Σημειώσεις Μαθηματικών 2012-13 (Δούδης Δημήτρης).pdf

177
Μαθηματικά Γ’ Λυκείου Θετικής & Τεχνολογικής Κατεύθυνσης Αλεξανδρούπολη 2012-2013 Ενημέρωση: Τετάρτη, 24 Απριλίου 2013
  • Upload

    -
  • Category

    Documents

  • view

    3.545
  • download

    11

description

Σημειώσεις Μαθηματικών Κατεύθυνσης Γ' Λυκείου 2012-13 (Δούδης Δημήτρης), Έκδοση: 24-04-2013

Transcript of Σημειώσεις Μαθηματικών 2012-13 (Δούδης Δημήτρης).pdf

  • &

    2012-2013

    : , 24 2013

  • (2012 - 2013)

    [1] , 3 , .

    [2] [http://www.study4exams.gr/math_k/][3] . , . , . , , 1, 2, .

    .[4] , , &

    , .[5] , , &

    1, 2, . .[6] , [http://lisari.blogspot.gr][7] , . ,

    .[8] , .[9] , [http://blogs.sch.gr/imavros/].[10] , .[11] , .

  • : 3 2012-2013

    & ()

    2:

    1

    -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

    .

    1

    . 2.1: . 2.2: C M. 2.2: C M

  • : 3 2012-2013

    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 .

  • : 3 2012-2013

    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);

    . . .

  • : 3 2012-2013-1-

    & ()

    2: 1

    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

  • : 3 2012-2013-2-

    ) 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

  • : 3 2012-2013-3-

    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 .

  • : 3 2012-2013-1-

    & ()

    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

  • : 3 2012-2013-2-

    (..) 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 ]

  • : 3 2012-2013-3-

    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

  • : 3 2012-2013-4-

    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 .

  • : 3 2012-2013-5-

    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

  • : 3 2012-2013-6-

    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

    -

  • : 3 2012-2013-7-

    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

  • : 3 2012-2013-8-

    ) 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 .

  • : 3 2012-2013-9-

    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.

  • : 3 2012-2013-10-

    ) 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

  • : [3 ] 2012-2013

    & ()

    1: 555 1

    1.2) - 1. ,

    .2. T

    .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

  • : [3 ] 2012-2013

    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 , .

  • : [3 ] 2012-2013

    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

  • : [3 ] 2012-2013

    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 .

  • : [3 ] 2012-2013

    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 .

  • : [3 ] 2012-2013

    & ()

    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

  • : [3 ] 2012-2013

    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 , .

  • : [3 ] 2012-2013

    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

  • : [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.

  • : 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 .

  • : 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 .( )

  • : 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

    . ()

  • : 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

    .

  • : 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.

  • : 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 ) .

  • : 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

  • : 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

    .

    ) .2. ... f -

    ) . - , x x x x7 6 9 8 .

    ) , - f() f()

    -

    [,] .) , f() f()

    -

    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.

  • : 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 .

  • : 3 2012-2013-5-

    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 .

  • : 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) .

  • : 3 2012-2013-1-

    & ()

    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

  • : 3 2012-2013-2-

    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.

  • : 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 )