(g 2 ++++++++====
.
) Cf, Cg.) Cf, Cg .
) ,
.
5. :f x)1x(fxx)x(f 2 ++++
x .
) f(x)=x2+x x .
) 2
1
y ====
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41
Cf (, f()),
2-=2+2
1.
6. :f
: 2)yx(f)y)x(f(f ++++++++====++++ y,x
)1x2(f)x(f 2 x .
) f.
) f(x)=x+f(0).
) f(x)=x+2.
) Cf Cg g(x)=lnx+3.
7. x
1xx)x(f
++++====
x)4(x2)x(g 2 ++++++++++++==== , .
) () Cf (1, 1).
) =1.
) () Cg.
8. )1xln(2)x(f ++++==== )1xln(2)x(g ==== .
) Cf Cg.
) f g.) Cf Cg ,
.
9. N(t) t
. 50
,
. 2,22ln
3ln .
10. ( /),
3x2-2x+1, ( /),
2x+3, x
.
5 6 .
11. . =300m,
=60m, ( ),
45 Km/h, .
-
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42 ROLLE
7. ROLLE
.
1. f
Rolle,
.
: , ,
>>>>
++++====
1xxx
1xx)x(f
2 Rolle [0,4].
2. :) f
(,),
Bolzano Rolle
f.
( f).
f.
: , 249, 1.) f
(,), f +1
(,) Rolle
.
: , 250, 3.
) f
(,), ) ).
: , 250, 7.
:
1.
.
2.
:
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43 ROLLE
-
1. .
2. Bolzano.3.
f(A).
4.
Rolle
f. (
f).
1.
f
.
2.
f
1,2
Rolle
.
.
8=42.
(
),
.3. x0 f(x0)
,
Rolle .
: f [2,3]
f(2)=5 f(3)=10. x0
(2,3) f(x0)=2x0.
. Rolle :) x0
f(x0)=, Rolle
(x)=f(x)-x.
) x0
f(x0)=x0, Rolle
(x)=f(x)- 1x1
++++++++
.
) x0
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44 ROLLE
f(x0)+f(x0)=0, , * ,
Rolle (x)= )x(fex
.
) x0 ),(
(x0-)f(x0)=f(x0), ],[ ,
Rolle (x)=x
)x(f
.
) x0 0
x0f(x0)=f(x0), Rolle
(x)=x
)x(f.
) x0
(-x0)f(x0)=f(x0), Rolle
(x)=(x-)f(x).
4. Rolle :
x0 (, )
Cf (x0, f(x0))
xx, Rolle f
[,].
5. x0 (, ) f(x0)
, f
[,].
: , 250, 5.
6. :
x0 (, )
Cf (x0, f(x0)) ,
f [,].
: f [1,2]
f(2)=f(1)+3. x0 (1, 2)
Cf (x0, f(x0))
y=3x+12.
7. x1, x2,,x f(x1),
f(x2),,f(x) ,
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45 ROLLE
[,]
f.
: f [,] f()=
f()=. x1, x2
(,) 21 xx
2)x('f)x('f 21 ====++++ .
8. . :
) )x(g)x(f
)x(g)x(f
, h(x)=f(x)-g(x)
[, x]
[x,] h(x)=0. ( ).
: ex ex x .
) )x()x(g)x(h
,
.
: , 249, 3.
:
.
9. . Rolle . :
: f [,]
f()=0 f()= )2
(f2 ++++
.
x0(,) f(x0)=0.10. f
f(x)
x . (
).
: , 256, 1, . 257
1.
11. f f
f,
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46 ROLLE
f. ( ,
).
: , 252, .
-
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47 ROLLE
. ROLLE
1. f : [0, 2]
f(1)=0 f(2)=-f(0).
) x1 (0, 1)
f(x1)=f(2).
) x2 (0, 2)
f(x2)=0.
2. ++++ ),0(:f
),0( ++++ f(x)>0 ),0(x ++++ .
(0,0) Cf
(,f()) (,f()) 00
Cf (x0,f(x0)) (0,0).
3. f : [, ] f()= f()=.) Cf
y=x+7.
) x0 (, )
2
)x(f 0
++++==== .
) x1, x2 (, ) 21 xx
2)x('f
1
)x('f
1
21
====++++.
4. f : [, ]
f()=, f()= f(x)>0 x [,].
) x0 (, )
00 x)x(f ==== .
) x1, x2 (, ) 21 xx
1)x('f)x('f 21 ==== .
) x)x(f ++++ x [,].
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48 ROLLE
5. f f(0)= f(x)=f(x)
x , .
) g(x)=f(x)e-x, x .
) f.6. (x) :
(0)0,(2)0.
) P(x)=0 .
) P(x)=0 .
) P(x)=0 .
) P(x)=P(x)
.
7. f : f(0)=1
f(x)f(x)-f2(x)=xe2x x .
) f.
) f(A).
8. f, g ),0( ++++ f(1)=g(1)=0
f(x)=-eg(x), g(x)=-ef(x) x>0.
) f, g
),0( ++++ .) f(x)=g(x) x>0.
) f.
9. f : f(0)=1
(f(x)+f(x))e2x=f(x)-f(x) x .
f.
10. f [, ], >0 f
[,] (,) f(x)0 (,)
f()=f()=0.
) f()=0.
) x0(, )
x0(f(x0))=f(x0) .
) Cf (x0, f(x0))
(0,0) xx
0 45 x0
0.
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49 ROLLE
11. f f()=f()=0,
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50 ROLLE
) f(x)=x x [,].
16. f f
[0, 1], f(0)=1 f(1)=-1.
Cf (0,f(0)) (1,f(1)) ( 2
1,y0).
) f(0)=f(1).
) x0 (0, 1)
f(x0)=2x0-1.
) x1, x2 (0,1) x11, f(x)=0
(0, 1).
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51 -
8.
.
1.
, , ,
.
f
f
.
:1. . (.x.
f(x)=-1/x).
2. f (, ]
( ) [,) f
x0=, (, ] [, )=
=(, ).
3. f
,
,
. (.x.f(x)=(x-1)3).
4.
,
.
5. ,
.
f(x)>0 x
f .
f()=0 f(x)>f()=0 x> f(x)
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52 -
f(x)=ex-1-x-2
x2, x .
2. ,
.
: , 256, 5, 6.
3. :
) f
(,),
.
) f
(,), f
(,).
) f
(,), ) ).
: , 257, 5.
:
f : 1 : f(x) >1,
f(x) (-1).
: f(x) >1. f(x)=
=(x-)(x) () 0. f(x)=(x-)-1(x)+(x-)(x)=
=(x-)-1[(x)+(x-)(x)]. f()=0 ()+(-)(x)=
=() 0. f(x) (-1).
2 : f(x)
(-1), >1 f(x)
f(x) .
: f(x)
. 1
f(x) (-1).
-1=-1 =. f(x)
.
:)
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53 -
x3-3x+2=0.
) x3+x2+x+1=0 >3
1
.
4. f
, f,
,
. (
f
).
: , 257, 6.5. f
:
) f, :
i)
f(x),
ii) f
( ).
) .
, f
, f,
( ) f(x)=0, f
. x0 f
f + -, f(x0)
, f - +, f(x0)
.
f .
f,
f.
f .
f (,),
f .
: , 268, 3, 4.
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54 -
. f
( ) f. f()=0
f()>0 f(), f()
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55 -
f(x)>g(x) ( f(x) h(x)>h() f(x)-g(x)>0 f(x)>g(x)
x>.
: , 269, 3.
) :
)x(g)x(f ( )x(g)x(f ) x,
h(x)=f(x)-g(x)
. h
x0= h()=0,
)(h)x(h 0)x(h )x(g)x(f x
.
: lnx x-1 x>0.
8. :
( x)
, ,
. .
: , 268, 7, 8,
9, 10, 270, 7, 8,
9, 10.
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56 -
.
1. ) 1xe
x
++++
x 1xxln x>0.
) e>e.
2. f(x)=x4+4x+13, x .
) f.
) 10)
2(f ==== , * .
) ,
(134+83+16)(4+4+13)=1004.
3. (x)=x100+x+ ,
Q(x)=(x-1)2.
) =99 =-100.
) 0)x(P x .
) (x)+P(x) 0 x , , .
) Cf
.4. f : (0, ++++ ) f(x)=x2+e2-lnx, >2
0)x(f x (0, ++++ ).
) .
) Cf.
(1,f(1)),
(2-4) /sec.
.
5. f :[1, 4]
f(2)
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57 -
) f .
) 0>> .
7. f : (0, ++++ ) x
xln1
)x(f
++++
==== .
) f .
) x, y >0, 1xy
ylnxln1
++++++++.
8. f:[, ] f()=,
f()= f(x)>0 x [,].
) x0 (, )
f(x0)=x0.) x1, x2 (,) 21 xx ,
f(x1)f(x2)=1.
) x)x(f ++++ x [,].
9. (x)=x4-4x+15, x .
) () (x).
) P(x)=0 .
10. ,
4x)4(e4 22x ++++++++++++ x .
) , .
) , )
.
11. f : (0, ++++ )
2)x(fxe 2)x(f ++++==== x .
) f(x) f(x).
) f .
) ( ) f-1.
12. f, g : 0)x(g
xe)x('g)x(f)x(g)x('f ++++==== x .
) )x(g
)x(f)x(h ==== .
) )x(g)x(f)x(g)x(f 22
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58 -
f(x)f(x)-f2(x)=xe2x x .
) f.
) f.
14. f :
f(x)>0 ef(x)+lnf(x)=x+e x .
) 1)0(f ==== .
) f f.
) lnx1.
) f .
) f.
) 2x3x 22xx2x2
++++ .
) x=-x.
16. :f :
1xx2
1e2)x(f)x(f 2x3 ++++====++++ x .
) f(0).
) e
x
-x-1=0.) f.
) f .
17. 2x
xln)x(f ==== .
) f.
) e2x xe2
x>0.
) 2x x2
x>0, =e.
18. :f : f(x)f(x)=x
x .
) f.
) f
.
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59 Cf
9.
, DE L HOSPITAL,
Cf
.
1. f
, f ,
f(x) f(x)>0 f(x)0 ,
. f(x)
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60 Cf
, f,
,
,
f. ( f
).
: , 278, 3.
4. f
x1, x2,,x,
f(x1)=f(x2)==f(x)=0.
.
: >0
xln
x
)x(f ====
x0=1.
5. f
, x0 f
,
f, x x0
f(x0)=0 .
f
.
: , 279, 5.
6. :
) Jensen : f : [,]
0)x(''f [, ], )2
(f
2
)(f)(f ++++
++++
0)x(''f [,], )2
(f
2
)(f)(f ++++
++++.
: 0)x(''f [, ].
x1 (,2
++++)
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61 Cf
)x('f2
)(f)
2
(f
2
)(f)2
(f
)x('f 11
====++++
++++
==== .
x2 ( 2 ++++
, )
)x('f2
)(f)
2
(f
2
)2
(f)(f
)x('f 22
====++++
++++
==== .
)]x('f)x('f[2
)](f)(f[)
2
(f2 21
====++++
++++ (1).
x3 (x1,
x2) 12
123 xx
)x('f)x('f)x(''f
==== .
)x(''f)xx()x('f)x('f 32121 ==== (2).
(2) (1) :
0)x(''f)xx(2
)](f)(f[)
2
(f2 321
====++++
++++
0)x(''f [, ] x10 x f(0)=0. 2
)(f)
2
(f
.
) :
f
f.
f(x).
Cf . f
.
: lnx x-1 x>0.
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63 Cf
. -
Cf
1.
.
:
, .
: , 285, 1.
2. ++++
(, ++++ )
( , ). -
,
, :
) >+1 .
) =+1 y=x+ ++++
.
) y=
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64 Cf
. ,
DE L HOSPITAL,
Cf
1.
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65 Cf
)2x2x
xlim
2
4
0x ++++.
)
2
x
x1e
xxlim
2x
0x
.
) )x
e
x
x(lim
x2
0x
.
)xx
1x
lim
2
0x .
7. f f(0)=f(0)=0 f(0)=2.
) x
)x(flim
0x
.
) x
)x('flim
0x .
) )1xln()x('f
)x(fxlim
2
0x ++++
++++
.
8. f : y=2x+3.
) x
)x(flimx
.
) ]x2)x(f[limx .
)
41x2x2)x(fx
3xx2)x(f3xlim
32
22
x====
++++
++++++++++++++++
.
9.
>>>>++++
++++
====
0xx
x)1xln(
0xx2
1e
)x(f2
x
.
f , , .
10. :f , f(0)=2
x0=0 0.
====
====
0x0
0xx
)x(f)x(g .
) x
)x('flim
0x .
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66 Cf
) 1x
)x(flim
20x====
.
) g(x).
) g(x) .
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67
10. A
.
1. :
. :
f, g
:
1. ==== dx)x(fdx)x(f , 0.
2. ==== dx)x(g)x(fdx))x(g)x(f( ,
, 0 ==== dx)x(g)x(fdx))x(g)x(f( .
3. ++++==== c)x(fdx)x('f , c .
. :
1. ====
dx)x(fdx)x(f , .
2. ====
dx)x(gdx)x(fdx))x(g)x(f( ,
====
dx)x(gdx)x(fdx))x(g)x(f( .
3. ==== 0dx)x(f .
4. ====
dx)x(fdx)x(f .
5. ========
)(f)(f)]x(f[dx)x('f .
6. f [, ]
0)x(f x [,], 0dx)x(f .
:
) 0)x(f x [, ], dx)x(f
Cf, x=, x= xx.
) ..x. >>>>====
250 01xdx f(x)=x
[0, 5/2].
7. f , ,
( ).
++++ ====
dx)x(fdx)x(fdx)x(f ( Chashles).
.
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69
:
x ++++ , 0 ,
.
:) dx)2x3( = c
3
)2x3(++++
.
) ++++==== ++++++++ cedxe 2x2x .
) ++++++++
====++++
c5
1x5lndx
1x5
1.
3. :
1. :
) ( ...),
.
` ) x
.
) x
.
: , 307, 1, 338,
1.
2. :
) :
++++ dx)x( = c
)x(++++
++++ .
++++ dx)x( = c
)x( ++++++++ .
++++++++====++++
c
)x(dx
)x(
12
.
++++++++====++++
c
)x(dx
)x(
12
.
) :
2(x)+2(ax)=1. 2(x)-2(ax)=(2x).
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70
2(x)(x)=(2x).
2
)x2(1)x(2
==== .
2
)x2(1
)x(2 ++++
==== .
)x2(1
)x2(1)x(2
++++
==== .
)x2(1
)x2(1)x( 2
++++==== .
1)x(
1)x(
22 ==== .
1)x(
1)x(
2
2 ==== .
) :
(x)(x)=2
1(2x).
=2
1[(+)+(-)].
=2
1[(+)+(-)].
= 21 [(-)-(+)].) :
: xdx , xdx .
,
.
: , 317, 5 i), ii).
: xdxx . ,
, ,
.
: , 317, 5 iii).
: xdx , xdx . =1 : dx
x
)'x(dx
x
xxdx ======== =
= cxlndx]'x[ln ++++==== .
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71
=2 : ======== dx)1x
1(xdx
22
cxxdxdxx
12
++++======== .
>2
.
: dx)x()x( , dx)x()x( , dx)x()x( .
.
: , 317, 6.
3. :) :
cg
fdx)'
g
f(dx
g
'fgg'f2
++++ ========
.
cflndx]'f[lndxf
'f++++======== .
cxlndxx
1++++====
.
c
xlndxx
1 ++++++++====++++
.
) :
.
: dx]'f[lndxf
'f==== =
cfln ++++==== .
: :
1= dx2x3x
3x22 ++++
2= dx
4x3x
1x3
2
++++
.
,
.
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72
: :
= dx6x5x
1x2 ++++
++++.
: 3x
B
2x
A
6x5x
1x2
++++
====++++
++++.
)2x(B)3x(A1x ++++====++++ .
:
: ( ) :
++++====++++ )2x(B)3x(A1x x+1=(A+B)x+(-3A-2B) .
+=1 3+2=-1 =-3 =4.
: ( ) :
)2x(B)3x(A1x ++++====++++ x 2 x 3. =-3
=4.
: = dx6x5x
1x2 ++++
++++
= c3xln42xln3dx3x
4dx
2x
3++++++++====
++++
.
, :
1
12 x
A...
x
A
)x(
K...
)x(
B
x
A
++++++++
++++
++++++++
++++
1, 2,,
1.
: :
= dx2x3x
1x3 ++++
++++.
,
((x)=(x)(x)+(x))
.
: :
= dx
6x5x
1x2x2
3
++++
++++.
4. :
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73
==== dx)x('g)x(f)x(g)x(fdx)x(g)x('f ,
:
====
dx)x('g)x(f)]x(g)x(f[dx)x(g)x('f , .
:
) :
.
) :
.
) :
,
.
) :
.
: , 316, 1.
) :
dxx
)x(A2
, dxx
)x(A2
(x)
.
)'x(x
12
==== )'x(x
12
==== .
) :
f,
,
-
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74
.
: , 340, 10,
11.) :
+1
.
: ====e1
dx)x(lnI .
) 1 IeI ==== .
) 4.
5. ( ) :
) : f(x)
[, ] x=(u) 1-1 [u1, u2]
(u1)= (u2)=, x=(u)
dx=(u)du ====
uu
2
1du)u('))u((fdx)x(f .
: ==== 11
2dxx1I .
)
dx)x('g))x(g(f , u)x(g ====
. du)u(f
g(x)dx=du.
: ====
uu
2
1du)u(fdx)x('g))x(g(f g(x)=u
u1=g(), u2=g().
:
: dx)x('f))x(f( dx))x(f(
)x('f
.
f(x)=u
: duu du
u
1 .
:
++++==== dx)3x()4x6x(I32 .
: dx)x('f )x(f dx)x('fe
)x(f . f(x)=u
: du u
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75
dueu .
: ==== ++++1
03x dxxeI
2
.
: dx)x('f))x(f( dx))x(f( . dx)x('f))x(f( dx))x(f( .
dx)x('f))x(f( dx))x(f( .
dx)x('f))x(f( dx))x(f( .
f(x)=u. x
dx du.
: :
) dxx
)x
1(
I2==== .
) dx)2x(I ++++==== .
: dx)x(f
)x('f .
f(x)=u
: c)x(flnculnudu ++++====++++ ==== .
: dxex
e1I
x
x
++++
++++==== .
:i) : dx)x('f)x(f dx
)x(f
)x('f
.
f(x)=u.
:
dx4xxI 2 ++++==== .
ii) : dx)x(g)x(f . u)x(f ==== .
:
dx1x2xI 2 ++++==== .
iii) : dx)x()x(B
)x(A
.
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76
(
) .
:
dx1x2x
xI 22 ++++++++==== .
:i) : dxx dxx
xdxx .
,
2x=1-2x 2x=1-2x x=u
x=u
.
: ==== xdxI3 .
ii) : dxx dxx
3 . , ,
: ======== dxxI
============ ====
ux
222
22 xdxxdx
x
1xdxx
2
1
2
1
22 I
1
xI
1
uIduu
====
========
I-2 .
: ==== xdxI3 .
iii) :
====
dx)x(gdx)x(f ====
dx)x(gdx)x(f
====
dx)x(gdx)x(f .
u=-x x2
u ==== (u=-x u=2-x).
: ====2
02
0 dx)x(fdx)x(f .
: dx)x(g)x(f f(x), g(x) . u
.
:
++++==== dx)4x()1x(I62 .
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77
: dx)e(f x dxx
1)x(lnf .
ex
u ,
lnx=u.
: ==== dxx
xlnI
2
.
,
u,
u x.
: ++++====
dx)x(fdx)x(f .6. ====
x dt)t(f)x(F (
) :
: f
====x dt)t(f)x(F
)x(f)x('F ==== x .
) . :
====x )x(f)'dt)t(f( . ====x )x(f)'dt)t(f( . ====)x( )x('))x((f)'dt)t(f( (
).
)x('))x((f)x('))x((f)'dt)t(f( 11)x( )x( 2221 ==== ( ).
++++==== )x( )x()x( )x( )x( )x( 2121 21 )'dt)t(f)(x(dt)t(f)x(')'dt)t(f)x((( ).
: , 338, 5.
)
, , ,
. , ,
-
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78
.
: , 339, 3.
)
. , (0/0),
De l Hospital.
x ,
m(-) dx)x(f (-) m
f [,] f
[,],
.
: , 339, 6.
: )dtt1
1(lim 1xx 2x ++++
++++
++++. ( )
: )dt)tt(x
1lim x02x
++++++++
.
( :
dx)x(fdx)x(f
f [,].)
f. ,
,
, f
x
.
.
: , 339, 5.
: f
====x4
t xxdt)t(fe x .
f.
7. :
: f
[, ] (, )
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79
)()(fdx)x(f ==== .
dx)x(f)(f
====
f [,] f .
) f . dx)x(ff
====.
: , 341, 1.
)
.
: 2)x(f >>>> x [1, 5],
>>>>51 5dx)x(f .
) 0x
(, )
,
Rolle
.
: ====1
0 2
1dx)x(f ,
0x (0,1) 00 x)x(f ==== .
8. :
0)x(f x [, ],
dx)x(f
Cf, x=, x= xx.
f [,].)
Cf, x=, x= xx.
dx)x(fE ====
f [,].
: , 349, 1.
) Cf, xx x=.
-
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80
f(x)=0 1 2
.
dx)x(fE 1==== dx)x(fE2
==== .
:
9x
1x)x(f
2
++++==== ,
x'x x=2.
)
Cf xx. f(x)=0
1 2 .
dx)x(fE 21==== .
: , 349, 3.
)
Cf, Cg x=, x=.
dx)x(g)x(fE ====
f(x)-g(x) [,].
:
1x
1)x(f
==== ,
4x)x(g ++++==== x=3 x=5.
)
Cf Cg.
f(x)-g(x)=0 1 2
.
dx)x(g)x(fE2
1 ==== . : , 349, 5.
)
.
.
.
: y=x+, xx
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81
y=0
.
: , 350, 6,
10.)
f,
yy y= y=.
( ),
2
, y=x, x=-y
y=-x, x=y, ).
: xy2 ==== , yy
y=2 y=-1.
9. :
f.
:
) f ),0( ++++
, : ++++====
x1 dt)t(fx
11)x(f
x ),0( ++++ .
) f
++++====++++x0
x2t e)3x(dte)t(f3
x .
) f(x)
====10 )x(fdt)tx(f2 x>0.
) f
++++====1
0xx1 e)x(fdx)x(fe x .
10. :
:
) f [, ]
0)x(f x [,], 0dx)x(f .
) f, g [,]
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82
)x(g)x(f x [,],
dx)x(gdx)x(f .
) f [, ],
dx)x(fdx)x(f .
) m f
[,] : m(-) dx)x(f (-).
:
1 :
.
-1x1 -1x1.
2 :
,
.
3 :
, [,],
.
: , 353, 10.
-
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83
.
1. f(x)=x
1 x>0 f(9)=1 f.
2. f,
f (x)=x+2x
2
f =6.
3. f,
f(x)=12x2+2,
(1,1) 3.
4. x>0, f :
xf(x)=ex -f(x) f(1)=
e
1 .
5. f(x)ef(x)=2x+1 x,
f (1, f(1))
3/5, f.
6. f f(x)lnx+x
)x(f=ln(x+1)
x>1 f(2)=0. f.
7. f f(x)>0, f(x)=-2x)x(f
x>0 f (2)=1. f.
8. f: )f(x)+f(x)=1+e-x
x ) xo=0 2.
= 1
0
dx)x(f .
9. f : f(0)=0
2f(x)=ex f(x) x .
) f(x)=ln
++++
2
e1 x.
) G g(x)=x2008f(x),
h(x)=G(x)G(-x), x
h.
10. :
i) dxx34 , ii) ++++ dx)4x3x(3 , iii) dxx .
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84
11. :
i)
dxx
xx 43 2, ii) ++++ dxxx
, iii) xdx2 .
12. :i) dx
x
12
, ii) ++++ dxe 5x3 , iii) ++++ dx)xe(
x .
13. :
i) ++++
dxe
1ex
x
, ii)
dxx
xln12
, iii) ++++
dxx
xxxx22
2
.
14. :
i) xdx3 , ii) ++++ dx)exe(xx , iii) dxxx
2 , iv) dxxx 3 .
15. :
i) dxx2
1, ii)
dx
3x
1, iii)
dx
x42
3.
16. :
i) ++++
dx3e
ex
x
, ii) ++++
++++dx
8xx
1x33
2
, iii) ++++
dxxlnx
xln1.
17. :
i) xdx , ii) ++++
dx
2x
x22
, iii) xdx3 .
18. :
i)
dxx1
x22
, ii) dxxlnx
1, iii)
++++dx
xe
)x1(ex
x
.
19. :
i) dxx
x, ii)
++++dx
7e
ex
x
, iii) ++++
dx
7x5x
5x22
.
20. :
i) dxxx
12
, ii) ++++
dx7x
1, iii)
dx
x21
x2
.
21. :
i)
dx1x
12
, ii) ++++
++++dx
6x5x
1x22
, iii)
dx
9x
4x32
.
22. :
i)
dxx4x
43
, ii) ++++++++
dx9x6x
x22
, iii)
dx)1x(x
12
.
23. :
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85
i) ++++
dx2x3x
12
, ii) ++++
dx)2x(x
13
, iii) ++++
dx
)2x(x
5x4x2
2
.
24. :
i) ++++
dx1x
1x, ii)
++++dxxx
1xx2
2
, iii) ++++ dx2x3x
x2
3
.
25. :
i) ++++
dxx
4x3x2
3
, ii) ++++
dxx
1x2x3x3
34
, iii) ++++
dx2xx
x2
3
.
26. :
i) dx2xx , ii) 1)dx-(2xx , iii) dxx
1x
2.
27. :
i) xdxx3 , ii) 2xdxx
2 , iii) ++++ dx2x)1x2x(2 .
28. :
i) dxxe x , ii) dxex
x3 , iii) ++++ dxe)1x(x32 .
29. :
i) dxxe 5
x
, ii) dx2xx , iii)
++++ dxex 7x3 .
30. :
i) xdxe x ,ii) 2xdxe
x ,iii) xdx3ex3 .
31. :
i) xdx2x , ii) (lnx)dx ,iii) xdxe
x ,iv) ++++
dxex
1xx1
.
32. :
i) xdxlnx3 , ii) 9)dxxln(x
2 , iii) dxx
xln2
.
33. :
i) ++++++++ dx)x1xln(2
, ii) 1)dx-log(xx22
, iii) xdxlnx .34. :
i) ++++ dx1xx22 , ii)
++++dx
)e1(
e2x
x
, iii) ++++
++++dx
)x6x(
3x42
.
35. :
i) ++++
dxx1
x22
, ii) dx)x-(3x2 , iii) dxxx .
36. :
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86
i) ++++
dx)1e(
e2x
x
,ii) dxeexx ,iii) xdx4 .
37. :
i)
++++
dxxe
1x2
, ii)
dxx4x
2
, iii) ++++
dx)1x3(x
52
.38. :
i) ++++
3
0 2dx
16x
x, ii) ++++ dx8xx
43 , iii) ++++
dxx1
x2.
39. :
i) ++++
dx1xx
1, ii) dx
x
e x, iii)
++++
++++dx
ee
eex4x3
x2x
.
40. :
i) ++++
dx1x2
x3,ii) ++++++++ dx)xx)(1x3(
1032 ,iii)
dx1x21-2x
14
.
41. :
i) dx1x33 ++++ , ii)
dx2x3
x, iii)
4
1
x
dxx
e.
42. :
i) xdx2 , ii) ++++ dx3x)x3(
2 , iii) dx2x-1 .
43. :i) xdx4x , ii) dx2xx3 , iii) xdxx
22 .
44. : ++++++++
dx5x3x4
1.
45. :
1
1
2dxx1 .
46. :
i) 2
0
2 dx1x , ii) (((( )))) ++++
2
1
dx1x4x1 , iii) ++++
2
1
2 dx3xx2 ,
iv) ++++
2
1
dx)1x3x( .
47. : 2
0
dx)x(f f(x)=
>>>>
1xe
x1xxe
2
2
x2
.
48. = xdxln :
) I=xlnx--1.
) 3 4.
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88
60.
x , y
f(x)=2+x-x2.
61. f(x)=lnx, xx yy
y=1.
62.
f(x)=3-x g(x)=x2+1.
63.
f(x)=2x g(x)=2x2.
64.
f(x)=xe g(x)=lnx x=
(>0). )(Elim ++++
, )(Elim0 ++++
.
65.
f(x)=x
(,0)
(0,0).
66.
f(x)=4x3+x2-2x, g(x)=43x-2x2
x=-1 x=2.
67. :
i)4
1xdxx
1
0
3 , ii)6
dx)x1(
12
3
4
2 ++++ ,
iii)0
e
1exdxln
x
1e
1
, iv)ln4 16lnxdxln
4
2
,
v)4 16dx23
1
x ,vi)2
1dx2
8
1 4
2
x ,
vii) 6dxx1
x
4
27 3
23
2
.
68. f:(0,+)
f(x)= ++++
++++++++
x
1
02
x
02
dtt1
1dt
t1
1 . f
.
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89
69. N :
F(x)= x
1
2 dt)12t6t6( .
70. f , xo=0
f(0)=3, 1xe
dt)t(xflim
x
x
0
0x
.
71. ====++++1x
x
dt)t(tf x3+3x2-1 f [-2,0],
f(-1).
72. F(x)= ++++2006x
x
dt)t(f : F(x)=f(x+1)-f(x).
73. f, g, f, g [,] f()=f()=g()=g()=0,
====
dx)x(g)x(fdx)x(g)x(f .
74. f R
x f(2
-x)=f(
2
+x) f(
4
)=-
2
1,
: 1dx)x(fx24
3
4
==== .
75. :
) ====++++
++++
++++
++++
dx)x(fdx)x(xfdx)x(xf .
) ====e
1
1
0
x dx)x(lnfdx)x(fe .
) ====2
2
4
3
4
1dxx
1dxx >0.
) ====++++2008
2008
0dx)2008x(xdx .
76. f ,
:
) : ++++====++++ 1
0
dx)x(fdx)x(f .
) 1dx)x(f2
==== :
1dx))]1x(([f
1
0==== ++++ .
77. P(t) , t , (t)=(t) (1),
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90
>0 .
(t) , :
) P(t).
) 1920 , 5.000.000
1990 10.000.000,
.
)
, 2010.
78. f(x)= ++++++++++++++++x1
2003200119991997 dtt)ttt(t .
f .
79. f(x)= 2x/6 dtt
t , x>0.
) f(x).
) x0(4
,
3
)
Cf (x0, f(x0))
y=x.
80. h(x)=(x-1)
x
2 dtt
lnt
,x>0.
) h(x) .
) Rolle h
[1,2].
) (1,2)
-1ln=
2 dtt
lnt.
81. f(x)=x+e-x,g(x)=x-e-x.
) f(x)-g(x) f(x)-x [0,+ ).
) Cf, Cg
x=0, x=2.
) Cf
x=0,x=2,y=x.
82. f
x= xx3
-
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91
x=4
. f [,],
dx(x)f .
83. f,
[, ], ,
, :
dx(x)f +
dx(x)f .
84. f(x)=1+2x
1.
) .
)
4
5
21 dx(x)f 2.
) Cf,
xx x=2 x=4.
) xx
.
85. f : R* (0, + )
f (x)=x2f(x) f (1)=e
2004.
) f f(x)=2004e x1-.
)
g(x)=2x
(x)f, xx
x=1 x=2.
86. h(x)=ex.
) f g R,
f(x)+g(x)=h(x).) f, g.
) ()
f,g x=0 x= >1.
) ++++
lim ().
87. f , g R
f(x)-g(x)=(x2+2x-1)ex xR.
) h(x)=f(x)g(x) Ch (0, -1).
-
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92
) Cf,
Cg.
88. f R
++++====
x
0
xt
4)x(f)1e(dt)t(fe2 xR.
f R .
89. f R f(1)=f(1)=0.
) ====10
210 dx)x(''fx2
1dx)x(f .
) 1x
6)x(''f
3 ++++==== =
10 dx)x(f .
90.) ex>x-1 xR.
) f(x)=x+xe-x
. N f
(0,f(0)) 2x-+7=0.
) :
i) f R
=x f
++++x .
ii) ()
f, =x x=0
x= (>0).
iii) )(Elim ++++
.
91. dte
)t(f)x(g x x==== f
[,], >0 0dx)x(f ==== .
(,) )(fdx)x(f ==== .
92. ==== dt)tx(f)x(F f
f(x)>0 x .
a) F
F(x).
) x0 F(x0)=0 F(x)=0
x .
93. f
++++====10
xx1 e)x(fdx)x(fe x .
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93
94. f ++++====x0
t0 dt)du)u(f1()x(f
x .
) f.
) f 1-1.) ==== ++++
xe0
t0 dt)du)u(f1(
= ++++)
x
1ln(0
t0 dt)du)u(f1( ),0( ++++ .
95. f 0)x(f
====10
22 dt)xt(tfx21)x(f x .
g(x) 2x)x(f
1)x(g ==== .
) )x(xf2)x('f 2==== x .) g .
) f 1x
1)x(f
2 ++++==== ,
x .
) ++++
++++
2xx
xdt)t(flim ,x>0.
96. f : ++++ ),0(
>>>>x2 )x(xfdt)t(f ),0(x ++++ .) ====
x2 dt)t(f)x(g
.
) x1dt)t(fx2 ====
(1, 3).
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94
11.
1. z : z4= 2)z( .
) |z|=0 |z|=1.
) z0 z
1z ==== .
) z0 z6=1.
) z z4= 2)z( .
) z
z0.
2.
2z
i2zw
++++
++++==== , z=x+i x, .
) w +i , .
) z, w .
) z, w
.
) z,
Re(w)=Im(w).
) z 2+2=4,
w =x.
3. z z3+|z|=0.
) |z|.
) z3+|z|=0.
) z1 z2 ,
z13 z2
3.
) =z12004+z2
2004.
4. z, w :
2
1
1x
2xi63wxi43zlim
2
2
1x====
++++
.
) z.
) w.
) |z-w|.
5. z=+i ln|z|=1-|z|(1).
) (1) |z|=1.
) (, ) z.
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95
) z
w=2+2i.
6. f, [0,2] z1=f(0)+i z2=1+f(2)i. |z1+z2|=|z1-z2|,
f(x)=0 [0,2].
7. f, [,]
(,) f()>>0. )(if
)(ifz
++++==== . z
f(x)=x
(,).
8. :
++++++++
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96
)
.
) :3
4i
3
5z ==== z C,
(1).
) .
) g(z)=z-3, z
(1).
g(z).
12. z1 z2 f
: f(x)=|z1x+z2|, x .
) g(x)=f(x)+f(-x).) Cf x+
(0, |z2|), 21zzw ====
.
) : 2+|z2||z1+z2|>>>++++++++++++
++++++++
0xwzxw3x
0xzwx22 .
)x(flim0x
w
z .
15. f [, ] f(x) 0
x [,].
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97
z : )(fz
1z ====++++ )(f
z
1z 2
22 ====++++ .
:
) x3f()+f()=0 (-1,1).
) |f()|
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98
) 1,2(,) : f(1)f(2)=4.
23. f : [0, 4]
f(0)=f(4)=2 f
[0,4].) 2)x(f x [0,4].
) f [0,4] 3,
x1,x2(0,4) 4)x('f
1
)x('f
1
21
==== .
) g(x)= x)x(f ++++ (x)= x2 x
[0,4]. 0x (0,4)
C Cg x0
.
24. f : f(0)=1
x22 xe)x(f)x('f)x(f ====++++ x .
) f.
) f.
) y=-x+1 Cf.
25. f :
x)x(f)x(f3 ====++++ x .
) f(0).
) f
.
) x)x(f)x('xf
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99
) f ),1[ ++++ .
) f.
) f(x)>2x2+lnx ),1[x ++++ .
28. f : : f(x)=f(x) x f(0)=f(0)=1.
) xe
)x(f)x('f)x(g
++++====
.
) f(x)+f(x)=2ex, x .
) f.
29. f(x)=ex+e-x-x2-2.
) f .
) f .
) f (-,0] [0,+).
) f(x)=0.
) ex+e-xx2+2 x .
30. f(x)=3x4+4x3-(10+)x2+. f
x0=1 f(x0)=2, :
) , .) f .
) f.
) f(x)=0.
) f.
) f.
) 3x4+4x3+3212x2 x .
31. f : :
f3(x)+3f(x)=x3+3x+3ex-3 x .
) f .
) f(x).
) f(x)=0.
) f.
) f .
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100
32. f, ,
22
22
zx
zxzx)x(f
++++
++++==== z=+i, , 0
.) )x(flim
x ++++, )x(flim
x .
) f, |z+1|>|z-1|.
) f.
33. 1x
x4x)x(f
2
3
==== .
) f.
) Cf xx.
) Cf
xx.
) Cf .
) f .
) f
f.
) x3-x2-4x+=0,
.
34. f : : xe
2
)x('f
1
)x(f
1====++++
x . f(0)=1, :
) ++++==== dxe))x('f)x(f(Ix .
) f(x)=ex, x .
35. F f : ,
: F2(x)F(x)F(-x) x , 0.
) F(0)=F().
) f(x)=0 .
36. f : :
==== dt)tx(f)x(g , x
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101
37. f
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102
) f(x)=0.
42. f, xe
1x2)x('f
====
f(0)=-1.
) f.
)
f g 1x2
)x(f)x(g
====
x=1.
43. f, g f=g, g=f
f(x)>0, x g(-x)g(x)xf(0), x>0.
) f, g .
) h(x)=e-x(f+g)(x) (x)=ex(f-g)(x)
.
) f(0)=1, f, g.
44. f :
2
e
)x('f)x(f)]x('f[
x2 ====++++
, x
, f(0)=1,f(0)=0.) f2(x)=ex+c1x+c2, c1, c2 .
) f2.
) xe)x(f x ==== , x .
) : dx1e
)x(''f)x(fI
x ==== , x>0.
45. f f(x)>0
lnf(x)+e
f(x)
=x x
.) f .
) f.
) f(x)=1 f(x)=e.
) ++++==== ++++ 1e
ee1
1 2 dx)x(fdx)x(fI .
46. f
++++====x1
3 1x2)x(lnxfdt)t(lnf , x>0.
) : ====2/1
0
x
14dx)x('fe .) : f(x)=3e2x-2, x .
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103
) Cf A(0, f(0)).
)
Cf, x=ln2.
47. f [1, +) f(x)1 f(x)-lnf(x)=x x1.
) f(1).
) f(x)=1 f(x)=e.
) f .
) f.
)
Cf-1, 1 x=e.
48. f * f(x)=1+xe-f(x)
1e
1xex
)x(f
++++++++
x 0.
) : xx)x(f xe)'e( ==== , x 0 f(x)=ln(ex-x-1), x 0.
) f * .
) ==== dxe)x('fxI )x(f2 .
49. f, g
f(x)>0 ====x
1 dt)xt('fx)x('g , x .) g(x)=f(x2)-f(x), x .
) f .
) (0,1), :2
)('f)('f 2 ==== .
50. f ====21
10 dt)t(fdt)t(f .
) ====20
10 dt)t(f2
1dt)t(f .
) ====x0 dt)t(fx
1)x(
Rolle [1,2].
) (1,2) ====10 dt)t(f)(f .
51. f
2f(x)+f(2004-x)= -x, x .
) f.
) g(x)= xln )x(f .
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104
) h(x)=x2f(x), , :
dx)x(xhh =h()-h().
52. f :
[f(x)]1821+[f(x)]3=-ef(x), >0, x .
) f(x)=c , x , c .
) g(x)=1e
)x(fx
.
53. f :
f(x)+f(x)=1, x f(0)=e+1.
) ++++y
lim (2x
lim
f(xy)).
) f .
54. f
g : ( fff )(x)=x
( ff )(x)=g(x) , x . Cg
(1,3) (2,2
9), :
) 1
0
dx))x(g(f =2
1.
) 2
1
dt)t)(fff( =23 ( f 2006
).
55. f
f(x)+f(x-1002)=0, x . :
) f(x+2004)=f(x), x .
) ++++2005
1
dx)2005x(f = 2006
2
dx)x(f .
56. f : :
f(x)]3-2003[f(x)]2-2003f(x)-2004 =0, x .
) f .
) : I= ++++++++
++++5
32
4
dx1xx
2004)x(fx,
J= ++++++++
5
82
4
dx1xx
x, K=
++++++++
3
82
3
dx2003)x(fxx
1xx.
IJK.
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105
57. g
f(x)= ++++1x2
1
dt)tx2(g .
) : f(x)+f(x)=2[g(2x-1)+2g(2x-1)] ,
x .
) g xx
g ,
h(x)=f(x)+f(x) .
58. ) : (x2 + x +1)11(2x2 x +1)7+
+(x2 + x +1)7(2x2 x +1)11=x22x.
) : 2ln
7x4x
9xx22
2
++++++++
+++++++++5(x23x+2)
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106
62. f [0, 1], f(x)>0
x[0,1] 1
0
2 dx)x(fln +5
1=2
1
0
2 dx)x(flnx .
) f(x)=
2x
e , x[0,1].) :
++++
1
0
dx)x(f)x1(f
)x(f.
63. f(x)=x44x3+5x2, >0.
) f .
) f.
) f(x)=0.
64. f(x)=x+x
a2, , .
) f(x)=0
(0,).
) a
0
dx)x(f =,
(0,) f()=.
65. ) f
f(x)=0 (),
f(x)=0 +1
.
) : 4x= x2+15 10.
66. f(x)=ex+x3+x, x .
) f
f1.
) f1(x)=0.
) f1
, f1 1.
67. ) f [,]. f
[, ], :
dx)x(f +
)(f
)(f
1 dx)x(f =f()f().
) f(x)=ex+x5. :
++++
1e
1
1 dx)x(f .
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107
68. ) f .
: f(x)= )x(f 1
f(x)=x.
) f(x)=x+
x
2004
t
dte2
, x , f
: f(x)= )x(f 1 .
69. f(x)=xxx
xxx
4e
53e
++++++++
++++++++, x .
) f ++++ .
) f .
) g(t)=)t(e , :
++++xlim ++++
++++1)x(f
)x(fdt)t2)t(g( .
70. f .
) x, y f(x+y)=f(x)+f(y)+xy(x+y),
,
dx)x(f =0( ).
) x 1x3dt)t(fx3
1 ,
33)x(f
x
x))x(f(dt)t(f ====
.
71. f [0,10]
g g(x)=2)x(f))x(f())x(f(6
3)x(f3))x(f(623
2
++++
++++++++
.
) f .
) g f, g.
72. :z2(2)z+1=0, (0,4 ).
)
.
) z1, z2 .
1 2
1 2
z zz z
3
+++++ =+ =+ =+ = z1, z2.
73. f(z)= z iz ,zC.
) :f(z)=2i.
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108
) 2)z(f ==== z .
) 1z ====
w=f(z) .
74. :z z+4Re[(12i)z]+4=0 (1).)
.
) z1, z2 ,
8zz 21 .
) t1, t2 z1 z2 (
. ) 21 zz ,
14
21
2
21 52)tt(10tt ++++====++++++++ N*.
75. f : : f3(x)+2x2f(x)=33xx .
x
)x(flim
0x====
:
) =1.
) :
i)x
)x(flim
0x .
ii)x
))x(f(flim
0x .
iii)2x3x
)xx(flim
20x
2
++++
.
76. f : ,
. 11x
)x(flim
1x====
:
) : x
)x(f
lim2x .
) : f(1)=0.
) ,
====
====
1x,
1x,1x
)x(f)x(g
.
) =1 g
(): y=2x xo(0,1).
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109
77. f : : 1x)x(f x .
) : 1x
)x(flimx
====++++
.
) :
i) x x
)x(flim
++++, *N .
ii) ))x
1(xf(lim
0x .
iii) )1x
x(f
x
2xxlim
2
2
2
x ++++
++++
.
78. f
: 1x
x
)1x
x
(f ====
1x .
) : 01x
xlimx
====++++
.
) f(1).
) xo6
(
,
2
) :f(xo)=0.
79. f [0, ++++ ) f(x)>0 x
[0, ++++ ).
f, xx, yy x=u
E(u)=euf(u) 0u :
) f(x)+f(x)=ex x[0, ++++ ).
) 0x
1)x(flimx
====
++++.
) f.
) f(x)= 2
1
x.
80. f f(x)>2007
x . Cf y=2006x+1
.
81. f (e, + )
f(x)=1+ x
0)t(f dte x(e,+ ).
) f(x)=ln(x+e).
) f .
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111
i) f.
ii) , xx=(exx)+(exx) x
[0,+ ).
iii) f, xx x=0 x=
4
.
87. f f(x) 0 x1
t 1xdt)t(fe
x .
) f(1)=e
1.
) f (x)>0 x .
) f(x)=x (0,1).
) z
2
5i)1(ef2z ====++++
.
88. f (x)= 2x1
12010
++++++++
, x .
) f .
) f(x)=
.
) : ++++
++++
1xx
xdt)t(flim .
89. f , f(x) 0 x ,
f(2005)=2
1, f(2007)=3 f(1)f(2)=f(3)f(4) .
) f()=1.) xo[1,2] f
2(x)=f(1)f(2).
) f .
90. f 0, f(0)=2006
xf(x)+x4(x
1)=(x), 0 *x .
) f .
) .
) )x(flimx
)x(flimx ++++
.
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112
) f(x)=0 .
91. f :
,
A(1,f(1)) , 0x f(x)
x
)x('f=xex.
) f(x)=xexexe 12
x2++++ .
) f(e2004 ) f(2004).
) g ,
g(0) )2
e1(gdx))x(f(g10 .
92. Cz , , , i
i)iz1(
++++
++++====++++ (1).
) z .
) z
.
) z
.
) 7i43z4
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113
ii) 11 zwzw ==== , w .
iii) w,
10zwzw 22 ====++++ ,
.
95. f 0)1(f >>>>
i)1(f
34
2
)0(f)1(fz ++++
==== ,
)i31(2
zz ==== .
) f(-1)f(0)f(1)=8.
) z)zRe(2 ==== .
) f(-1)
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114
) Cf,
xx x=1 x=e+12
3E ====
..
98. f :
2
1
)x(f1
5)x(xf2
====++++
++++ x f(0)=3.
) 9xx)x(f 2 ++++++++==== , x .
) g(x)=ln(f(x)) :
i) g.
ii) g
dx9x
1I 40 2 ++++==== .
iii) J+9I=K, dx9x
xJ 40 2
2
++++
====
dx9xK 402
++++==== .
iv) J+K=20.
v) J, K.
vi) g g-1
.
99. z=x+yi x ,z
1zw 2 ====
f .
) x0
z
1z2 .
)
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115
i) () y=5x-8.
) , 3,
().
2m/sec , .
101. f(x)=x+ex-1.
) f .
) ex=1-x.
) g :
x g(x)+eg(x)=2x+1.
i) g .
ii) g(0)=0.
iii) (gof)(x)>0.iv) f Cf-1
(e,1).
v) Cf-1 .102. dt
1t
)1tln(()x(f 3x2 2
2
++++
====
) f.) f .
) 05x
)x(flim
5x====
.
) Cf xx .
) z=f(3)-2i.
z .
103. f : 0[ , ++++ )
dt)t)t(f1(1)x(f x1 ++++++++==== x>0.
) f(x)=xlnx+x,x>0.
) f(0)=0.
) f .
) Cf
.
) 21e
dx)x(f1
e
1
.
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116
104. :f
dt)e1
t1)x(f xx )t(f
2
++++
++++==== x .
) f
====40160 dx)2008x(fI .
) f .
) 3
x)x(f
3
==== , x .
) 1z
izw
++++
++++==== , Cz , 1z
.
i) z y=-x-1.
ii) M(a,f(a)) Cf -1
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117
) 1x
1)x(f
++++==== .
) f
.
) ()
f, xx
x=, x=+1 >0, )(Elim ++++
.
108. *:f 0)x(f *x
1-1, =(0, ++++ )
:)x(f
1)x(f 1 ==== 0x .
) f(f(x))=x1 1)
x1(f)x(f ==== 0x .
) f(1)=-1 f(-1)=1.
) f-1(x)=x .
) f , :
i) f(x)0 f(x)>0 x
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118
) f(x)=f(1)
(0, 1).
) )1('f
)1(fdx
1)x(f
110 2
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119
12.
. 10 :
1. : z 1=+i, z 2 =+i :
) 21 zz ++++ = 1z + 2z , ) 21 zz = 1z - 2z , ) 21 z.z = 1z . 2z ,
)
2
1
z
z=
2
1
z
z.
:
. : 21 z.z = (((( ))))(((( ))))ii ++++++++ =
= (((( )))) i ++++++++ =()(+)i 1z . 2z =(i)(i)=
=ii=()(+)i.
. .
2. z 2 +z+=0, , ,
, 0 C.
:
z 2 +z+=0 , , 0 : z 2 + z+
=0
z 2 +22
z+
2
2
4
=
2
2
4
-
z 2 +2
2
z+
2
2
4
=
2
2
4
4-
2
2
z
=24
.
1: >0
z 21, = 2
-
. 2 : =0
z=2
-.
3 :
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120
3.
.
: : ) z = z = z , ) z 2 =zz , ) 21 z.z = 1z 2z ,
)2
1
z
z=
2
1
z
z, ) 21 zz 21 zz ++++ 1z + 2z , ) v21 z.....z.z =
= 1z 2z vz vz = z v.
), ) :
) 21 z.z2 = 1z
22z
2 (z 1z 2 ) (((( ))))2.1 zz =z 1 1z z 2 2z z 1z 2 1z 2z =
=z 1z 2 1z 2z .
)
2
2
1
z
z=
2
2
2
1
z
z
2
1
z
z
2
1
z
z=
22
11
zz
zz
2
1
z
z
2
1
z
z=
22
11
zz
zz
.
20 :
1: :
4. .
: f,
[,]. : ) f [,] )
f() f() f() f()
x 0 (, ) : f(x 0 )=.
: f()>====
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121
:
x x 0 : f(x)-f(x 0 )=0
0
xx
)f(x-f(x)
(x-x 0 ).
0xxim [f(x)-f(x 0 )]= 0xx im (((( ))))
0
0
0 xxxx
)x(f)x(f=
0xxim
0
0
xx
)f(x-f(x)
0xxim
(x-x 0 )=0 f x 0
0xx
im
0
0
xx
)f(x-f(x)
=f(x
0) .
0xxim
f(x)=f(x 0 ), f x 0 .
: . f(x)= x =
=
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122
) f(x)=x v, 1,0{ . x 0 x x 0 :
0
0
xx
)f(x-f(x)
=
0
v0
v
xx
xx
=
(((( ))))(( ))
0
1v00
2v1v0
xx
x...x.xxxx
++++++++++++ =
= 1002v1v x...xxx ++++++++++++ . 0xx
im
0
0xx )f(x-f(x) =
=x 01v . (x v)=x 1v .
) f(x)= x , x 0. x 0(0, + ) x x 0 :
0xxim
0
0
xx
)f(x-f(x)
=
0x2
1. ( x )=
x2
1,x>0.
) f(x)=x, x . h 0 :h
)f(x-h)f(x ++++=
= (((( ))))h
xhx ++++ =h
x-hxxh ++++ =x(((( ))))h
1-h +
+xh
h
0him
h
h=1,
0him
h
1-h=0. :
0him
h
)f(x-h)f(x ++++=x. (x)=x.
) f(x)=x, x . h 0 :h
f(x)-h)f(x ++++=
=h
x-h)(x ++++ = hx-x.h-x.h =
=xh
1-h-x
h
h.
0him
h
1-h=0
0him
h
h=1
0h
im
h
f(x)-h)f(x ++++=-x. : (x)=-x, x .
) (x)=
x
x
=(((( )))) (((( ))))
x
xx.-xx2
=
= x xx 2
22 ++++= x12 x -{x=+/2, }.
) (x)=
x
x=
(((( )))) (((( ))))
x
xx.-xx2
=
=x
x-x-2
22
=-x
12
x - {x=, }.
7. f, g x 0
: (f+g)(x 0 )=f(x 0 )+g(x 0 ).
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:
x 0 :(((( )))) (((( ))))
0
0
x-x
)x(gfx)(gf ++++++++=
0
00
x-x
)g(x)f(x-g(x)f(x) ++++=
0
0
x-x
)f(x-f(x)+
0
0
x-x
)g(x-g(x).
f, g x 0
0xxim
(((( )))) (((( ))))
0
0
x-x
x(gfx)(gf ++++++++=
0xxim
0
0
x-x
)f(x-f(x)+
0xxim
0
0
x-x
g(x-g(x)=
=f(x 0 )+g(x 0 ). (f+g)(x 0 )=f(x 0 )+g(x 0 ).
8. : ) (x )=x 1- , -,
)( x )= x n , >0, x , ) ( xn )= x1 , x *.
:
) y=x =e nx u= nx y=e u .
y=(e u )=e u u=e nx x
1=x
x
=x 1- , x ),0( ++++ .
) y= x =e nx u=x n y=e u .
y=(e u )=e u u=e nx n = x n ,>0 x .
) x>0, (((( ))))nx = x1 .
x
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, f()=0. f(x1 )=f(x2 ). ,
f(x 1 )=f(x2 ).
10. f, g
, : ) f, g ) f(x)==g(x) x c
x : f(x)=g(x)+c.
:
fg
x : (fg)(x)=f(x)g(x)=0. fg
. , c,
x : f(x)g(x)=c : f(x)=g(x)+c.
11. f
:
) f(x)>0 x , f
.
) f(x)0 f(x 2 )-f(x1 )>0 f
.
f(x)
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:
f x 0 . x 0
f ,
>0 , : (x 0 -,x 0+) f(x) f(x 0 ), x(x 0 -,x 0+). , , f
x 0 , : f(x 0 )= 0xxim
0
0
x-x
)f(x-f(x)=
++++ 0xxim
0
0
x-x
)f(x-f(x).
x(x0-, x
0), :
0
0
x-x
)f(x-f(x) 0
: f(x0)=
0xxim
0
0
x-x
)f(x-f(x)0 (1). x(x 0 , x 0 +),
:0
0
x-x)f(x-f(x)
0 :
f(x0)=
++++ 0xxim
0
0
x-x
)f(x-f(x) 0 (2). (1), (2)
f(x0)=0.
3: :
13. f F
f G(x)==F(x)+c, c f
G f G(x)=F(x)+c, c .
:
G(x)=F(x)+c, f
, G(x)=(F(x)+c)=F(x)=f(x), x.
: G f .
x : F(x)=f(x) G(x)=f(x), G(x)=F(x), x.
c , G(x)= F(x)+c,
x.
14. f [,]
G f [, ] :
f(t)dt=G()G().
:
F(x)= x
f(t)dt f [, ].
G f [,] c
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G(x)=F(x)+c (1). (1) x=,
G()=F()+c=
f(t)dt+c=c. c=G().
G(x)=F(x)+G(). (1) x=, G()=F()+G()=
=
f(t)dt+G().
f(t)dt=G()-G().
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.
10 :
1. . 87.
.2. C . . 88,
. 89.
3. ( i). . 90.
4. . . 97.
20 :
1: :
5. . . 13