Контрольные работы по математическому анализу

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  • . . .

    2005

  • 2

    517 -

    : . / . . . , . . . . : , 2005. 34 c.

    -, .

    , - , , . : -

    . . , - . .

    : -

    . .

    . . . , 2005

  • 3

    1. 4

    2. 10

    3. 14

    4. 16

    5. 22

    6. 26

    7. 30

    8. 33

  • 4

    . b )(xfy = , Xx , a ,

    ,...,2,1, = nXxn , axnn

    =

    lim -

    )}({ nxf )(xfy =

    ,...,2,1, =nxn b

    bxf nn

    =

    )(lim ,

    bxfax

    =

    )(lim .

    , a b + :

    +=

    )(lim xfax

    , bxfx

    =

    )(lim , =+

    )(lim xfx

    ..

    1sinlim0

    = x

    xx

    ,

    . . :

    1. 32

    3

    3542limxxxx

    x ++

    +

    2. 623lim

    2

    23

    2

    ++ xx

    xxxx

    3. 324238lim

    35

    35

    +

    ++ xx

    xxx

    4. 125

    103lim3

    2

    5

    x

    xxx

    5. 62

    642

    65453limxxxxxx

    x +

    +

    6. 2

    3

    1

    23limxxxx

    x +

    7. 32

    32

    354543limxxxxxx

    x

    ++

    8. 1212lim

    2

    2

    1

    + xx

    xxx

  • 5

    9. 457

    567

    42435limxxxxxx

    x +

    ++

    10. 27

    32lim3

    2

    3 +

    + x

    xxx

    11. 522

    43lim5

    32

    +

    ++ xx

    xxxx

    12. 1

    12lim3

    2

    1 +

    ++ x

    xxx

    13. 563

    739lim23

    4

    +

    + xx

    xxx

    14. xxxx

    x 224lim

    2

    2

    2

    +

    15. 32

    32

    425723limxxxx

    x +

    +

    16. 5

    5245lim2

    5

    xxx

    x

    17. 63552

    21

    32

    lim24

    34

    +

    ++

    xx

    xxx

    x

    18. xxxx

    x 224lim

    2

    2

    2 +

    ++

    19. 3 23 5

    6 36 2

    5

    32limxx

    xxx

    +

    20. 32

    9lim2

    2

    3

    xx

    xx

    21. 458

    568

    244lim

    xxxxxx

    x +

    ++

    22. 1

    2lim3

    2

    1 +

    x

    xxx

    23. xx

    xxxx 25

    5lim3 5

    3 23

    ++

    24. 64

    1252lim3

    2

    4

    x

    xxx

    25. 345823lim

    25

    25

    +

    ++ xx

    xxx

    26. 1

    lim3

    3

    1

    x

    xxx

    27. xx

    xxx +

    + 3 5

    4 24 5 423lim

    28. xx

    xx 2

    8lim2

    3

    2 +

    +

    29. 345

    54lim3 5

    3 23 5

    +

    ++ xx

    xxxx

    30. xx

    xxxx 85

    273lim5 3

    3 433 2

    +

    ++

  • 6

    B. :

    1.

    +

    1212lim

    2

    2

    3

    xx

    xx

    x

    2.

    96

    31lim

    23 xxx

    3.

    31 13

    11lim

    xxx

    4.

    42

    21lim

    22 xxx

    5.

    12

    11lim

    21 xxx

    6.

    +

    + 44

    21lim

    22 xxx

    7.

    +

    +xxx

    x3lim 2

    8.

    +

    + 15

    13lim

    31 xxx

    9.

    +

    + 274

    32lim

    33 xxx

    10.

    +

    22lim axxx

    11.

    + 11lim

    2

    2

    3

    xx

    xx

    x

    12.

    251

    54lim

    25 xxx

    13.

    +

    +1lim 2 xxx

    x

    14.

    + 163

    42lim

    24 xxx

    15.

    645

    87lim 28 xxx

    16.

    +

    +xxx

    x2lim 2

    17.

    812

    21lim 32 xxx

    18.

    +

    xxx

    x41lim 22

    19.

    xtgxx

    x

    22

    2cossinlim

    20.

    166

    41lim

    24 xxx

    21.

    +

    +xx

    x3lim 2

    22.

    13lim

    2

    3

    4

    xx

    xx

    x

    23.

    ++

    +xxxx

    x22 1lim

    24.

    166

    41lim

    24 xxx

  • 7

    25.

    ++

    +53lim 2 xxx

    x

    26.

    363

    68lim

    26 xxx

    27.

    +x

    xx

    x 1lim

    2

    3

    28.

    2sin4

    1sin

    1lim220 xxx

    29.

    ++

    +xxxx

    x342lim 22

    30.

    ++

    + 2

    22

    4)23)(12(

    123lim

    xxxx

    xx

    x

    C. , :

    1. xxtg

    x

    5lim0

    2. xx

    x 4sin3sinlim

    0

    3. xxtg

    x 27lim

    0

    4. xx

    x 3sin8sinlim

    0

    5. xx

    x 37sinlim

    0

    6. x

    x

    x2

    sinlim

    0

    7. xtgx

    x 25lim

    0

    8. xx

    x 4sin9lim

    0

    9. xtgxtg

    x 65lim

    0

    10. xxctgx

    5lim0

    11. 2

    3lim0

    xctgxx

    12. xctgxtgx

    43lim0

    13. xctgxctg

    x 72lim

    0

    14. xctgx

    xtgx 3

    2lim 20

    15. 2

    2

    0 45sinlimxx

    x

    16. xtgxtg

    x 65lim

    0

    17. 2

    2

    0 53cos1lim

    xx

    x

  • 8

    18. x

    xx 4cos1

    7lim 23

    0

    19. xtgx

    x 75sinlim

    0

    20.

    3

    2lim2

    2

    0 xtg

    xx

    21. xxtg

    x 43lim

    0

    22. xctgxctg

    x 36lim

    0

    23. x

    xx 5

    2cos1lim2

    0

    24. x

    xx 3sin

    7lim2

    0

    25. xctgxtgx

    75lim0

    26. xctgx

    tgxx 43lim 20

    27. xtgx

    x 53sinlim

    2

    0

    28. 204

    sinlim

    x

    x

    x

    29. 20

    6lim xxctgx

    30. x

    xx 3cos1

    2lim 22

    0

    D. , :

    1. xxx

    x 2

    3

    0 sin34lim

    2. xtgx

    x 65sinlim

    0

    3. )1ln(

    3lim0 x

    xtgx +

    4. x

    ex

    x 4sin1lim

    0

    5. 1

    lim 20 xx ex

    6. )1ln(

    3sinlim0 + x

    xx

    7. )1ln(

    3lim0 x

    xtgx +

    8. xx

    x

    3sinlim0

    9. x

    xx ln

    1lim1

    10. 20cos1limx

    xx

  • 9

    11. xxxtgx

    x sinsinlim

    0

    12. xx

    x

    lnlim

    13. xtg

    tgx

    x 3lim

    2

    14. ctgxx

    x

    lnlim0

    15. 30sinlimx

    xxx

    16. bxax

    x cos1cos1lim

    0

    17. xee bxax

    x sinlim

    0

    18. 30lim

    xarctgxx

    x

    19. xtgx

    x 2cos1lim

    4

    20. 311lnlimxx

    x

    21. xx

    x 42sinlim

    0

    22. xctgx

    x 43lnlim

    0

    23. x

    e x

    x 3sin1lim

    2

    0

    24. )3ln(

    5sinlim0 + x

    xx

    25. 1

    2lim 32

    0 xx ex

    26. 230 232lim

    xxxarctgx

    x +

    27. x

    ex

    x 5sin1lim

    0

    28. 20 43cos1lim

    xx

    x

    29. xxxtgx

    x 3sin42sinlim

    0

    30. 2

    2

    04

    sinlim

    x

    x

    x

  • 10

    .. )(xfy = x

    )()( xfxxfy +=

    x , - :

    xy

    xxfxxf

    xx

    =

    + 00

    lim)()(lim

    , )(xfy = - x ; - . + , , )(xfy = x -, , - .

    y , )(xf , dxdy

    , dxxdf )(

    . -

    . A. :

    1. 2634

    24

    += xxxy

    2. xx

    xy += 33

    2 5

    3. 32 3121xxx

    y +=

    4. xx

    y 21 +=

    5. 2321 3 += xx

    y

    6. ( )2xxy += 7. ( )2xay +=

    8. 2

    311

    =x

    y

    9. 4 33 2

    53

    xxy =

    10. xtgxy cos32 =

    11. xexy 5sin7 +=

  • 11

    12. ctgxxy +=

    13. 5 32ln xxy +=

    14. 3 xxey x =

    15. 4

    cos1 xy =

    16. tgxxy 6sin2 =

    17. 3

    ln5

    xeyx=

    18. ctgxxy += 4

    19. xx

    ey x ln31

    21

    +=

    20. 32sinx

    xy =

    21. 43 2 xxy =

    22. 4

    ln xtgxy +=

    23. 5

    sin2 ctgxxy =

    24. 53 235 xxxy ++=

    25. xxey x sincos +=

    26. 2

    2 12

    +=

    xxy

    27. 2

    13

    +=

    xxy

    28. 44 53

    32 xxx

    y ++=

    29.

    2

    43 2

    11

    =xx

    y

    30. 3

    ln1 xx

    ey x ++=

    B. :

    1. xxy cossin =

    2. xxxy ln)( 2 +=

    3. xetgxy = 4. xctgxy cos=

    5.

    += xxxy

    2sin

    6. ( )2cossin xxy =

    7. 11

    +

    =xxy

    8. xxxy

    sin22

    =

    9. 2

    cos+

    =xxy

  • 12

    10. xtgxy =

    11. xxey

    x

    cos3 4

    =

    12. 1

    132

    2

    +

    =xxxy

    13. ( )2sin xexy += 14.

    ( )x

    xyln

    1 2=

    15. xxxey

    x

    cossin2

    =

    16. tgx

    xxy 3ln =

    17. xxtgxy cos+=

    18. xxy sin3 4=

    19. xexy 2)1( +=

    20. 14

    322 +

    =xxy

    21. ctgx

    xxy 4sin =

    22.

    += 2

    3cos xxxy

    23. tgxxy = sin 24. xctgxy cos=

    25. xexy 2)23( +=

    26. xxxtgxy

    2cos

    3 +

    =

    27. ( )2cos xexy += 28.

    ctgxxtgxey

    x

    =2

    29. 2sinxexyx

    =

    30. )( 2 xx

    ctgxy

    =

    C. :

    1. 4

    3sin3 xy =

    2. xy 5cos2=

    3. xxy 5cos3sin =

    4. ( )22cos3 xxtgy +=

    5. )2( 23 xxctgy =

    6. 3 3sin xy =

    7. )23( 2 += xxy

  • 13

    8. xxy

    4cos5sin

    =

    9. 1

    32

    +

    =

    xeey

    xx

    10. )23ln( 4 += xxy

    11. 5)14(sin += xy

    12. 3 2)3sin2(

    1

    xxy

    =

    13. 1cos1sin

    +

    =xxy

    14. )21ln(

    12

    xxy+

    =

    15. 7)4cos1(1x

    y+

    =

    16. )3ln(52

    2xxtgy +=

    17. )(2sin1

    2xxxy

    =

    18. 41

    1bx

    axy+

    =

    19. 125ln

    +

    =xxy

    20. 1cos

    )12sin(+

    =xxy

    21. 3)34( += xtgy

    22. xx

    y2cos

    )13sin( +=

    23. 3)2cos1(1

    xy

    +=

    24. )3sin(12ln

    xxy +=

    25. 3 4)32(ln

    1

    xtgxy

    =

    26. xxexy 324)3sin( =

    27. )ln(

    212

    4

    xxxy

    =

    28. xxtgy

    5cos31 2+

    =

    29. 3 3 32sin

    = xexy

    30. ))23ln(cos( 2 xxy =

  • 14

    -

    . )(xfy = (

    ) 0x , )( 0xf - . ..

    )(xfy = 0x ,

    )(xfy = 0=y . ..

    )(xfy = 0x 0x

    , , 0x ,

    0x :

    y + -, 0x - ;

    y - +, 0x - ;

    y , . A. :

    1. 21 xxy+

    =

    2. 211x

    y

    =

    3. 12

    =xxy

    4. 12

    2

    =xxy

    5. 322 )1(32 = xxy

    6. 241 xx

    y +=

    7. 22 1x

    xy +=

    8. 2

    3

    3 xxy

    =

    9. 3)1( xxy =

    10. 3

    2

    )1()1(

    +

    =xxy

  • 15

    11. 2)1(12

    =xxy

    12. 2

    3

    )1(2 +=

    xxy

    13. 43 )1( xxy =

    14. 32 )1( = xy

    15. )2)(1( 2 = xxxy

    16. xxy = 3 2

    17. 3223 )4( = xxy

    18. 422 xxy =

    19. xxy = 1

    20. 2

    3 2

    +

    =xxy

    21. 3

    1362

    +

    =xxx

    22. 3

    43xxy =

    23. 3 2)4(1 = xy

    24. )1(2 xxy =

    25. 345

    5xxxy +=

    26. )4)(1(

    =xxxy

    27. 23 )2( += xxy

    28. )1(

    )1(2

    2

    +

    =xxy

    29. )1(422 = xyx

    30. )2)(1(22 = xxyx

  • 16

    -

    . )(xFy =

    )(xfy = ,

    x )()( xfxF = ..

    . dxxf )( CxF +)( -

    )(xfy = ( C - ) :

    CxFdxxf += )()( . , - :

    = vduuvudv , )(xuu = )(xvv = - , -

    )(xu )(xv . . A. :

    1.

    + dx

    xxx 242 2

    2. + dxx 2)2(

    3.

    dx

    xx 13 2

    4. + dxxx3cossin

    5. xdx

    2cos

    6. + dxx

    x 23

  • 17

    7. + dxxx )( 3

    8. dxx

    x3

    22 )1(

    9.

    + dx

    xx 4 311

    10.

    dx

    xeex

    x21

    11.

    +

    dx

    xeex

    x2cos

    1

    12. dx

    xxctg

    2

    2

    cos23

    13.

    + dxxx 3253 2

    14. dx

    xxtg

    2

    2

    sin4

    15. dx

    xx

    2

    3

    coscos2

    16. dxxx

    3

    2)1(

    17. dxxxx

    4

    24 3 )(

    18.

    dxx

    x2

    31

    2

    19.

    ++ dx

    xxx 32321

    20. + dx

    x

    xxx3 2

    4 54 34 2

    21. xdx2sin

    3

    22. dxx

    x4

    22 )9(

    23. dxx

    x3

    23 2 )1(

    24. dxxxx

    3

    4 34 23

    25.

    dxxx

    2

    43

    25

    26.

    + dx

    xx

    23 2 1

    27.

    + dxxxx 32 31

    28.

    + dxxxx

    423 53

    29. dx

    xx

    2

    3

    sinsin1

  • 18

    30.

    + dxxxx

    23

    B. , :

    1. + dxx 4)23( 2. dxe x3

    3. dxx32sin

    4. xdx

    4cos2

    5. +13xdx

    6. dxx53

    7. + 34xdx

    8. + dxx3 57

    9. + )16(sin2 xdx

    10.

    dxee

    xx 23

    11. dxxctg )3(

    12. )92(cos2 xdx

    13. ( ) + dxx4 357

    14.

    dx

    xee

    xx

    3

    22 1

    15. + dxx 53

    )15(

    16. + dxxtgx )3cos(

    17. ++ dx

    xx

    )14(sin)14(sin5

    2

    3

    18.

    dxx

    xtg

    3sin

    33

    2

    2

    19.

    + dxx3

    2

    31

    25

    20. dxxtg )12(

    21. )12(sin3

    2 xdx

    22.

    + dxeexx

    3

    32

  • 19

    23. ( ) + dxx5 237

    24. + 3)14( x

    dx

    25. ( ) + dxx34

    53

    26. dxxctg )25( 27. dxx )14sin(

    28. xdx

    83

    29. 6)23(4xdx

    30.

    + dxx7

    5

    23

    C. :

    1. xdxxsin 2. xdxx 2cos 3. dxxe x3 4. xdxx 2sin)4( 5. dxxe x

    6. dxxx 2sin 7. dxxx )13cos( 8. xdxx 5sin2 9. dxex x22

    10. xdxln 11. dxxx )1ln( 12. + xdxx sin)3( 13. xdxx cos)2( 14. dxex x2)5( 15. dxxx )52sin(2 16. + dxxx )14cos(2 17. + dxx )1ln( 2 18. + dxxx )1ln(2

  • 20

    19. dxxx 2sin2

    20. dxex x2)13( 21. xdxx 5sin)42(

    22. dxxe

    x

    2

    4

    23. + dxxx )57cos(

    24. xxdx

    4cos2

    25. xxdx

    5sin 2

    26. + dxex x3)42( 27. xdxx 4cos)3( 28. + dxex x2)4( 29. + dxex x2)1( 30. xdxx 3sin7 2

    D. - :

    1. + dx

    xx

    92

    2

    2. + dxxxxx4

    143

    2

    3. ++ dxxxx

    )12)(1(

    4. ++ dxxxx

    xx)4)(3)(1(

    91412 2

    5. dx

    xx

    163

    2

    6. + dxxxxx

    254

    3

    2

    7. + dx

    xx

    9)2(

    2

    8. dxxxx

    232 2

    9. dxxx

    x3

    2

    41

    10. + dxxxx

    23 24

    11. + dxxx

    x65

    5224

    2

    12. +++ dxxxxxx

    )12(23

    2

    2

    13. dxxxx

    )64(3

    2

    2

    14.

    +

    xdx

    xx 2

    12

  • 21

    15. + dxxx

    x23

    3 1

    16. + dxxx

    x)2)(3(

    12

    17. + dxxx

    xx4

    83

    45

    18. ++ dxxxxx

    22

    2

    )1)(9(62

    19. +++ dx

    xxxxxx

    22

    23

    )2()3(1425

    20. dxxxx

    )1()1( 225

    21. dxxx

    x)9)(2(

    4

    22. ++ dxxxx

    272

    2

    23. + dxxxxx

    3

    2 323

    24. + dxxx

    x2

    3)1(

    25. dxaxx

    33

    23

    26. +++ dxxx

    x102

    252

    27. ++ dxxxx

    xx23

    2

    2152

    28. + dx

    xxxx

    5215723

    29. +++ dxxx

    x44

    124

    30. ++ ))(( bxaxdx

  • 22

    -

    . )(xf = -

    ],[ ba , - - :

    )()()( aFbFdxxfb

    a

    = ,

    )(xF )(xf =

    ],[ ba . A. :

    1. +1

    0

    2 )123( dxxx

    2. 4

    12xdx

    3. 1

    43xdx

    4.

    2

    2

    )1(cos dxx

    5. 9

    1

    )(3 dxxx

    6.

    +

    2

    1

    21 dxx

    x

    7. +9

    4

    )1( dxxx

    8. 2

    1

    3 )( dxxx

    9. +3

    0

    )1( dxex

    10. 8

    13 2x

    dx

    11.

    +

    2

    14

    2 1 dxx

    x

    12. a

    dxaxx0

    2 )(

  • 23

    13.

    +

    8

    13 2

    3 2 1 dxx

    x

    14. +4

    0

    2)1( dxx

    15.

    +0

    )3(sin dxx

    16.

    6

    8

    2 2cos xdx

    17. +

    1

    03 4)83( x

    dx

    18. +1

    0

    1 dxx

    19. a

    aaxdx

    2

    2

    20.

    13

    25 4)3( x

    dx

    21. 3

    0

    3sin xdx

    22. dxex

    3

    0

    3

    23.

    1

    024 x

    dx

    24. 2

    1

    22 dxx

    25. 4

    0

    4sin xdx

    26. +1

    014x

    dx

    27.

    6

    9

    2 3sin xdx

    28. +

    1

    03)31( x

    dx

    29. +8

    3

    1 dxx

    30. dxee xx

    4

    0

    24 2

  • 24

    B. - :

    1. 1

    0

    dxxe x

    2. +3

    22 232 xxdx

    3.

    3

    4

    2sin xxdx

    4. +2

    12 127xxdx

    5.

    +1

    0

    )1ln(e

    dxx

    6. 2

    12 54xxdx

    7.

    3

    4

    2cos xxdx

    8. 3

    12 43xxdx

    9.

    0

    sin xdxx

    10.

    +3

    0

    )3ln(e

    dxx

    11. ++1

    02 56xxdx

    12. 8

    02 2sin xxdx

    13. +4

    12 23 xxdx

    14. 2

    0

    2dxxex

    15. 1

    31

    2 16 xxdx

    16. 2

    0

    cos xdxx

    17. 2

    12 32xxdx

  • 25

    18.

    0

    2 sin xdxx

    19. ++1

    02 3103 xxdx

    20. 4

    2

    22 dxex x

    21. ++3

    22 78xxdx

    22. 2

    0

    2sin xdxx

    23. +

    4

    23

    2

    952 dxxxxx

    24.

    +5

    0

    )5ln(e

    dxx

    25. ++5

    22 363 xxdx

    26.

    +4

    0

    )4ln(e

    dxx

    27. ++

    3

    12 134

    12 dxxx

    x

    28.

    +4

    24

    cos)1( dxxx

    29. +++2

    1)3)(2)(1( xxx

    dx

    30. ++3

    22 12xxdx

  • 26

    , )(xyy = ,

    1xx = 2xx = , =2

    1

    )(x

    x

    dxxyS

    , )(yxx = ,

    1yy = 2yy = , =2

    1

    )(y

    y

    dyyxS

    , , -

    )(xyy = , 1xx = 2xx = ,

    =2

    1

    2 )(x

    x

    dxxyV

    , , -

    )(yxx = , 1yy = 2yy = ,

    =2

    1

    2 )(y

    y

    dyyxV

    A. , :

    1. 0,42 =+= yxy

    2. 0,32 =+= xyx

    3. 0,4,1,4 ==== yxxxy

    4. 0,2,1,22 ===+= yxxxy

    5. 0,,0,sin ==== yxxxy

    6. 0,5 2 ==+ xyx

  • 27

    7. 01,122 =+= yxxy

    8. xyxy == 8,7

    9. xyxy == 2,2

    10. 2,4 2 +== xyxy

    11. 2,42 == xyyx

    12. xyxy == 5,4

    13. 22 2, xyxy =

    14. 0,422 =+= xxy

    15. 0,8,32 === xyxy

    16. 0,6,1,

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