Σημειώσεις Μαθηματικών 2012-13 (Δούδης Δημήτρης).pdf
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Σημειώσεις Μαθηματικών Κατεύθυνσης Γ' Λυκείου 2012-13 (Δούδης Δημήτρης), Έκδοση: 24-04-2013TRANSCRIPT
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-1. z i , Im(z) i .2. z w w z 3i , Re(w) z .3. 1z i 2z i 1 2Im(z z ) 0 ,
1 2Im(z ) Im(z ) 0 .4. .5. z Re(3z 2) 3Re(z) 2 6. z z 0 , Re(z) 0 Im(z) 0 .7. z,w 2 2z w 0 , z w 0 .8. 2 2z (x 1) (x x) i z 0 , x 1 .9. z i .10. z Re(z) Im(z)
y x .11. 1 2z ,z , Re(z Re(z Re(1 2 1 2z ) ) z ) .12. .13. z Re (z) = 2, z
x = 2.14. Im(z i) 8 , z
y 8 .15. y x
z i, .16.
.17. z,z, z, z .18. 1 2 3z 3 4i,z 3 4i,z 3 4i .19. z,z y y .20. z 2 3i : y x 4 .21. z, z
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22. 1 2Im(z .z ) 0 , 1 2Im(z ) Im(z ) 0 , 1 2z ,z .23. : 2 2( i) ( i) .24. : z z 0 z .25. : z z 2 Im(z) .26. 1 2z z , 1 2z z .27. 22 2z z z z 2zz .28. : 40 40(2 3i) (2i 3) .29. vi 1 v 4k 1,k .30. z i w i : z iw .31. z,w z w z w .32. z z z 4 , z x 4 .33. z z z 1 , .34. 1 2z ,z 2z z 0 0 0 ,
1 1 2 2z z ,z z .
35. 2x 2x 10 0 1 3i 3 ii .
1. : 3z 2 4i z i 6i 2. : 3z 4 (2z 5) i 2iz 3. : 2z z 2 0 4. : z z 3iz 3i 5 0 5. : 3z z 10 0 6. z , 2zIm 0 2z
, z 2 , . z .
7. , , z 2i 2w 1 (1 )i .
8. z 1 ( 2) i , [0,2] . M(z)
.9. z : y x - 3
w 2iz (2 i) z 3 .10. 2z 2z 1 0, [0,2) .
) ) .
11. z 5 3i 1z 2z 1 : y x 1 , 2 : y 2x -1 .
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12. z 1 6i 1z 2z 1 : y x 3 , 2 : y x .
13. 1 2 3z ,z z 1 1 2 2 3 3z z z z z z 1 1 2 3z z z 0 . :
) 1 2 2 3 3 1z z z z z z 0 ) 2 2 21 2 3z z z 0 ) 3 3 31 2 3z z z
14. (..) z, 21,iz,1 z .
15. z x yi . 3i zw z 2 , ..
M(x,y) z .16. 3i zw z 2
, z z 2 , M(x,y) z : 3x 2y 6 0 .
(C) z (q), , (q), (C);
. . .
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1. z 02. z w z w , ().3. z 0 z 0 4. z iz iz iz iz 5. 2z , 2z .6. z z , 7. 1 2z z z z , , 1 2z z z z 2 21 2z z z z .8. 2 21 2z z 0 , 1 2z z (;).
() ()1. :
) z z ) z 0 ) 2z z z 2. 1 2z ,z 1 2 1 2z z z z .3. z , z z .4. i 1 .
5. z 1 , 1z z .6. 1 2z ,z 1 2z z , 1 2z z .7. z , i z O 0,0 , .8. z z z z .9. k z k z k z .10. w z z , z , w z z .11. z z 0 z 0 .
1. z 0 , z;2. :
. 2.3:
2
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) z Im(z) ; ) z Re(z) ;3. ;4. 2z zz z 2012 .5. z 3 yi z 5 , y;6. :
) 2 2z z ; ) 2 2z z ; ) 2 2z z ; ) 2 2z z ; ) 2z zz ;7. z ;
) 2 2z z ; ) 2 2z z ; ) 2z zz ; ) z i z ;8. z , 2z z,z 0 , z .
- 1. z :
) Re(z) z ) Im(z) z) z z Re(z) ) z z Im(z)
2. 12z 2z z , *z x yi,x . 2 1Re(z ) 4 .
3. z,w z w z w 2 , 2012 20122012z wu (z w)
.4. * z 14 11z 27z , 25z 5. z, w z w z w , Re(zw) 0 .6. ** z,w z w z w zw
.7. z : 1 - z > z , Re (z) < 12 .
8. z : z-1 z-2 , Re (z) < 32 .
9. z,w 2 2 2z w 1 z 1 w . -
1. z,w 1 1z 2 4 3 2zw 6z 3
16z 3 0 z 2 . 1 3w .2. z , : 2z i z 2i . 2z 2 i z 1 2i ,
z 1 .3. z, w : 2z 3w 2z 3w , : 522z w z3
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4. 1 2z ,z :22 2
1 2 1 23 z 2 z 3z 2z , : 1 2 1 2z z z z .5. z,w z 1 : z 1 1 2z 1w z 1
. w .
6. z z 1 2i 2z 2 i , z 1 .7. z z i 2 z i , 3z 5i .8. z 7 7(1 2i)z (z 2) , :
) 5 z z 2 ) 2z 1 5 9. 1 2z ,z 1 2z z 3 , :
) 4 41 2
41 2
z zw (z z ) 1 2z z , . )
1 2
1 2
z zu (z z ) 1 2z z ,
.10. * 1 2 3z ,z ,z 1 2 3z z z 1 31 2
2 3 1
zz z 3Re z z z 2
.
1 2 3z z z 0 .11. 1 2z ,z 1 2 1 2z z 2 z z , 1 2 1z 2z 19 z .12. 1 2 3z ,z ,z 1 2 3z 1 z 1 z 1 3 1 2 3 3z z z 2 ,
1 2 2 3 1 3 1 2 3z z z z z z 3 z z z 3 .13. z : z z i 1 .14. z : z 1 2i z 2 i z 1 3i i 2 2 .15. * z,w 2 2 22 z w 6 z 3 w .16. z,w z 1 w 1 , z w 1 zw .17. z,w : 3 z 32
5z 2w 2z 3 , 5 2w .
18. z z 7 3 z 1 , z 2 .19. z : z = 1z = z - 1 .20. * z 2z 1 1 z 1 1 , z 1 .21. * z : 8 8 8z (z) 128 , z 2 .
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2: 1
1. 0z - z = , > 0 0k(z ) .
) 0z - z , 0 0k(z ) .
) 0z - z , 0 0k(z ) .
) 1 0 2 z - z 0k(z ) 1 2.
2. 1z - z = 2z - z , z C, 1 2M(z ),M(z ) , z1 z2.) 1 1z - z z - z , 1 2z ,z , M(z) z xx
(;).) 1 1z - z z + z , 1 2z ,z , M(z) z yy
(;).) z - z + , * z , M(z) z -
yy (;).) 1 2z - z z - z , -
. 1 2M(z )M(z ) 1M(z ) .) 1 2z - z z - z , -
. 1 2M(z )M(z ) 2M(z ) .) ,
.3. 1 2z - z z - z 2 , z1, z2 0 1 2z - z 2
z1 z2, 1 2M(z ),M(z ) 2.4. 1 2z - z z - z 2 , z1, z2 0 1 2z - z 2
z1 z2, 1 2M(z ),M(z ) 2 (;).5. 1 2z - z z - z 2 , z1, z2 0 1 2z - z 2
z1 z2, 1 2M(z ),M(z ) 2.
. 2.3: - -
3
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(..) 1 2z z z z
1 1 1z x y i 2 2 2z x y i
M z , 1 z , 2 z z - 1 1 x ,y 2 2 x ,y .
.. z - .
1 2z z z z , 1 1 1z x y i 2 2 2z x y i
[1 2z z z z 1 2z z z z 1 2z z z z ]
M z , 1 z , 2 z z - 1 1 x ,y - 2 2 x ,y . .. z - (, ), .
0z z , >0 0 0 0z x y i .
[ z 0z 0 ]
M z 0 z z , .
.. z . 2 2 20 0x x y y
[ 2 2 2x y ]0z z , >0
0 0 0z x y i .
[0z z 0z z 0z z ]
[ z 0z 0 ]
M z 0 z z , .
.. z - . 2 2 20 0x x y y
[ 2 2 2x y ]
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1 2z z z z 2
1z 0i 2z 0i
, >
2 2 M z , 1E z 2E z z - E(,0) E(, 0) .
.. z - , - 2 .
222 2
yx 1
1 2z z z z 2
1z 0 i 2z 0 i
, >
2 2 M z , 1E z 2E z z - E(0,-) E(0,) .
.. z - , - 2 .
222 2
yx 1
1 2z z z z 2
1z 0i 2z 0i
, - >
[ -
2 1z z z z 2 ]
2 2 M z , 1E z 2E z z (, 0) ( , 0) , . .. z , .
222 2
yx 1, x 0
1 2z z z z 2
1z 0i 2z 0i
, >
2 2 M z , 1E z 2E z - - z (,0) ( ,0) , . .. z - , . 222 2yx 1
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1 2z z z z 2
1z 0 i 2z 0 i
, >
[ -
2 1z z z z 2 ]
2 2 M z , 1E z 2E z z (0, -) (0 , ) -, . .. z , . 2 2
2 2y x 1, y 0
1 2z z z z 2
1z 0 i 2z 0 i
, >
2 2 M z , 1E z 2E z - - z (0,-) (0 ,) , . .. z - , . 2 2
2 2y x 1
1. 1 2 2 1z z z z , 1 2z ,z , .2. 1 2 2 3 1 3z z z z z z , , 1 2 3z ,z ,z 1 2 3 1z z z z -
, (;).3. 1 2 1 2z z z z (0,0) , 1 2M(z ),M(z ) -
, 1 2OM M (;).4. 2 3 3z z z z z z 1 2 3z z ,z ,z , , 1 2 3z ,z ,z , -
z (;).5. 2 2 21 2 2 3 1 3z z z z z z , 1 2 3z ,z ,z 1 2 3 1z z z z -
(;).
() ()1. 1 2z ,z 1 2 1 2z z z z .2. 1 2z ,z 1 2 1 2z z z z .3. 1 2z ,z 1 2 1 2z z z z .4. z , z z 2 z .
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5. z z i z 1 .6. z , z 3 , 2 z i 4 .7. z , z 1 , z 12 5i 13 .8. z z 2 , 2z 4iz 12 .9. 0z z 2 2 2 4 .10. 1 2 3z z z z z z .11. 1 2z z z z .12. 1, 2 z1 z2 xx
12, z1 = 2z .13. 1z - z = 2z - z , z C,
(z1) B (z2).14. z 2 z i (2,0) (0,1) .15. 1z - z = 2z - z z C z1, z2 C .16. 0z - z = , > 0 -
(z0) .17. 2 3i -
z 4 .18.
z-2 1 .
1. z1 z2 -
, ;. z1 = - z2 B. z1 = 2z . z1 = - 2z. m (z1) + Im (z2) = 0 E.
2. z -z - 2 = z - i :. yy B. y = x . xx. (2, 0) (0, 1)E. (0, 2) (1, 0)
3. (2, 1) 3 z . z - (2 - i) = 3 B. z - (1 2i) = 3
. z - (2 i) = 9 . z - (2 i) = 3 E. z (2 i) = 3
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4. z . z 1 < 1 z i < 1
B. z 1 < 1 z i < 1 . z 1 > 1 z i > 1
. z 1 < 1 z i < 1 . z 1 < 1 z i < 1
5. z . z 2 < 2 z 3 < 1 B. z 2 < 2 z 3 > 1
. z 2 < 2 z 3 > 1 . z 2 < 2 z 3 > 1. z 2 > 2 z 3 < 1
6. z 2 = z i y = x, . 1 B. - 1 . 2 . - 2 E. 4
7. z1, z2, z3 , 1z z = 2z z = 3z z z . 2 B. 3 . 1 . 4 . 0
1. :
) z 1 2 ; ) z 1 2 ; ) 1 z 2 ;2. z 1 1 & z i 1 ;3. :
) z 2 , Im(z) 0 & Re(z) 0 . ) z 2 & Re(z) 0 .) z 2 & Im(z) 0 . ) z 2 2 & Re(z) 0.
4. 1z 3 2z 4 3i , 1 2z z ;5. 1z 2 2z 5 , 1 2z z ;6. -
;7. 2 + 3i
3 + 2i ;8. z
2 4 , z
-
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1. (..) z : ) z 1 i 3 ; ) z 1 i 4 ; ) z 1 i 2 ;
2. .. z : z 4 z .3. .. z : z 2 3i z 1 4. ) .. z : z 3 z 3 10 .
) z .., z.) 3 4z ,z .., 3 4z z .
5. .. z :) z 5 z 5 8 ; ) z 5 z 5 8 ; ) z 5 z 5 8
6. z - : 1z 3 2z 3 , 21 .
7. z z 2 2i 2 ,) .. z ) z .
8. .. z, : z 1 z 4i . .., .
9. ) .. z : z 2i 3 z 2i .
) 1 2z ,z 2i 1 21 2
z 2i z 2i 3z 2i z 2i ,
1 2z z .
10. z x yi 0 zw z . .. w.
11. z 1 1 z 2 1 , Re(z) 0 1 z 3 .12. z i z i , Im(z) 0 .13. z : 22 2z z 2 z z z 0 . z , -
.. .14. z (0,0) = 1,
2z iw iz 2 .
15. x , .. z, 1 xiz x i .
16. z,w z 5 w 12 9i . :) 10 z w 20
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) 40 z 3w 50 ) 5 2z w 25 .
17. z z 3 2i 7 , 2 z 2i 12 .18. z z 7 4i 3 , 7 z 1 2i 13 .
19. ** , *z,w z w 1w z , ( ) .
- 1. z (2x 3) (2y 1)i,x,y . ..
M(x,y) 2z 1 3i 3 , .
2. z w z 1w z i . .. z,
.. w :) ) ) w 1 .
3. z w 2w z z
, . M(z) z z , w M(w) -.
4. 1 2 3z ,z ,z 1 2 3z z z 0 1 2 3z z z , - .
5. ) .. z : (1 i)z 2 2 .) .. w : w 2i 1w 2 4i
.
) z w . (: )6. z,w . :
) .. z z 2z 2i .
) .. w : w w 1 i .) z w . (: 2 )
7. z z 3 4i 2 , :) .. z.) 3 z 7 .) z ;) 1 2z ,z .., 1 2z z 4 .
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8. z z 4 z 4 10 , :) .. z.) 3 z 5 .) 1 2z ,z .., (0,0), 1 26 z z 10 .) z 4z 15 .
9. z 3 (5 1)i, w z (2 i) .) .. .) z (0,0).) z, w .
10. z, w w1 w z zi *1 1w , .) : * , 1w w , z .) z.
11. z 2iz 2 6i 2 z 5 3i (1).) .. z (1).) z .
12. 1 2 3z ,z ,z - , , , . , :) 1 2 1 2z z z z , , , .) 1 2 2 3 3 1 1 2 2 3 3 1z z z z z z z z z z z z , , , .
13. *z,w . :) wwz z
wwz I Iz .
) w Iz , z, w .14. *z,w 2 2z w 0 . z,w -
.15. z 3z 2z ,
.16. z 13z 2z ,
.) z 3 , z 4i 1 .) .. z, z 3 .
- 1. z z 3i 2z 3i , :
) .. z.
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) 7 z 6 5i 13 .) z ).) 1 2z ,z .., 1 2z z 6 .) 1 2z ,z .. , 1 2z z 6 ,
1 2z z 6 .2. z w , (z 5) 2(z 5)i 6 5 iw 2 5i 4
) .. z.) .. w.) .. , .) z w 20 .) z w , -, z w 20 .
3. z,w zw 0 2 2z zw w 0 (1). :) z w 3 3z w .) z w z w .) z w 120 .)
2011 2011z w 1w z .
4. . .. z 2 z 5i 2 .. .., :
) xx,) xx,)* xx
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.3. , A {x / f(x) } .4. .5. f(A) {y / x y f(x)} .6. f :
f f .
f 0x 0x f .
7. 0f x , f 0x .8. f : A B .
1 2x x , () 1 2 1 2f(x ) f(x ), x ,x A . ; 1 2f(x ) f(x ) , 1 2 1 2x x , x ,x A ;
9. , x A : x A f( x) f(x) yy (;).
10. , x A : x A f( x) f(x) (;).
11. , x A : x T,x T A f(x ) f(x) f(x T)
1.2) -
4
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12. fC f y f(x) , 0 0 f 0 0M(x ,y ) C y f(x ) .
13. xx fC .14. f,
f x .15. 2f x x x , 0
x 2 K ,2 4
. fC 0 0
16. fC , : fC fC xx. fC fC fC . gC g x f x c, c . fC c
c 0 c c 0 . gC g x f x c , c . fC c
c 0 c c 0 .17. f, g
f x g x , . f x g x , xx Cf
Cg1.
y f x . f R f x , f x / f x ., :i) f(x) , fA ii) g(x)f(x) h(x) , fA {x / h(x) 0} iii) kf(x) g(x) , k , k 2 , fA {x / g(x) 0} iv) f(x) ln g(x) , fA {x / g(x) 0} v) f(x) g(x) , fA vi) f(x) g(x) , fA vii) f(x) g(x) , f A {x / g(x) 0} {x / g(x) k ,k }2 viii) f(x) g(x) , fA {x / g(x) 0} {x / g(x) k,k } ix) h(x)f(x) g(x) , fA {x / g(x) 0 & h(x) } f hA {x D / g(x) 0} x) f , .
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2. f(x) f(x), f x g x 0 f x g x 0 f x g x .
3. f g . : f(x) g(x) , 2 21 2f x x g x x 0 , 1f x x 2g x x .
4. fC xx , f x 0y 0
. f x 0 .
5. f g C , C , y f xy g x
f x g x .6. x , fC ( ) xx ,
f x 0 ( f x 0 ).7. x fC gC ,
f x g x .8. f
: , f y / x A y f x .
i) f.ii) y f(x) (1) x ( ).iii) (1), () y
(1) x ( ).iv) x f,
fx A , () y.v) f(A) y ()
()
, .
. .
f yy f.
, , , 1. : (. 1, 145)
i) 4 3 xf(x) ln x ii) 3 2f(x) x 4x 3 iii) f(x) log x 3 iv) 4 xf(x) log x 2
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v) x xf(x) 2 vi) f(x) 2x 6 vii)
f(x) x 6 viii) x 2f(x) x 1
ix) xf(x) x x x)xf(x) 2 x xi)
xx xef(x) e e xii) 3
1f(x) 1 x
2. 3f(x) x x 2 2g(x) x 5x 6 . (. 2,3 , 145)i) fC gC .ii) fC gC .iii) fC gC .iv) gC xx.
3. f(x) log(5 x) g(x) 1 logx .i) fC ;ii) fC gC .
4. f(x) x 2 g(x) x .
5. f :i) f.ii) -1
.iii) f( 1) .iv) f.v) f(x) 0 .vi) f(x) 0
f(x) 0 .6. f
:i) f.ii) 0
.iii) f(2) .iv) f.v) f(x) 0 .vi) f(x) 0
f(x) 0 .
7. : (. 6, 1, 5, 145-8)i) 2f(x) (x 2) ii) 1f(x) 1x iii)
2f(x) 1x 1 iv) f(x) ln x 1 v) x 1f(x) e 1 vi) f(x) ln x 1 vii) f(x) 1 x viii) 3f(x) x 1
8. 5 -3 f(x) 2 x 1 .
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9. f. f f x 6 .
10. :i) 2f(x) x x ii) xf(x) x 2 iii)
2f(x) 1x 1 iv) f(x) ln x 1 v) f(x) 1- ln(1 x - 4) vi) 3f(x) x 3 vii)
xx
ef(x) 2 e viii) f(x) 2 x 2 ix) 25xf(x) x 3 x) f(x) 3 lnx 1
11. f :i) 2f(2x 1) 4x 2x 5 x .ii) 2f(ln x) x x 2 x 0 .iii) 3 6 2f(x ) x 2x 1 x 0 .
12. f , g 2 2[f(x)] [g(x)] 2 2(f g)(x) x , f , g .
, 13. :
i) 3f(x) x 4x ii)24 xf(x) x
iii) x1 1f(x) 21 2 iv) 2f(x) ln x 1 x 14. :
i) 2f(x) x 1 x ii)22
1 xf(x) ln 1 x
15. f : . :i) f(x) f( x)g(x) 2
.
ii) f(x) f( x)h(x) 2 .
iii) f .
16. f : f(x) 2x . T .
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1: 555 1
1.2) , , 1. (2) :
f gD D A ( ) x A : f(x) g(x) (
)2. : ,
. 2f x x 4g x x x A 1,0,1 , .
3. .. f x x g x x .
4. , : f gf(D ) g(D ) ( ) f gC C ( )
5. f g ( Df Dg x f x g x ).
6. f g , f g D D x : f(x) g(x) . , f g ., f g , ( ) f g .
7. f(x)g(x) 0, x A f(x)=0 g(x) 0 , x A , x f x 0 x g x 0 . f(x) 0 x A g(x) 0 x A( f(x)g(x) 0, x A f(x)=0, x A g(x) 0, x A ).
.. 1 ,x 0f x 0 ,x 0
0 ,x 0g x 1 ,x 0
, f x g x 0 x
1.2) , ,
5
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8. , x A 2f x 1 , f x 1 x A f x 1 x A . x f x 1
x f x 1 .
.. 1 ,x 0f x 1 ,x 0
, 2f x 1 x .
***********************9.
.10. .11. ff g, f g, f g, g , :
k f, k k f fD D (k f)(x) k f(x) , f , * ffD D
(f )(x) f(x) ***********************
12. .13. f: g:
f g , x / g x g(B) A . g f , x A / f x B f(A) B .
14. f g .
15. : gD , fx D : f(x) , g f
f g. fD , gx D : g(x) ,
f g g f.16. .
f g g f f g g f .
, , ! f : g(x) x, x . f g g f
17. , , . f g h f g h , .
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1.
.
2. f g : f,g , . x / g x x / x g(x)
f g . f g f g x f g x .
3. f g : 1 1
2 2
f x ,x Af x f x ,x A
1 1
2 2
g x ,x g x g x ,x
.
f g f g :
1 1 1 1 1 1
1 2 2 2 2 1
2 1 1 1 1 2
2 2 2 2 1 2
f g x ,x x / g x f g x ,x x / g x f g x f g x ,x x / g x f g x ,x x / g x
1 2 1 2 , , , .
17. f = g .
f g , f(x) g(x) .(. 7, 146)
i) 2f(x) x - x - 6 g(x) x 2 x - 3 ii) x xf(x) g(x)x 1 x -1 iii) x 1f(x) x g(x) x x 1x 1
iv) f(x) x 2 x -1 g(x) x 2 x -1
18. h , f , g x :f(x)[f(x) g(x)] g(x)[g(x) h(x)] h(x)[h(x) f(x)] 0
(. 8, 146)19. : ff g,f g, g :
i) 2 2f(x) x x - 2 g(x) 4 x ,ii) xf(x) g(x) 1 2xlnx
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: [3 ] 2012-2013
iii)2x 1 , 2 x 4 x 1 , 1 x 5f(x) g(x)=5 , 4 x 7 2x 3 , 5 x 6
20. f g ,
: (. 10, 11, 12, 146-7)i) f(x) 2 - x g(x) ln x , f g g fii) 2xf(x) g(x)= x 1x 2 , f g , g f
1f f
iii)x
xef(x) g(x)=ln(x-1)e 1 , f g , g f , f f , g g .
iv) 2x+1 , 4
-
: [3 ] 2012-2013
31. f : f(xy) xf(x) yf(y) xy , x , y , .
32. * A f : f(x) x f(x y) f(x) f(y) , x , y , f(x) x .
33. f : , x , y : f(x y) f(x) f(y) : i) f(0) 0 ii) f .
34. f : , :f(x) 0 x f(x y) f(x y) 2f(x) f(y) , x, y . : i) f(0) 1 ii) f .
35. f : , x 2 1f f (x) x 4 .
: i) 2 21 1f x f (x)4 4
x , ii) 1 1f 2 2
36. f : , 2f f f (x) x 3x 4 , x , y , f(2) 2 .
(. 4, 45, 2, 3, 4, 9, 145-8)37. x 2K(x) 4 2x . x
(x) x 5 , :i) x.ii) .
38. ( A 90 ). (B) 4 (AB) x , x.
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: 3 2012-2013-1-
& ()
1: 555 1
1. 1 2A ,A
( ) , 1 2A A ... ) 1f(x) x ) 2
x, x 0f(x) x 3, x 0
2. 1 0A (,x ] 2 0A [x ,) , 1 2A A (,) .
3. - .
4. f (..) , ! 1 2x ,x 1 2x x
1 2f(x ) f(x ) .5. f , :
fC f xx (1) f(x) 0 .
6. f f(x) 0 , .
7. f g , f(x) g(x) .
8. f fC - y k, k (1)
9. f : 1 2x ,x 1 2x x . :) 1 2
1 2
f(x ) f(x ) 0x x 1 2x x 1 2f(x ) f(x ) f
) 1 21 2
f(x ) f(x ) 0x x 1 2x x 1 2f(x ) f(x ) f
*****************10. f 0(,x ]
0[x ,) , f (,) 0x x 0f(x ) .11. f 0(,x ]
0[x ,) , f (,) 0x x 0f(x ) .
1.3)
6
-
: 3 2012-2013-2-
12. f (,) , f (,) .
13. f [,] , f - .
14. f [,] , f - .: , (, ), .
15. f : A f(A) . :) [,] , minf maxf ,) [,) , minf maxf ,) (,] , minf maxf ,) (,) , minf maxf .
16. ) maxf 0 , ff(x) 0, x A .) minf 0 , ff(x) 0, x A .
17. ! f : ff(x) , x A maxf . maxf , f(x) fA !
1. [ : ]
f : A , : 1 2f(x ) f(x ) 1 2x x .[, f A : 1 2 1 2x x f(x ) f(x ) ].
2. [ : ] f : A , : 1 2f(x ) f(x ) 1 2x x .[, f A : 1 2 1 2x x f(x ) f(x ) ].
3. [ : ] f : A ( ) , : 1 2f(x ) f(x ) 1 2x x [, : 1 2 1 2x x f(x ) f(x ) ].
4. , - .
5. , - .
*****************6. f ,
( ) f .(.. (,) ( , ) ): .
-
: 3 2012-2013-3-
7. f , ( ) f .(.. [,] [ , ] )
*****************8. f 0x , 0x
, 0f x . ( ).9. f 0x , 0x -
, 0f x . ( ).: , .
1.
: -
. : 0 ,
0 , 2 1 2 1 , 2 2 *0 , 2 2 * 0 , 0 , 1 1
0 , 1 1
0 ,
2 12 1 2 1
2 1
f x f x , x ,x x xx x .
f -.
[ , ]. .
. - .
2. - :
-
: 3 2012-2013-4-
- . 2 0, , 0 , 0, , 0 , 0, , 0 , 1 2, 0 , 1 ,
1 2, 0 , 1 .
. ( ) ( ).
-
: 3 2012-2013-5-
39. : (. 1, 4 156-7)
[) ) ]i) 2 xf(x) ln(x 1) e ii) f(x)=-1+2 3-x iii) f(x) lnx x 3 iv) 2f(x) x 3
v)23x 2 , x 0f(x) x 2 , x 0
vi) 3x 1 , x 0f(x) x 2 , x 0
vii) 2
5f(x)= 9-x viii)1- xf(x) 1 x
ix) xf(x)= 2 x40. ) f .
f .) f,g .
f+g .) N xf(x) e x,x 0,
41. f : A (0, ) g : A (0, ) , fg .
42. ) f , , [0,], 0 , f(0) ,f() 0 .i) f [ ,0] [0,] .ii) f f [ ,0] [0,] .
) 22h(x) 1 1 x [ 1,1] .43. )
) : i) 7 3x x 2 0 ii) x ln x 1 .44. )
x2f(x) 2x3 .
) x x4 2 2x9 3
45. ) f(x) ln x x (0, ) .
) 2 2ln(x x 1) x ln(x 2) 1 46. ) f(x) x x2
[0,] .
) e2e e 1 .47. 2 2f(2x x 3) f(3x x ) , xf(x) e x .48. 2 2x ln(x e ) 4e .49. ) g x
f(f(x)) g(x) 0 (1) , f .
-
: 3 2012-2013-6-
) f(f(x)) x 0 (1) x , f .50. ) f ,
3 3g(x) f (x) 3f(x ) 2 .) 33x xh(x) e 3e 2 .
51. ) f g - f() , f g .
) 3g(x) x x 1 [ 1,1] .) 3g(x) x x 1 [0,] .
52. f : 5 x 3f (x) e 2 6f (x) x . f .
53. f: . fC xx yy 2 1
) f .) g , g g f g .
54. 7f(x) 2x 3x 5 fD [0, ) .) f .) fC xx.) 2 3f(x ) 04 [0, ) .
55. :
i) 2f(x) 3x 2x 1 ii) f2f(x) 1,A [2,6]x iii) (x) 7 x 10 56. f A(0,2),B(1,3)
2f(x) 5 1 , f .57. 3f(x) 4 16 2x ,
) f ) .
58. ) 2f(x) 25 x .) 41821f(x) x 1821 .
59. f,g : 2f(x) g(x) x 3 x . fC gC .
60. f,g : 2f(x) 3 (g(x) 2x) x . g y 2x , f .
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: 3 2012-2013-1-
& ()
1: 555 1
1. x A ,
x () .2. [ ] 1-1 :
y f x y x A , y f(A) f(x) y x. (y k ) . fC f .
3. , 1-1 .
4. ! . 1-1 , - . ( ), :
5. f 1-1, ( 3).6. 1-1 fD , fD .
( )7. 1-1 (. f(x) 0 ).
*****************
8. [ ] 1f f : , f(A) f, , f fA , 1f x y f y x
, f x y, 1f y x . , 1f f ,
1f f x x fx A 1f f y y y f .
1.3 1-1
7
-
: 3 2012-2013-2-
9. ! 1 1f f
11 1f x f xf x .
10. f , 1f 11f f .11. f , f 1-1 ( ).12. f , f ( -
).13. f , f
( ).14. -
. x y f x . .
[]1. f 1-1.2. f , 1f .3. f fA , 1f
f(A) .
4. fC 1fC 1y f(x)y f (x)
1f(x) f (x) .5. f: :
i) 1f .ii) 1f x f x f x x x f , . fC 1fC , y x .( , , 1f(x) f (x) f(x) x []).
6. f: fC 1fC y x .( )
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: 3 2012-2013-3-
1. f 1-1 :
1 2 fx , x A 1 2f(x ) f(x ) 1 2x x( )
1 2 fx , x A 1 2x x 1 2f(x ) f(x )( ).
f . y y f x .
2. f 1-1 1-1 ( ) : , , , 1-1 , , , -
1 2x , x , , .
3. 1-1 : 2 1 2 fx ,x A 1 2x x 1 2f(x ) f(x ) xx fC .
4. 1-1 .
5. f, 1-1 f(A) .
6. 1f y f x x. - y x.
7. f 1-1 (.x. 1f x x (.x. f x x ).
-
: 3 2012-2013-4-
1-1 (. 2 156)1. 1-1 :
i) 2x 1f(x) x 3 ii) f(x) 3 2 x iii)
xxe 1f(x) e 1
iv) f(x) x 3 2
v) xf(x) 1 2x 3e vi)x
32f(x) e x vii) 23x 2 , x 0f(x) -x 1 , x 0
viii) 2x 1, x
-
: 3 2012-2013-5-
(. 2 156)13. ( ) ()
i)2x 1f(x) 2x ii) 1 xf(x) ln 1 x
iii) f(x) 2 1 x iv)2x - 4x , x 2f(x)= -x-2 , x 2
()
v) 3f(x) x 1 vi) f(x) 1 2 ln(x 2) vii)x xx xe ef(x) e e
14. :
i) 4 2f(x) x 5x 3 ii) 2f(x) x 2x 3 iii) xf(x) e15. f(x) (2 1)x 3 , , 1f f .16. f : 3(f f)(x) x , f -
.17. * f : ( -
f(x) ), (f f)(x) x f(x) , x .18. A 3f(x) x x : ) 1f 2 ) x 1f (x) 3 .19. f : A B g : B , g f -
.20. f f f , f -
.21. f(x) 0 f(f(x)) xf(x) x 0 , : ) 1f ) f(1) 1 ()22. -
:i) f(x) x 2 ii) 2f(x) 2x x ,x 1
23. f : (0, ) 2
21 23f(x) 2f 3x 5x x
x 0 .24. f : (0 , + ) f( ) f f , , (0,+ ). :
i) f(1) 0ii) 1f f(x)x
iii) f x=1 , f .
25. f : [2, ) 2f(x) x 4x 1 1-1 1f .26. f 1-1,
1f .27.
22xf(x) x 1 A [0, ) .
) .) .) .)
28. f : , (f f)(x) xf(x) (1) x .) f(0) 0 .) f(x) 0 x 0 f . ()
29. 3 2f(x) x 6x 12x 10 .
-
: 3 2012-2013-6-
) .) (f f)(x) 3 . ()
30. x 2 212f(x ) 4f (x) 9 , f .31. f A(2,5)
B(3,2) .) .) 1f (5) 1f (2) .) 1 2f 3 f(x 2x) 2 .
32. 2x xf(x) e e 1,x .33. f : f(x y) f(x) f(y) x,y , :
i) f(0) 0ii) f iii) f(x) f(x), iv) f(x) f(x), v) f(x) f(x),
34. f : f(x y) f(x) f(y) , (1).) 1f(0) 1, f( x) f(x) x .) f(x) 0 , x .) f 1-1, 1 1 1f x y f x f y ,x,y 0 . ()
35. f : f( ) . :) f .) f -1 .) 1f(x) f (x) f(x) x ( -
) ()36. 3f(x) x 2x 2
. ()37. x 3 2f(x) e x e .
) .) 1f(x) f (x) .) 1 1 2f (2 ln x 2) f (ln x 1) .
38. ) f(x) 3 x 3 .) 1f,f .) 1f(x) f (x) .
39. f 3f (x) f(x) x, x f( ) .) f 1-1.) 1f .) f(0,0) C) f(2,1) C) f .) 1f (x) f(x)
-
: 3 2012-2013-1-
& ()
1: 555 1
1.4 0x (, , ) 1.5 (, )
1. f 0x . -
x 0x .2. 0x x : x 0x , -
, . x 0x .
3. 0x x : x 0x , - , . x 0x .: 0x x
4. 0x x 0x . :
0x f , .
00x xlim f(x) f(x ) .
5. 0x x
lim f(x) , 0x ., 0x - 0x ( ) x 0x ( , . 158-160).
6. f(x) 0x , .7. :
0 0 0x x x x x xlim f(x) lim f(x) lim f(x) . ,
, ( ) .
, ;8. [ ]
0 0x x x x
lim f(x) lim f(x) 0
00x x h 0lim f(x) lim f(x h) ( 0 0x x h x=x h ),
0
0x x h 1lim f(x) lim f(x h) ( 00x h x=x hx 0x 0 )
8
-
: 3 2012-2013-2-
9. .10.
.. 1f x 2x x 2 2 1g x x x 2 0x 2 . : 2x 2 x 2lim f(x) g(x) lim(2x x ) 8 .
11.
0x xlim f(x) 0 , f(x) 0 0x . ( 1, . 165, )
, f(x) 0 0x , 0x x
lim f(x).....0 ! [ < 0]( )
12. 0x x
lim f(x) 0 , f(x) 0 0x .13.
0 0x x x xlim f(x) limg(x) , f(x) g(x) 0x . (;)
, f(x) g(x) 0x , 0 0x x x x
lim f(x)..... limg(x) ! [ < 0]14. f, g 0x f(x) g(x) 0x ,
0 0x x x xlim f(x) limg(x) .
( 2, . 166, )15. f 0x f(x) 0 0x ,
0x xlim f(x) 0 . (;)
16. 0x x
lim f(x) 0 0x x x xlim f(x) lim f(x) . ( 2, . 166, )! . ,
0x xlim f(x) -
0x x
lim f(x) ( )
17. [ 0 ] 0 0x x x x
lim f(x) 0 lim f(x) 0 . ( )
18. 0
2x xlim f (x) 0 0x xlim f(x) 0 . ( )
1.
0x xlim f(x) :
x 0x . x 0x -
00 , :
) f ,)
, .( g f 0x 0x ).
-
: 3 2012-2013-3-
, . - , .
, - , 0x -.
, , , - .
. 1. f
. :)
x 2lim f(x)
)
x 1lim f(x)
)
x 1lim f(x)
)x 1lim f(x)
)
x 1lim f(x)
) x 2limf(x)
)x 3lim f(x)
y
23
4
1
-2
-2
1-1 2 3 x
. 0/0 /02. , , ( ):
i) 2x 1x 1lim x 1
ii)
mnx 1
x 1lim ,m,nx 1
* iii)1x
lim
3x-1
3- x-11 iv) 2x 1 1 5lim x 3 3x 13x 12
v) 3 2x 1x 3x x 3lim 1 x
vi)1x
lim 1-x
1-x1-x2
2 vii) 3x 01 x 1lim x viii) 3 3x 2
x 2lim 2 x
ix) 3 2x 1x xlim x 1
x)
m
nx 1x 1lim ,m,nx 1
* xi) 2x 15x 4 3x 1 3lim x 1
()
xii) 3 2x 13x 2 2x 1lim x x
() xiii)
33x 1
10 x 3x 5 1lim x 1
() xiv)3
2x 1x 3x 1 2x 1 2lim x x
3. , , ():
i) x 4lim f(x) x 5lim f(x) , x 2x 2012f(x) 2e x 5 27 x 2 , ii) 2x 3
x 3 5xlim x x 9
() iii) 2x 2x 2lim x 4
iv)2
3x 29x 6x 1 x 5lim 1 x 7
v)
2 2
x 3x x 5 x 2x lim x
vi)
2
2x 1x 1 2x 3x 5lim x 1
vii)2
x 3
x 1 7x 11lim x 3
, viii) x 2
x 3 4 x 1 3lim x 2
(), ix)2 2
x 42x 9x 4 x 4xlim x 4
()
-
: 3 2012-2013-4-
. - 4. 2 21x 2
2x (5 2 )x 1lim 151x 2
, .
5. 4 3
2x 1x x 2lim 1x 1
, .
6. * 2x 2
x x 3 2lim ex 2 .
7. 22x x 2, x 1f(x) 3x 2 6, x 1
. , , x 1lim f(x) 3 . ()8. ) f ,g : . x 2lim 2f(x) 3g(x) 2 x 2lim 5f(x) 7g(x) 4 , x 2limf(x)
x 2limg(x) .) f ,g : . x 2
f(x)lim 3x 2 3x 2lim g(x) x 8 4 , x 2lim f(x)g(x) .) f ,g : x 0
f(x)lim x 3 2 2 34f (x) xf (x) 2x f(x) 3x . x 0limf(x) .
9. , x 1limf(x) , :2 2 2
55x x , x 1f(x) 2x 4x 2011 lnx, x 1
()
10. 2
2x 2x 2f(x) x 1
. x 1limf(x) 2 , , . ()11. 2g(x) 4x (2 )x . , x 2lim g(x) 18 ,
22x 2
x 11x g(x)lim 1x 5x 14 . ()
12. 2 x 3 xf(x) x 2 22x 2x x 2xg(x) x
. x 0 x 0limf(x) limg(x) . ()
13. z, w z 3w , z 2iz 2i z i, , :
10 53
2 Im(w)x Re(w)x 2012 ln( x), x 1f(x) 2 Im(w)x Re(w)x, x 1
. .. z x 1lim f(x) . ()
14. 32x 1, x 1
f(x) x 3 , x 14
, ) ... f 1f .
) ( ) 1 1
x 1f (x) f (1)lim x 1
. ()
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: 3 2012-2013-1-
& ()
1: 555 1
1.5 ( , , )
1.
.
2. x , .
3. x, x
x 180 ,
180 x .
: x 180 x . 0 0 x 0
x 0 0 x lim lim 0,01745.. 180 x 180 .
4. : x x x ( x 0 ) x x x * .
[]1. x 0
xlim 1x .
2. x 0xlim 1x .
3. x 0xlim 1x .
4. f x g x 0x 0x x
limg x 0
, 0x x
lim f x 0
.5. () x () = ()
0x x
lim f(x) 0
g 1 0x , 0x x
lim f(x) g(x) 0
.( )
1 f > 0, x A f(x) , .. f(x) h(x) f(x) h(x) .
9
-
: 3 2012-2013-2-
1. (3) 00
, -
0x xlim f(x)
, -
, (x)x 0 x 1
x
.
. 1. 2f(x) 2x (x 5) x , x 5limf(x) f(5) .2. f : , : 5 6 4x f(x) x 3x x (1). :
i) x 0limf(x) ii) 2x 0f(x)lim x iii) x 0
f(x)lim x3. A 2 2f (x) 2f(x) x 0 , xR, x 0limf(x) 1 .4. f : f(A) ( 5,2) . ..: x lim x f(x) 0 .5. f , g , f x 0 g x 0 0x .
0x x
lim f x g x 0
, 0x xlim f x 0 0x xlimg x 0 .6. 2 x 2 f(x) x 3 x 2 , :
i) x 1lim f(x) ii) x 1f(x) 2lim x 1
iii)
22x 12f (x) 8lim x 3x 2
()
7. f : , 1 xx f(x) x x 0, 2 (1).
i) x 0
f(x) 1lim x
ii) x 0limf(x) f(0) , x 0lim f(x) ()8. f : , x 0
f(x)lim 2x . ()i) x 0limf(x)ii) x f(x) 0 , 0.iii) f, g 2 2 2x f(x) g(x) x x f (x) , 0, x 0limg(x) .
9. f : , : 2f(x) 1 x 1 , x x 0f(x)lim ,x .
i) .ii) 22 xf(x) f (x) 2 , x , 2x 0
f(x) 1lim 2x . ()10. f f(x) lnx 1 x .
i) f.ii) f .iii) f ( ,0] , :
10 f (x) 1 x 0 .
-
: 3 2012-2013-3-
iv) 2 1x 0lim x f (x) . ()11. f : , : x 1 f(x) x 1 , x (1).
x 1f(x)lim ,x 1 , :
i) x 1limf(x) .ii) . ()
12. ) g 5x 7 7g(x) x A .
x 0limg(x) 0 .) f : . f , -
5
x 0x 7 7lim f(x) 0x
()
13. f : , : 2 2f (x) x , x . :i) 2x 0lim f (x) 0 . ii) x 0lim f(x) 0 iii) x 0limf(x) 0 ii) 22 xf(x) f (x) 2 , x , 2x 0
f(x) 1lim 2x . ()14. f : , : 1f (x) x , x
2x 1lim f (x) 1 .i) : f(x) x , x .ii) : 2y 1y 2
, y .iii) x 1limf(x) . ()
15. f : , 3f x f x x 0x 0 . - f 0x 0 .
. 16. , , :
i) 3x 0xlim x x ii) x 0
1 1lim x xx iii) 2x 0
2 1lim 1 x x
iv) 2x 0 2 xlim x 4 2 v) x 0
xlim 3x vi) x 3(2x 6)lim 3x 9
vii) x 0limf(x) , 1 xf(x) x fA {x |x , }
viii) x 0xlim x ix) x 0limf(x) ,
2x , x 0f(x) 1 x , x 0x
x) x 0limf(x) ,
x x xf(x) x 7x , f
A {x |x , }7 ()
xi) 2x 0x xlim x
() xii) 2x 1(x 1)lim x 4x 5
() xiii) x 1
(x 1) 1lim x 1 ()
-
: 3 2012-2013-4-
xiv) x 2lim f(x) , 1f(x) (x 2) x 2 , fA ( 2, ) ()
xv) 2011x 0x 1lim x x () xvi) x 0limf(x) x 0
1limf x , 2 2
1 xf(x) x 2x fA
xvi) x 0limf(x) x 01limf x , 2 2
1 xf(x) x 2x fA
. ()17. f : f(A) ( 5,2) . : x lim x f(x) 0 .18. f(x) x 1 2x , x 0
f(x)lim x ()19. : x 0
x 2x ... xlim 28x . ()
20. : x 0x 2x ... xlim 120x
. ()
. 21. f :
0x xlim f x
. 0x x
lim f x
.22. f :
0x xlim f x
. 0x x
lim f x
.
23. f : * x 2
f 4 xlim 0f x
. .24. f f x y f x f y x,y f 0 0
x 0lim f(x) 1 : ) f(0) 1 ) 0 0x xlim f(x) f(x ) 25. f f x y f x f y *x,y
x 1limf(x) 0 , : ) f(1) 0 ) 0 0x xlim f(x) f(x )
26. : ) x 2xlim 2x )
2x 0
xlim 2x x .
27. f : x 0 f xlim 1x . 22x 0 f x xlim x x , .. 28. f f(x) ln x 1 x .
) f.) f .) f ( ,0] , : 10 f (x) 1
x 0 .) 2 1x 0lim x f (x) . ()
-
: 3 2012-2013-1-
& ()
1: 555 1
1.6 0x
1. . , :
.
, , , , 0 , , .
1. :
) g(x) f(x) 0x 0x x
limg(x) 0x xlim f(x) .) g(x) f(x) 0x
0x xlim f(x) 0x xlimg(x) .
. 1. ( ) ,
x0= - 4 , -2 , 0 , 1 , 2 , 4 , 6 . [- 6 ,10] , -;
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
-d0-g0
-h0-u0-v0
0
10
-
: 3 2012-2013-2-
. - 00 , (: )
2. :)
23 2x 3
9 xlim x 9x 27x 27
) x 2 3xlim x x2
) x 0 5 3xlim x 4 2 x 1
3. 3 2x 5f(x) x 3x 4 .
) f.) x 2limf(x)) x 1lim f(x)
4. :) 2x 1
1 3lim x 1 x 1 ) x 0
2 2lim x x ) x 0
1 1lim x x
5. : ) xx 0lim e lnx , ) x 0 ln xlim x6. ) 3x 0lim log x ) 1x 0 2
lim log x7. ) x 0
x lim x ) x 2
x lim x
) x 2limx
) x 0limx
. - 8.
22x 5x 6f(x) ,,x 3 , x 3limf(x) 14 .
) x 3 18 15 6 .) , x 3 .
9. , , 2
2x 1x ( 1)x lim x 2x 1
. ()
10. , R : 3 2 3x x 2 1 x 3 f x x 3x 2
, 0x 1 .11.
2 22x +f(x)= x -2x+1 , , x 1limf(x) ;
12. R 2x 9 2x 5limx
.
-
: 3 2012-2013-1-
& ()
1: 555 1
1.7
1. .2. x , x , x , x .3. 0x
: , .
4. .
5. x , x (, ), 0 , -: 2x x x .
6. , x , x ( ,),
-
: 3 2012-2013-2-
. [3 (i,iii,v), B4(ii), 186]
4. 2x 3f(x) 16x 4x
) f .) xlim f(x) xlim f(x) .) x 0limf(x) 1x 4
lim f(x)
. ()
. [2 (v), 3 (ii,iv,vi), 186]5. :
) 2 2xlim x 2x x x 1) 2xlim x 3x x 2) 2xlim 4x x 2x 3) 2 2xlim x 3x 4x x x () ) 2 2 2xlim( x 2x 1 4x 1 9x 3x) ()) xlim( 9x 1 4x 1 25x) ()
. - / [1,2,3, 187]6. 2 3 2xlim ( 2)x x ( 2)x 3 .
7. P(x)f(x) Q(x) , 2P(x) ( 1)x x 5 3 2Q(x) ( 2)x x 2 .
xlim f(x) . ()
8. , :4 2 3
3x( 1)x ( 1)x x 5lim (1 )x x 1
. ()
9. 2xlim x x 1 x .10. , 2xlim ( x -4x+2-x-)=3 .
11. f(x)=2x +1 -x+x+1 , xlim f(x) =1.
12. xf(x)lim 1x x
3f(x) - 8xlim 2x 3f(x) .
13. xxf(x)lim x-1 xlim f(x) .
14. f (0, ) . 2xlim 2x 4x 1 3f(x) 6 , xlim f(x) .
-
: 3 2012-2013-3-
15. f : (0, ) 2 2
2xx f(x) x 1lim 2x 3
:
i) xlim f(x) ii) , xxf(x)+x+1lim =1x+2
16. P(x) : 2xP(x)lim 3x 2x 1 0 2x x
P(x)lim 3x 2x 1 0x .()
17. f xf 2xlim 5f x . , -
, : (i) xf 8xlim f x (ii)
xf 8x f xlim f 2x f x
. 18. ) x : 2 2x 2x f(x) 3x 2x 2 , xlim f(x) .
) x 2xf(x) x 1 1 , xlim f(x) .
19. f 2
23x 5 3x 10f(x)x x
x 0 . xlim f(x) .
20. 22x
5x 2xlim x 100 . ()
21. 2xxf x , x Rx 1 ) : 2 2x xf xx 1 x 1 x R .) xlim f x xlim f x .) x f xlim x
22. f : R R 2 2f x x 1 x x ) 2 1f x x 1 x ) xlim f x .
. 23. : ) xxlim 3 ) xxlim 5 ) xxlim 0,5 ) xxlim 0,524. : ) 5xlim log x ) 0,3xlim log x25. : ) 2x
2-xlim ln 2+x )x
xlim ln(e x 1)
-
: 3 2012-2013-4-
26. 22x f(x) ln 2x
0 .
) f.) x 0limf(x) .) xlim f(x) .) xlim f(x) ln x . ()
27. ) x
x xx5lim 3 2
.
) x xx xx
5lim 2 5
. ()
28. : )x x
x 1 x 1x2 lim , 0 2
)
x 1 xx 2 xx 1lim , 0, 0 2
()
29. x 2 xx x 1e 2f x ln e 2
x .
30. x x 1 xxlim 9 3 2 3 .31. x 0
3 2 log xlim 1 2log x .
. 32. :
) x1lim x ) x
1lim x ) xxlim x ) x
xlim x ) x1lim x x
) 3 3x
1lim x x )4
3x1lim x x )
23x
4x xlim x 8 ) x4x xlim x x
) x 2
2xlim x 1+x
33. x x
Ox 1.) x.) ONOM , .) (ON) (OM) , .()
34. :
33 2
25 4 2 1 5 2 1 3 1
.
-
: 3 2012-2013-1-
& ()
1: 555 1
1.8) [ ]
. 1. [ ]
f 0x A - (3) ():) f 0x A (
0 0x x x xlim f(x) lim f(x) )
) 0f(x ) ( f 0x 0x )) :
0 00x x x x
lim f(x) lim f(x) f(x ) .2. , f 0x A :
0
0x xlim f(x) f(x )
00x xlim f(x) f(x ) 0
0 0h 0limf(x h) f(x ) 0 0 0h 1lim f(x h) f(x ), x 0
3. 0x .
4. 0f : [x ,) 0x ()0
0x xlim f(x) f(x ) ,
5. 0f : (,x ] 0x ()0
0x xlim f(x) f(x ) .
6. f ( ) 0x A :)
00x xlim f(x) f(x ) ( )
) 0x A .
7. ! 0x A , f 0x , 0x .
8. f 0x , - 0x .
12
-
: 3 2012-2013-2-
9. f 0x , : 0x A
0x xlim f(x)
0x x
lim f(x)
00x xlim f(x) f(x )
. 10. [ ]
f : A , , (,
00x xlim f(x) f(x ) 0x A )
11. , - .
12. - (/ ).
13. , .
f A (,)
f
0 0( ) ( ,x x ),
f
0 0( ) ( ,x x ),
f - .
1 1 0 0( ) ( , ) ,x x ,( )x x
1.
.2. ,
( ).
3. f 0x 0f(x )
0x xlim f(x) , 0 0x xlim f(x)=f(x ) .
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x1 x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x0
f(x0 )
-z
limf(x)x x0
-o0
x1
-
: 3 2012-2013-3-
. . [1,2, 3 .197-198]1. N ln(x+2) , x 2f(x) 2(x+2) , x 2
2 .
2. N x-2e , x [0,2)
f(x) 0, x 212x - 3 - , x (2, )ln(x - 2)
0x 2 .
3. A f 0x 1 x 1(x 1)f(x) (x-1)lim 4x 1
, f(1) .
4. A 2x 0xf(x)-xlim =x -x , f 0x 0 .
5. A f 0x 2 , f(2) , 3 3 22x 8x x 2 f x 3x 6x 4x 8( ) ( ) , x .
6. A x f x 3) 5( )g(x g 2 g(2) 5 , f 2.
7. f : R R : 2f x x x R :) f 0x 0 .
) f x x 0g x x0 x 0
0x 0 .
8. f,g,h : A . :) f g h f, g 0x A , h
0x A .) xh(x) x f(x) e g(x) , h 0x A g -
0x A , f 0x A .9. f : : 3 2f x f x f x x ,
x R . f 0x 0 .10. f : R R 0x 0 , x R
2 2 2 x 2xf x f x x x x 2f x .11. f,g : : 2 2 2f x g x 2f x 5 4g x x ,
x . f, g 0 x 2 .
. [3 .198]12. x 2 f x 2xlim 1x 2
f , 2
x 2xf x 2x 3f 2 6xlim x 2
.
-
: 3 2012-2013-4-
13. 2 2x, x 2f(x) f (6 x ), x 2
.
z. (). [1,2 .198]14. 2 3 2 x x 2 ,x 1f x ln x x ,x 1
, , f
.
15. 2x +x-5 , x 1f(x) x-1
7 , x 1
, , f
.. 16. f : f(x y) f(x) f(y) x,y , -
:) f(0) 0 .) f 0x 0 , f R.) f 0x , f R.
17. * *f : f(x y) f(x) f(y) *x,y , - :) f(1) 1 .) f 0x 1 , f * .) f *0x R , f *R .
18. *f : 0x 0 :f(x 3y) f(x) f(3y) (*) x,y .) f(0) .) f . ()
19. f 24f x f x( ) ( )2 3x 6x , x , x 1limf(x) 5 . f 1, f 1 .
. 20. f : 2,[ ) , x 2
x 2f(x) x 2 3x 2 .21. f : , x
x x f x x x .22. f : 2x 1 f x x x 2 , x
f 1,3 C .) , R .) f x x 2 , x .) 2x
1lim f x f x
-
: 3 2012-2013-1-
& ()
1: 555 1
1.8 &
1. [,] , .y
( )O()
a x
y
[ ]O a x
63
() () , - [, ] !
2. Bolzano .
3. Bolzano , x (,) , 0f(x ) 0 .
4. Bolzano .5. Bolzano 0f(x ) 0
( ). .
6. . Bolzano: f x x xx .
7. : 0x f(x) 0 0x f 0x , f xx
13
Bolzano
x0
-qf()
f()
-w
x1 x2
-
: 3 2012-2013-2-
8. Bolzano, - f [,] f() f() -.
9. f , x x , - . [ ]
y
f (x)>0
O a x
()
y
f (x)
-
: 3 2012-2013-3-
15. Bolzano.
16. . , f() f() !
17. :
y [f(),f()] - fC .
18. : f() f .
19. f() f (-).
20. - :
. , f [,] f [,] [m,M] , minm f maxM f .
21. ! .
=(, ) f()=[m, M]
=(, ) f()=[m, M)
=(, ) f()=(m, M)
=[, ] f()=[m, M]
-g
f()
f()
f()
=
=
f()f()
f()=
f()
f()
f()=
-l0
y=
-g
f()
f()
f()
=
=
f()
f()
f()=
f()
f()
f()=
-l0
y=
-j1
((
( ( (
) ) ) )
)
m
M
-e2
x1 x2
f(x1
f(x2
)
)=
=
x
f(x)
-q3
f()>0 f()>0,
-b4-q4
m
M
-g
f()
f()
f()
=
=
f()
f()
f()=
f()
f()
f()=
-l0
y=
-j1
((
( ( (
) ) ) )
)
m
M
-e2
-g
f()
f()
f()
=
=
f()
f()
f()=
f()
f()
f()=
-l0
y=
-j1
((
( ( (
) ) ) )
)
m
M
-e2
-g
f()
f()
f()
=
=
f()
f()
f()=
f()
f()
f()=
-l0
y=
-j1
((
( ( (
) ) ) )
)
m
M
-e2
-
: 3 2012-2013-4-
1. . Bolzano f
x. , :) f.) ,
f . f .
2. ( ) Bolzano ( 0 ) . .
3. , , ( ) 1-1 .
4. [ . Bolzano]. f(x) 0 (, ), . Bolzano
[,] . (, ), . Bolzano
[,] [,] . (,) , f() f() 0 f f 0 . [,] , 2
.
[, ] [, ) (, ], f() f() 0 . f() f() 0 f() 0 f() 0 f() f() 0 . Bolzano.
5. 0x (,) 0 0f(x ) g(x ) , - Bolzano [, ] h(x) f(x) g(x) .
6. f gC , C 0x (,) 0x [,] , -
f g - f(x) g(x) f(x) g(x) 0 . , f gh(x)=f(x) g(x), x D D , Bolzano .
-
: 3 2012-2013-5-
7. f , x [,] , : f [,] f()
f() , f [,] -
f.
8. . Bolzano ( - ), - . , .. - .
-
: 3 2012-2013-6-
: , - . - , , - , .
. Bolzano [ 6,7,8,9, 4,5,7*,8, .198-200][, , , - ]1. Bolzano
1 x 3x 1 , 3 x 1f(x) 2 x 1 x 1 , 1 x 5
[ 3,5]
2. , Bolzano -
2
2x-, 1 x 1f(x) 5x-3, 1 x 2
x -+1, 2 x 3
[ 1,3] .
3. 1x x xx .4. , 0 , x x ( )
.5. xe 1 lnx .6. N
x 2 x 3 4 x 1 x 3 7 x 1 x 2( )( ) ( )( ) ( )( ) 0 (1,3) .
. & [ 10, 6, .199-200]7. 2f x 4 x , x 2 ,1 .8. f x 4 x 2 x -
f x 0 .9. f : [ 1,2] R 3f x x x 2 .
) f.) f x 0 [ 1,2] .
. [ 9, . 200]10. f f : [2,4] . N -
0x ][2,4 0 f(2)+2f(3)+3f(4)f(x ) 6 .
-
: 3 2012-2013-7-
. 11. 2012 20147x 3x 2 ( 1,1) .12.
22 2 0x x 1 x 1 ,, 0 ,
1 2 , 1,1 2 221 2
1 1
.13. f : x
3 2 xf x f x f( ) ( ) ( )x xe x , (0,1) .14. 2lnx x 0 , 1
(0,1) .15. f f x f 2 x 0 x,
f x 0 16. xe x 2 0 .17. * , x x -
(0, ] .18. f : [0,4] f(0) f(4) h(x) f(x) f(x 2) .
) h) h .) f(x)=f(x+2) [0,2] .
19. f : , 3 2 6f x 3x f x x 1 0 x .
) f x 0 x .) f(0) .) f(x) 0 x .
20. f [0,] 3 2 x f 1(x) x ][0, , f (0,) .
21. f : f(2) 1 1, 4 f(x) 0 . 3xlim[(1 f(3))x 2x 3] .
22. f : . f ( ,2] , [2, ) , x xlim f(x) lim f(x) , f(2) 0 .) f.) f(x) 0
23. *f : : 2 4 2f (x) 6f(x) 5 x 4x , x . :) f(1)) f) x
xlim f(x)
-
: 3 2012-2013-8-
24. f f(x) f(x 3) 0, x . 0x [0,3] , 0 0f(x ) f(x 2) .
25. f(x) g(x) : 2g(x) 1 f (x) , x f(-2) f(1) 0 . g B 2, , f .[ ]
26. f : f(1) 2f(2) 3f(3) 0 f(x) 0 x , f .
27. 1 ,1 f 2 3x x f x 5 . f 1 ,1 .
28. f R. x R 2f x 3f x 2 0 . f .
29. f [0,5] f(1) f(2) f(3) 0 . f(x) 0 [0,5] .
30. 1z 1 i 2z ( )z 5 0, , . (0,1) 3 ( ) e 0 .
31. f , x 0lim f(x) 2 . 0x (0,1) 0f(x ) 1 .
32. f f(5) 7 . f(x) 6 , f .
33. 3 2x x 0 , , R , 0 , 1 0 . 1 ,1 .
34. f x 0,2 0 f x 2 x 0,2 . - 0x 0,2 2 0 0 0f x 2f x x 0 .
35. f : 0, R , f 0 f . 0x 0, , 0 0 f x f x 2 .
36. f [1,2] f 2 6 , f 1 f 2 8 , - , 0x 1 ,2 20 0 0f x x x .
37. f : me f(x) x , x . fC A(3,2) , :) f(x) x , x .) ( 1,1) , : f() 1 .
38. f :R R f 1 1 , 6f f x x f x 0 x R . N f (1) f (0).
-
: 3 2012-2013-9-
39. 2 ef(x) x x 1 , [2,5] . 2012 f(x) 0 [2,5] .
40. x 1 2f x xe 1( ) x x . 1 2 3 [x ,x ,x 1,0] 1 2 3A x , 1 ,B x ,0 , x( ) ( ) ( ),1 f.
41. f : [1,3] x 1limf(x) 2 f(1)f(3) 10 . N f(x) 4 (1,3) .
42. f : [,] [,] . f 0x [,] y x .
43. f [ 1 ,2] f(1) f(2) , 0x ( 1 ,2) 03f( 1) 4f(2) 7f(x ) .
44. f : [,] , f() f() 0 ,,, N* (,) . 0x (,) ,
0( )f(x ) f() f() f() .
45. 2 2 x x 12 ,x 1
f x 5 ,x 1x ,x 1
, , .
) , f 0x 1 .) f xlim f(x)
i) h(x) f(x) g(x) , g(x) ln(x-1) .ii) hC xx .iii) 2005 x f x y = , R
R .46. f : : f f(x) f (x) 3
f(1) 4 .) :
3
2xf 1 x 2x 1lim f 4 x x 2
.
) x 5lim f(x) 8 , fC 6.) g(x) x f(x) 30 (x) .
gC xx (4,7) .
) 0x [1,7] , 10f(x) 2f(2) 3f(3) 5f(5) .47. f : (0,1] 1f(x) lnxx
) f) (0, 1] 2 ln 2 3 .
-
: 3 2012-2013-10-
48. : . , . . .
!
, :
!
-
: 3 2012-2013-1-
& ()
2: 555 1
2.1
1. f 0x ,
: 0x . 0 0,x x , 0,x 0x , -
. 0x .
2. H 0x : x = S(t)
0t 0 0(t ) S (t ) , - .: , 0t 0
0
S t S t 0t t
, 0(t ) 0 ., 0t 0
0
S t S t 0t t
, 0(t ) 0 .
, (t) 0t 0 0(t ) (t ) , .
[]1. 0x , 0x .
[ . 217 ]2. 0x 0x .
[ ]
1. , ... -
, , - .
14
-
: 3 2012-2013-2-
2. . , , , , - .
3. f 0 fx D - f 0x 0x .
4. , , 0x , : 0f x x 0x . 0
0
f x f x x x
.
..
0
00x x 0
f x f xlim f xx x
.
5. f, (.. ) , (. 0x ), : f x
f x f lim f x
00
f x f x x x
0x x
. 0x x h . :
f - 0x x h , g -
0
x h,h 0x
6. , -
0
0x x 0
f x f xlim x x
f -
.
-
: 3 2012-2013-3-
[ .219-221]
1. ) 2f(x) x x 0x 1 0x 2 .) ( ) 1 x1 2g x , x , , 0x 0 .
2. 2+x , x 0f(x) x+ x +4 , x 0
, , f 0.
3. 2
2
x +x+ , x>1f(x) x-1x +2x- , x 1
, , , f 0x 1 .
4. g(0) 1,g (0) 2 . , 2g (x) , x 0f(x) x+ , x 0
, 0x 0 .
5. f 1 x : 3f x 3 x 2 x f x . , , f x = 1 .
6. f 0x 1 x 1f(x) 2lim 3x 1
,
f 0x 1 .7. f 0x , x
f(x)-f()lim x- .8. A g 0x , f x x) x( ) g(
0x g() 0 .9. f, g 0x 0 f(0) g(0) 0
( )x)f g(x x , f (0) g (0) .10. x f x 2) 1( )g(x 0g(x ) 1 0g 0(x ) , f
0x .11. f, g 0x 1 f(1) g(1)
2( ) ( )f x x g x x , x , : f (1) g (1) 1 .12. f : 2 22x x f (x) 2x f(x) x ,
x . f (0) 1 .13. A x 2 2 6 3f (x) g (x) x 2x 1 f, g -
0x =-1 , :) f( 1) g( 1) 0 ) 2 2f ( 1) g ( 1) 9
14. f : f (1) 0 . f(x 1) , x 2g(x) f(3x 5) , x 2
2.
15. f 1, 2 3 3x f (x) 2f(x) x 1 , x . N f 1.
-
: 3 2012-2013-4-
16. f 0x :) 0 0 0h 0
f(x 2h) f(x )lim =2f (x )h
) 0 0 0h 0f(x h) f(x 3h)lim = - 4f (x )h
)0
0 00 0 0x x 0
x f(x) x f(x )lim f(x ) x f (x )x x
) 0 00 h 1f(x h) f(x )1f (x ) limx h 1
17. f : 3f (x) f(x) x x , x . :(i) f x x x , x R .(ii) f 0.
18. f, g : 2 2 2 2f (x) g (x) x x , x . :) f x x x g x x x , x .) f g 0.
***********19. f : ( ) ( )f x y f x f( )y 2xy x, y f(0) 0 .
N ) f 0, f .) f f() 0 , f .
20. f : 0, 0, :f(x y) f(x) f(y) f(x) f(y) , x,y . f x 0 , f .
21. f x,y :2 2f(x) y f(x y) f(x) y . f 0x .
22. f , 0x 0 , 2 4xf(x) x x x . f
0x 0 .23. f 1 f (1) 2 , x,y * -
f(x y) f(x) f(y) (1) . f - 0 0
0
2x *, f (x ) x .24. f : 0x 0 .
x 0f(x) f(x)lim ( ) f (0)x
, , .
25. f x,y 2f(x) f(y) x y (1) . :
) 2f(x) f(y) x y x,y .) f 0x , 0f (x ) 0 .
-
: 3 2012-2013-1-
& ()
2: 555 1
2.2
1. f (x) f(x) . -
f., 0f (x ) 0f(x ) . 0f (x ) f 0x , 0f(x ) 0, 0f(x ) .
2. .3. f , f .
4. f , f .5. f(x) x 0, 0, .6. f 0 fx D ( () 0f (x ) ), -
( 1)f 0x 0 0(x ,x ) 0(,x ] 0[x ,) , 0 .
7. ! ()f (x) ( ) f (x) (), :
() ( 1)f (x) f (x) 1f (x) f (x) f(x) .
15
-
: 3 2012-2013-2-
[ 1,2,4,5 1,2 .227-228]
1. :x
x 0e 1lim 1x
.
2. x 1ln xlim 1x 1 .
3. f : 0x 0 :f( ) f() f() , . :) f(0) 0) f (x) f (0)x x
4. f : , f(x y) f(x) f(y) x, y .) f(0) 0 .) f(x y) f(x) f(y) .) f .) f 0 , f .
5. f: 0, f(x y) f(x) f(y) , x,y (0, ) 1 x1 x f(x) x , f 0x (0, ) .
6. f : 0x 1 f(1) 1 f (1) 1 . - :
) 2x 1 f(x) 1lim x 1
) x 1
xf(x) 1lim x 1
) x 1
xf(x) 1lim x 1
7. f , g : f 0x g 0x . (f g)(x) 0x , - 0g(x ) 0 .
-
: 3 2012-2013-1-
& ()
2: 555 1
2.3 -
1. x
.: .
2. - f, g 0x , 0 0 0(f g) (x ) f (x ) g (x ) 0 0f(x ) g(x ) 0 0f(x ) g(x ) 0 0 0f(x ) g(x ) . .
3. f 0x , f g , f g fg 0x . 0x .
4. f, g 0x f g f g fg 0x .: x x 0f x 0 x 0
, x x x 0g x x x 0
0x 0 , f g (f g)(x) x 0x 0 .
16
-
: 3 2012-2013-2-
5. f , :i) f , -
f.ii) f ,
0x , , f , , .
: .6. f ,g f(x) g(x) f (x) g (x) . f (x) g (x)
f(x) g(x) .7. f , 1f
f() f (x) 0, x f() , :
1 11f x , x f f f x : x f() 1f f (x) x :
1 1 1f f x x f f x f x 1 1 11(f ) (x) , x f()f f (x) (1) . (1) , 0 0f(x ) y 0f (x ) 0 , 1 0 0
1(f ) (y ) f x .
[]1. f , f -
.2. f , f
.******
: , . , , .
-
: 3 2012-2013-3-
( )
f f f
f1) f(x) c (c) 0 2) f(x) x (x) 1 3) f(x) x , {0,1} 1(x ) x 4) *f(x) x , * 1(x ) x *
5) f(x) x , [0, ) , 0 ,(0, ) , 0 1(x ) x [0, ) , 1 ,(0, ) , 1
6) f(x) lnx (0, ) 1(ln x) x (0, )
7) f(x) logx (0, ) 1(log x) x ln10 (0, )
8) f(x) ln x * 1(ln x) x *
9) f(x) x [0, ) 1x 2 x (0, )10) xf(x) e x x(e e) 11) xf(x) , 0 x x( ) ln 12) f(x) x (x) x 13) f(x) x (x) x
14) f(x) xfA {x /x 0}
{x / x , }2
22
1(x) x(1 x)
f fA A
15) f(x) x fA {x / x 0}{x / x , }
22
1(x) x(1 x)
f fA A
16) 1f(x) x* 2
1 1x x
*
17) f(x) x 1, x 0x 1, x 0
*
-
: 3 2012-2013-4-
: *f(x) x , , .
, :f(x) x fA [0, ) .
, :
( x) , x 0f(x) xx , x 0
fA . , -
.
( )
( )1) (f g) (x) f (x) g (x) 2)
( f) (x) f (x) 3)
1 1 2 2 1 1 2 2 ( f f ... f ) (x) f (x) f (x) ... f (x)
4)
(f g) (x) f (x) g(x) f(x) g (x) (f g h) (x) f (x) g(x) h(x) f(x) g (x) h(x) f(x) g(x) h (x) ( 3 - )
5) 2f (x) g(x) f(x) g (x)f (x)g g (x)
( )
g f g(), f g :
(f g) (x) f (g(x)) g (x) f(g(x)) f (g(x)) g (x) u g(x) , : f(u) f (u) u
y f(u) u g(x) , : dy dy dudx du dx ( )
1 2 3 y f(u (u (u (....u (x)....))))) , : 1 21 2 3
dudu dudy dy ...dx du du du dx
-
: 3 2012-2013-5-
( V )
f(x) , :
u f(x) , f , :
1) 1f (x) f (x) f (x), {0,1} 1) 1u u u , {0,1} 2) f (x)f(x) , f(x) 02 f(x) 2) uu , u 02 u 3) f(x) f(x) f (x) 3) u u u 4) f(x) f(x) f (x) 4) u u u 5) 2 21 f (x)f(x) f (x) f(x) f(x)
5) 2 21 uu u u u
6) 2 21 f (x)f(x) f (x) f(x) f(x) 6) 2 21 uu u u u
7) f(x) f(x)e e f (x) 7) u ue e u 8) 1 f (x)ln f(x) f (x)f(x) f(x) 8) 1 uln u uu u 9) f (x)log f(x) f(x) ln 9) ulog u u ln 10) f(x) f(x) ln f (x) 10) u u ln u 11) 1f (x) f (x) f (x), f(x) 0, {0,1} 11) 1u u u , u 0, {0,1} ! g(x)(x) [f(x)] f(x) 0 , g(x) lnf(x)(x) e
g(x) g(x) lnf(x) g(x) lnf(x) g(x) (x) [f(x)] e e g(x) ln f(x) [f(x)] g(x) ln f(x) ...
: - ., .
-
: 3 2012-2013-6-
[ 1,2,3,4,6,12,13,14,15 7,9 .238-240]
1. :) x x xe ln xf x , g x x x , h x ,x 1 1 x1 x
) x1 x 2 xlnx 2x 1f x , g x , h x , x1 x x 2 1 x e ) 2 22 x x x 1 x x 1 ef(x) x 2 2 e x ) xxxx x 3f(x) x , x 0, g(x) (x) , x 0, , h(x) (x 1) , x 1, (x) 3 , x2
) 2 2x 1f(x) x , x 0, g(x) log (x), x (1,2) (2,), h(x) ( x)( x) .2. -
Leibniz ( ):) (x) ln(x), x (0,) ) 4 2k(x) (3x 1)
3. , :) xf(x) (e 1) ln(x 1) ) 2g(x) ln(1 x )
)xe eh(x) (2x 1)ln x
4. N f 0x : f(x) x x 0x 0 .5. f : 3 1f (x) xf (x)x
x 0 . N f (1) .
6. A f , :) f (0) ) g (0) , f (0) =1 g(x) f(x)x f(x)
7. f(x)=22
x1+ x ,
f 3f4 4 .
8. xf(x) e (x) , 2 2f (x) 2f (x) f(x) 0 .9. N :
) 2f x x ln x , 22f(x) xf (x) x 0 ) xxy e , dyx x 1 y 0dx ) xy e x x , xy 2y 2e x 0
-
: 3 2012-2013-7-
10. f x 2f(x )=xf(x) , f (1) 0 .
11. f x :f(2x 3) f(x) . f 0x 3 xx.
12. A 64 27( ) f x x x , 0x (0, )2 0f 0(x ) .
13. A 2f(x) x g(x) x ,) (g f) (1) g f(1) f (1) (g f) (1) .) (g f) (0) g f(0) f (0) . (g f) (0) ;
14. P 2 x x x .15. P(x) x : 2P x 4P x
P(0) 4 .16. N P(x) , P(0) 1 2(P (x)) P (x) 8P(x) .17. ) (x) 2 , 2x -
- . , 2 x x x 0 .
) 2x 1 1 1 2 2f x x x 1 x 2 , 2.
) , 4 2 x x 4 x 1 x 3 4 2x 2 .
18. 3 2f x x x x 1 2 3 , , . :i) 1 2 3
f x 1 1 1x x x f x
1 2 3x , ,
ii) 31 21 2 3 0f f f
iii) 1 2 3f 0 1 1 1
f 0
iv) 1 2 3
2 22 2 2 2
f 0 f 0 21 1 1f 0 f 0
19. :
) x 2x 3x x1S 1 e e e e ) x 2x 3x x2S e 2e 3e e , .
-
: 3 2012-2013-8-
20. f : , y xf(x y) e f(x) e f(y) , x,y, , :) f(0) 0 ) xf (x) f(x) f (0)e , x .
21. *f : f(xy) f(x) f(y) *x,y R f 0 . :
) yf (x)f (y) x ) f (1) f ( 1) 0
22. f : , :f (0) 1 f(x y) f(x y) 2f(x)f(y) x,y . : f (x) f(x) x .
[ f() ]23. :
) 1f(x) x
() 1
( 1) !f )x( x ,
) 1f(x) x 1 ,()
1( 1) !f (x) (x 1)
* x { 1} .
) f(x) x () ( ) +x2f x ,
) f(x) x , () f (x) x 2
) xf(x) xe , () xf (x) e (x ) .24. 1 1 1 0f x x x x ,
0 1 1 , , , , 0 . () f (x) ! ( )f (x) 0 .
[ f-1]25. . f : (,) R , . f -
0x (,) 0 f (x ) 0 1f 0f(x ) : 1f 0f(x ) 1 0
0
1(f ) f(x ) f (x ) .
. xf(x) e x , 1(f ) (1) .26. x 3f(x) e x x , x .
(i) f 1f-(ii) 1f- 1fD - ,
1 1(f ) (1) 2 .
27. A f(x) x , x ,2 2
12
1(f ) (x) , x ( 1,1)1 x
.
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: 3 2012-2013-1-
& ()
2: 555 1
2.1 - 2.3
1. fC f 0 0(x ,y )
0 0 0 0(x ,y ) (x ,f(x )) - 0f (x ) f 0x ., : 0 0 0y f(x ) f (x )(x x )
:
f , :) 0 0A(x , f(x )) -
: .) : , 0 0M(x , f(x )) ,
0x , 0x - .
1) (x0,f(x0)) Cf. [5 220, 7,B1,11 239]
) 0f (x ) 0f(x ) .) 0 0 0y f(x ) f (x )(x x ) .
2) ( ) , Cf. [10 239]) 0 0M(x , f(x )) o -
(): 0 0 0y y f (x )(x x ) .(, , 0x )
) .
) 0x .
3) [3 228, 8,9,B2,6 239]) 0 0M(x , f(x )) o .) 0 f (x ) , 0x .) , , .
17
-m
x0
M
Cf
f(x0)
x0
M
Cf
f(x0 )
Cg
g(x0)=
x0
M
Cff(x0)
Cgg(x0)
=
N
A
B
-r0
y=x+:
x0+
-
: 3 2012-2013-2-
4) [A11, 2 239] : y x fC 0 0M(x , f(x )) fC - :) (),
0 0( )f x x .) ()
fC , 0f (x ) .5) [3 239]
0 0M(x , f(x )) . - f g 0x :) 0 0f(x ) g(x ) ,
fC gC , y f(x) y g(x) ,
) 0 0f (x ) g (x ) , fC gC .
0x -.
6) () f g. () fC gC , ,f( ()) ,g( ()) :) f () g () ) fC ,f( ()) ,
,g( ()) . , -.
7) () [4,10 239] () fC ,f( ()) gC . ,g( ()) gC :) f () g () g A gC -
().) gC
,f( ()) .
-m
x0
M
Cf
f(x0)
x0
M
Cf
f(x0 )
Cg
g(x0)=
x0
MCf
f(x0 )
Cg
g(x0)
=
N
A
B
-r0
y=x+:
x0+
-m
x0
M
Cf
f(x0)
x0
M
Cf
f(x0 )
Cg
g(x0)=
x0
MCf
f(x0 )
Cg
g(x0)
=
N
A
B
-r0
y=x+:
x0+
-m
x0
M
Cf
f(x0)
x0
M
Cf
f(x0 )
Cg
g(x0)=
-m
x0
M
Cf
f(x0)
x0
M
Cf
f(x0 )
Cg
g(x0)=
x0
MCf
f(x0 )
Cg
g(x0) =
N
A
B
-r0
-
: 3 2012-2013-3-
1. 2f(x) x x 3 0,1 .
fC .2. 2f(x) lnx x 3 , , R : 2x y 4 0 -
fC A 1,f(1) .3. f 3 2 3 4f(x) x f(x) 2x 4x -
x . fC A 1,f(1) .4. f : 2xlnx f(x) x x x . -
0x 1 fC M 1,f(1) .
5. 2f(x) x x 1 . - fC :) (1,1)) (2,1) .
6. 2f(x) x 4x 33 , 2g(x) x 1 .
7. f 3 2f(x) x x 2x 5 () : 2x y 1 .) ()
f .) () fC .
8. x1f(x) x e 3 2g(x) x 3x 5x . - fC A 1,f(1) , gC .
9. 2f(x) 2x x 2g(x) x 4x 1 .
10. f y 2x 1 fC 1 .
2
x 1f (x) 1lim x 1
.
11. 2f(x) x 2x 6 7 , .) , f
.) , fC xx.
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: 3 2012-2013-1-
& ()
2: 555 1
2.4
1. y f(x) 0x .
y x 0x 0f (x ) y x f (x).
2. x,y y f(x) f x, :) y x , f (x) 0 .) y x , f (x) 0 .
[ ]3. S S(t) -
t. H S t .
4. S t 0t 0S (t ), S t 0t , () 0t 0(t ) . 0 0=(t ) (tS ) ., , (t) S (t) .
5. t 0t - 0 (t ) , t 0t , () 0t 0(t ) ., , 00 0 t =( S) = ( ) (tt ) ., , - . (t) (t) S (t) .
6. , :) S(t) 0 , .) S(t) 0 , .) S(t) 0 , .) S(t) 1 , ( ).) S(t) 2 , ( ).) (t) S (t) 0 , .
18
-
: 3 2012-2013-2-
) (t) S (t) 0 , .) (t) S (t) 0 , .) (t) S (t) 1 , .) (t) S (t) 2 , .) (t) S (t) 0 , .) (t) S (t) 0 , .) (t) S (t) 0 , .) (t) S (t) 1 , .) (t) S (t) 2 , .
[ ]7. , , ()
x .8. : ( ) ( ) ( )P t t K t (1),
: 0 (x ) -
x, 0x x 0x . 0E (x ) -
, x 0x x 0x . 0P (x ) P
x, 0x x 0x .9. (1) : ( ) ( ) ( ) P t t K t .10. , :
x , K(x)K (x) x .
( ) x , E(x)E (x) x .
x , P(x)P (x) x .
[ ]11. y x [y y(x) ] x t ( x x(t) ), y
t [ y(t) y(x(t)) ].12. (g f) (x) g f(x) f (x)
dy dy dudx du dx , y g(u) u f(x) , .
13. dydx , x .
14. dydx x , y .
-
: 3 2012-2013-3-
1. ( -
)2.
(x,y) (x,y) 0 - t. : x(t),y( (t)) 0 . t.
3. x ,y y y y, 2y 2px ,
2 2x y 1 . , ,) t: x x(t) , y y(t)) t.
[ ] 3 m ' . 0,1 m/sec. 2,5 m :) .) .:, x, y, - t. : x(t),y(t) (t) , x (t) 0,1 m / sec .) (t) .
. x 1 1 1 (t) x(t) (t) x(t) (t) (t) x(t)x (t) (t) ...3 3 3 3
) y (t) . 2 2x y 9 [. ]., : 2 2 0 00
0
x (t ) x(t )x (t) y (t) 9 x (t) x(t) y (t) y(t) 0 y (t ) y(t )
, 0t -
0y(t ) 2,5m . 20 0x(t ) 9 y (t ) ... 2,75 .
-
: 3 2012-2013-4-
[ .243-245]
1. t - S 3 2S(t) 2t 21t 60t 3 S t sec. :) .) .) 24 m/sec.) 18 m/sec2.) .)
.2. (x) (x), x
3 21 x x 20x 600x( ) 10003 x(x) 420.) N
.)
.) (x) (xP ) )K(x .
3. 22 cm / sec . - 1,8 m, .
4. p(t) p(t) p (t) 0 .
5. 2 2x y 1 .
1 3,2 2
, y 3 sec.
N x - .
6. x y () () 12cm . 0t - 8cm/sec () 3cm :) T ) T ()
-
: 3 2012-2013-1-
& ()
2: 555 1
2.5) Rolle
1. Rolle -
.2. f Rolle
[,] , , , : 0x (,) 0f (x ) 0 . f (x) 0 (,) . f (,) .
[ Rolle] f xx
0A(x ,0) 0x (,) . 0x (,) -
f 0 0M(x ,f(x )) xx.[ Rolle]
( f(x) c ) 0f (x ) 0 0x (,) [ ].
3. , , 1t 2t , 0t 1t , 2t ( ).[ Rolle]
4. Rolle ! , - f , Rolle [ ].
20
-
: 3 2012-2013-2-
5. f (x) 0 (,) . - .
6. Rolle .7. [,] , [,]
. Rolle f() f() .
[ ]1. f , f .2. f , f , -
f.3. f , f -
f .4. , f - ( , 1)
1 , ()f ( ) .5. f (x) 0 x , f .6. f (x) 0 x , f .7. 1-1
f (x) 0 .8. f (x) 0 x , f 1-1 .
1. Rolle
: f(x) 0 (,) (,) (,) (,) (,) (,)
-
: 3 2012-2013-3-
! , Rolle f , () F, F -: F (x) f(x) fx A .
2. .Rolle f , - f . , f () 0 f () 0 , - Rolle f f .
f F f F
0 c f (x) f(x)1 x f(x) f (x) 21 f (x)2x
+1x+1
f (x) f (x) 11 f (x)+1
1x 2 x
f (x)f(x) 2 f(x)
x x f(x) f (x) f(x)x x f(x) f (x) f(x)
xe xe f(x)e f (x) f(x)e1x ln x
f (x)f(x) ln f(x)
xx
lnf(x) f (x)
f(x)ln
f (x)g(x) f(x)g (x) f(x)g(x) f(x) x f (x) x f(x)
2f (x)g(x) f(x)g (x)
g (x) f(x)
g(x) 2f(x) xf (x)
x f(x)
x2[f (x)] f(x) f (x) f(x) f (x) g(x)[f (x) f(x)g (x)]e g(x)f(x)e
f (x) g (x) f(x) g(x)
21x
1x 2
f (x)f (x) 1
f(x)
21
x x 2f (x)
f(x) f(x)
21
x x 2f (x)
f(x) f(x)
-
: 3 2012-2013-4-
3. (,) - . , x, , -, . - f Rolle.
I . Rolle1) f f (x) 0 [f(x) x] 0 (x) f(x) x
2) f f f (x) f(x) 0 f (x) f(x) 0f (x) f(x) 0 f(x) 0
-x -x
-x -x -x
e ee e e
x(x) f(x)e
3) f ()( ) f() f (x)( x) f(x) 0 f (x)( x) f(x)( x) 0
f(x)( x) 0
(x) f(x)( x)
4) f ()( ) f() 2
f (x)(x ) f(x) 0 f (x)(x ) f(x)(x ) 0f (x)(x ) f(x)(x ) f(x)0 0x (x )
f(x)(x) x
5) 1f () 1 f (x) x 0 f (x) (x ) 0 [f(x) x ] 0 (x) f(x) x
6) f () f()
2
xf (x) f(x) 0 xf (x) f(x) 0x f (x) (x ) f(x) 0x f (x) (x ) f(x) f(x)0 0x x
-1 -1x x
f(x)(x) x
7) f ()f () 0 2f (x)f (x) 0 2 f (x)f (x) 0f (x) 0
2(x) f (x)
8) f () f() 0 2 2 22f (x) f(x) 0 2f (x) f (x) 2f (x) f(x) 0f (x) f(x) 0 f (x) f (x) 0
22(x) f (x) f (x)
9) () ( 1)f () g ()f ()
() ( 1)
g(x) g(x)() ( 1)
g(x) g(x)( 1) ( 1)
g(x)( 1)
f (x) g (x)f (x) 0f (x)e e g (x)f (x) 0f (x) e f (x) e 0f (x)e 0
g(x)( 1)(x) f (x)e
, :10) 2f () f () 0 f(x)(x) f (x)e
11) f()f () 0 lnx f(x)(x) f(x)e x
-
: 3 2012-2013-5-
4. - [,] , . f() f( ) . - , , x . Rolle. f( )f()
-
. Rolle f(x)(x) x .5. , , f(x) 0 , -
: Bolzano ( 0 f() ) ( f(x) 0 x ) Rolle f ( f g , g(x) =
f (x))6. f(x) 0 ,
Rolle.7. f(x) 0 -
: Rolle
8. f(x) 0 , Rolle .
9. f(x) 0 (,) , .
10. (,) f () 0 (3)f () 0 Rolle -.
11. Rolle F , - Bolzano f , f , f ( ).
-
: 3 2012-2013-6-
[ Rolle ] [1, 249]1.
2x x ,x 0f(x) 3 ( )x ,x 0
,, -
Rolle [ 1,1] . fC .2. f f (x) 0 , x .
f xx .3. f [,] ,
(,) f() f() 0 . :) f(x)g(x) x c , c [,] 0x (,) 0g(x ) 0 .) 0x (,) fC 0 0 x ,f(x )
(c,0) .4. f [,] , (,) f (x) 0
x (,) . f() f() .
[ 1 ] [1,2,3, 249-250]5. ,, 05 3 ,
4 2x x 0 - (0,1) .
6. f 0, 2
f 1 f(0)2
.
0 x 0, 2
: 0 0f (x ) x .7. f [,], [,] .
, , 21z f ()[f() 1] 2i 22z f () i . 3
1 2Re(z z ) f () , f (x) 0 - (,) .
8. f [ 2,2] f(1) 0 2g(x) f(x)(x 4) , 0x ( 2,2) 0g (x ) 0 .
9. f : [,] (0, ) lnf() lnf() . (,) , f () f() .
10. f : [,] , [,] (,) f() f() c( ) . (,) , : f () c , c .
11. f : [,] , : 2 2 2 2f () f () . - (,) f() f () .
-
: 3 2012-2013-7-
12. f (0, ) 1f(e) f(1) e (1). 0x (1,e)
20 0 0x f (x ) 1 lnx 0 (2).
13. f, g [,] f() f() 0 . - (,) , f () f() g () .
14. f [1, e] f(1) 1 ,2f(2) 4 ln2, f(e) e 1 . (1,e) , 21f 2 .
15. f [0,] f(x) 0 x [0,] . - [0,] f () 0f()
.
16. f f(x) 0 x f 2012 ef 2011 .
f (x) f(x) (2011,2012) .17. f [,] (,) f() 0 . -
(,) f f .18. f [1,e] ,
(1,e) f(e) f(1) 1 . x f (x) 1 (1,e) .
19. f : [,] , , , f() f() . , , : c f f , c .
20. f : [,] , [,] (,) f() f() 0 . (,) , -: f () f() .
21. f : [,] , [,] (,) f() f() 0 . (,) , -: f () f() .
22. f : [,] , [,] (,) f() f() . (,) , : f () f() .
23. f : [,] , [,] (,) f() f() . (,) , : f () f() .
-
: 3 2012-2013-8-
24. f,g : [,] , [,] (,) h()h()e f() e f() . (,) -, : f () h () f() .
25. ) f . f (x) 0 , f(x) 0 .
) f . - f (x) 0 , f(x) 0 - .
[ 1 ]26. 2x 2x 2ln(x 1) 0 . []27. f , -
fC : 2x y 1 0 x . fC y 2x .
[ 1 ]28. f : xf (x) e , x 0 f(x) 1, x [0,1] .
0x0 0x (0,1) : f(x ) e 1 .
[ ] 1 .
[ ]29. x 2e x x .
[ ] [7, 249]30. 2xx x x [ ,] .31. 4 3 2x x 5x x 0 *, R 0
.
[ F F()=F() ( )]32. f : [,] (0 ) f(x) 0
x [,] . f() f() 0x (,) 0 0 0 0f(x ) ln f(x ) x f (x ) .
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: 3 2012-2013-1-
& ()
2: 555 1
2.5) ( )
1. (...)
.2. f ...
[,] , , , -: 0x (,) 0 f() f()f (x )
.
f() f()f (x)
(,) .[ ...]
( )f (x) [f() f()] 0 - (,) .
0x (,) - f 0 0M(x ,f(x )) , A(,f()) B(,f()) . [ ...]
3. - 1 2[t , t ] 0 1 2t (t , t ) .[ ...]
4. ... !5. ... .6. f [,] , ...
f [,] f .
7. f f() f() , - Rolle.[ Rolle ]
8. f f() f() , - (,) f () 0 . - f M(,f()) , - xx .
21
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: 3 2012-2013-2-
9. f f() f() , - (,) f () 0 . - f M(,f()) , xx .
1. ..., , :
) f() f()
f() f() .) f () .) -
, .
) 1 2 1 2 f (x ), f (x ),..., f (x ) x ,x ,...,x (,) .) 1 2 1 2f (x ), f (x ) x ,x (,)
[,] . , , - [,] [,] , [,] , 2
.
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) . - , x x x x7 6 9 8 .
) , - f() f()
-
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-
f() f()
f ..3. f (,) [,] 0x -
(,) .. 0[,x ] 0[x ,] .
4. [,] - 1 2 , , , 1 1 2 2 f() f() f ( ) f ( ) f ( )
1 1 2 2 f ( ) f ( ) f ( ) f () . f. [,] -.
-
: 3 2012-2013-3-
), , [,] 1 2
.
, , , 1, 1 1 2 , .. 1 2 1 ... ,
) ... .
[]1. f [,] -
: f [,] , f() f()f () f ()
[ 3, 249]
f [,] , f() f()f () f ()
2. f [, ], (, ) f (,) + f() f()f 2 2
f (,) + f()+f()f >2 2
3. f [,] . f f f 2 2
, , f 0 .
[ ... ] [2 .249]1.
2
3x + , x 1f(x) x -x+ , x 1
. ...
[ 1,2] :) , .) ,f() [ 1,2]
: 2x y 3 0 2. f [4,10] f(4) 6 f(10) 0 .
(4,10) , fC A(,f()) 0 135 xx.
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: 3 2012-2013-4-
3. f f 1-1, fC fC .
[ ...]4. x x x x7 6 9 8
[ ...] [3, 4,5 .249-250]5. N : , .6. f [1,5] f(1) 2 f (x) 2
x (1,5) , : 10 f(5) 6 .7. f x 1 x f(1)=0, f x x 1 , x .8. f R R .
: f 2009 f 2012 f 2010 f 2011 .9. f 2 f .
f(2x 3) f(2x 7) f(2x 1) f(2x 5) x .10. N : x xx 1 e xe 1 , x .11. : xe x 1 , x .12. : ln x x 1 , x (0, ) .13. ) N : x 1 lnx x 1x
, x 0, .
) : x 1lnxlim 1x 1 .
14. ) N : x ln(x 1) xx 1 , x 1 x 0 .
) : i)x 0
ln(x 1)lim 1x
ii) 3xln(x 1)lim 0x
15. : e 2 ln e .16. : x 1x e 1 (x 1)e x (1,2) .
[ [, ] ...]17. f [1,11] f(1), f(6), f(11)
, (1,11) , f () 0 .
18. f :f(2x) 2f(x), x .) : f(2) f(0) f(1)2
.
) (0,2) , : f () 0 .
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19. f [,] (,) f() f() . 1 2x ,x (,)
1 2f (x ) f (x ) 2 .20. f ... [0,3] , -
1 2 3 , , (0,3) 1 2 3f ( ) f ( ) f ( ) f(3) f(0) .21. f f ln f ln .
ln ln ln , ,, 0 2 e ,
1 2 , 1 2f ( ) f ( ) 0 .22. f , f( 1) 1 - , f(1) 1 .
) 1 21 1 1 2f ( ) f ( ) 2 .) 1 21 1
1 2
1 1 2f ( ) f ( ) .23. f -
A(4,11) B(19,5) . 1 2 , , 1 22f 3f 2 .
24. f [,] (,) f(x) 0 x [,] . 1 2 0 , , (,)
1 2 0
1 2 0
f f f 2f f f
.
25. f [,] - fC , :) fC -
.) (,) f () 0 .) f(x)-f()( x) x- . Rolle
[,] .) (,) f()-f()f ()= - .
26. f 2 [,] f() f() 0 . (,) f() 0 , (,) f () 0 .
27. f 2 f (x) 0, x , fC .
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: 3 2012-2013-6-
[ [, ] Bolzano] ... [,] , - Bolzano28. f [,] , (,) f() , f() , ,
:) (,) f() .) 1 2 , (,) 1 2 1 2f ( )f ( ) 1 .
29. f [,] , (,) , -: f() f() . :) 0x , 0
f f f x
, , .
) 1 2 , , , : 1 2
f f f
, 1 2 .
[ ... Bolzano Rolle]30. . f : [0,2] . (0,2) :
2f(0) 3f(2)f() 5
, f(0) f(2) .. 1 , 1 2x ,x (0,2) 1 2x x 1 22f (x ) 3f (x ) .. f (0,2) 2g(x) f(x) x x
. Rolle [0,1] [1,2] , , .. f (0,2) , p (0,2)
5f (p) f(0) f(2) .31. f [,] f() f() , f () 0
f () 0 . f (x) 0 (,) .
[... - F()=F()]32. f [,] , (,) , f(x) 0 x (,) .
(,) :( )f ()
f()f() ef()
.
[... ]33. f(x) f (x) , x ,
0 , xlim f(x) .
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& ()
2: 555 1
2.6 ) ( )
1. .2.
, f gC ,C x -, , yy, c , c 0 c 0 .
3. To .: , , . [ .252]
4. 1 2f (x) g (x), x , : 1 1f(x) g(x) c , x 2 2f(x) g(x) c , x
5. f (x) 0 fx A - ;
6. c, , x, , .
[]1. f f (x) f(x) , x , ,
c xf(x) c e , x . = 1 [ 252] f ( ) ( ) f x f x , x xf(x) ce , c .
2. f (x) 0 , f , f .
22
y
O x
y=g(x)+c
y=g(x)
22
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1.
f - . f f (x) 0
x . f , f(x) c x . c f.
2. . . .
3.
- f ,f , f - x. . , , -
f(x) g(x) x .[: , ] , f(x) g(x) c , x .
1 2 U , , : 1 12 2
g(x) c ,x f(x) g(x) c ,x
.
c 1 2c , c f.
4. f(x) 0 , ln(f(x)) .5.
. . , ,
. f . f 0.
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: 3 2012-2013-3-
[ ] [1, B1 .256-7]1. f,g : 2f (x) g (x)
2g (x) f (x), x .) 3 3h(x) f (x) g (x) .) h, f(0) 1 g(0) 2 .
2. f : f (x) 2f(x) , x f(0) 1 f(x) 0 , x . N :) G(x) ln f(x) 2x .) 2xf(x) e , x .
3. f, f (x) 0 x (0,1) (1,2) [1,2] , f [1,2] .
4. f : f(x)f (x) 2 x , *x .
5. f x,y 2f(x) f(y) (x y) . :) 2f(x) f(y) (x y) x,y .) f .
6. f : f x f x 0( ) ( ) , x .) f(0) f (0) 0 , :
i) H 2 2h (f )