Программа и экзаменационные задачи по математическому анализу для студентов 1 курса

Fbgbkl_jkl\hh[s_]hbijhn_kkbhgZevgh]hh[jZah\ZgbyJN ey bkke_^h\Zgby g_ij_ju\ghklb y′(x ) \ lhqd_ x0 = 0 jZkk...

Author: Мешков В.З. | Половинкин И.П. | Попов А.В. | Астахов А.Т.

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Fbgbkl_jkl\hh[s_]hbijhn_kkbhgZevgh]hh[jZah\ZgbyJN
NZdmevl_lijbdeZ^ghcfZl_fZlbdbbf_oZgbdb DZn_^jZ^bnn_j_gpbZevguomjZ\g_gbc

Ijh]jZffZbwdaZf_gZpbhggu_aZ^Zqb ihfZl_fZlbq_kdhfmZgZebam ^eyklm^_glh\dmjkZnZdmevl_lZIFF

KhklZ\bl_eb <AF_rdh\ BIIheh\bgdbg :<Ihidh\ :L:klZoh\

IGZqZevgu_k\_^_gbybal_hjbbfgh`_kl\bfZl_fZlbq_kdhceh]bdb 1. Fgh`_kl\Zbhi_jZpbbgZ^gbfbK\hckl\Zhi_jZpbcgZ^fgh`_kl\Zfb 2. \Zhij_^_e_gbybbo wd\b\Ze_glghklv G_h[oh^bfh_ b ^hklZlhqgh_ mkeh\b_ kms_kl\h\Zgby ij_^_eZ qbkeh\hc ihke_^h\Zl_evghklb IVIj_^_ebg_ij_ju\ghklvnmgdpbb 16. Ij_^_evgu_ ]jZgbqgu_ bahebjh\Zggu_ \gmlj_ggb_ lhqdb fgh`_kl\ L_hj_fZ ;hevpZgh-<_c_jrljZkkZ\l_jfbgZoij_^_evguolhq_dfgh`_kl\ 17. Hij_^_e_gb_ij_^_eZnmgdpbb\lhqd_ih=_cg_bihDhrbbbowd\b\Ze_glghklv 18. G_ij_ju\ghklv nmgdpbb \ lhqd_ JZaebqgu_ nhjfu aZibkb mkeh\by g_ij_ju\ghklb H^ghklhjhggb_ ij_^_eu b h^ghklhjhggyy g_ij_ju\ghklv G_h[oh^bfh_ b ^hklZlhqgh_ mkeh\b_ kms_kl\h\Zgby ij_^_eZ nmgdpbb \ lhqd_ :jbnf_lbq_kdb_ k\hckl\Z ij_^_eh\nmgdpbc 19. Ij_^_ebg_ij_ju\ghklvkmi_jihabpbbnmgdpbc 20. Djbl_jbcDhrbkms_kl\h\Zgbyij_^_eZnmgdpbb\lhqd_ 21. I_j\ucaZf_qZl_evgucij_^_e

22. bnn_j_gpbZevgh_bkqbke_gb_nmgdpbc\_s_kl\_gghci_j_f_gghc 36. Ijhba\h^gZy nmgdpbb \ lhqd_ >bnn_j_gpbjm_fZy \ lhqd_ nmgdpby G_h[oh^bfh_b^hklZlhqgh_mkeh\b_^bnn_j_gpbjm_fhklbnmgdpbb\lhqd_>bnn_j_gpbZe nmgdpbb\lhqd_G_ij_ju\ghklv^bnn_j_gpbjm_fhcnmgdpbbH^ghklhjhggb_ijhba\h^gu_ 37. =_hf_ljbq_kdbc kfuke ijhba\h^ghc b ^bnn_j_gpbZeZ nmgdpbb \ lhqd_ MjZ\g_gb_dZkZl_evghcd]jZnbdmnmgdpbb\lhqd_Nbabq_kdbckfukeijhba\h^ghc 38. Ijhba\h^gu_kmffuijhba\_^_gbybqZklgh]h^bnn_j_gpbjm_fuonmgdpbc 39. Ijhba\h^gZyh[jZlghcnmgdpbb 40. Ijhba\h^gZyb^bnn_j_gpbZekeh`ghcnmgdpbbBg\ZjbZglghklvnhjfui_j\h]h ^bnn_j_gpbZeZhlghkbl_evgh\u[hjZi_j_f_gguo 41. Ijhba\h^gu_ihdZaZl_evghceh]Zjbnfbq_kdhcbkl_i_gghcnmgdpbc 42. Ijhba\h^gu_ljb]hghf_ljbq_kdboh[jZlguoljb]hghf_ljbq_kdbonmgdpbc 43. Ijhba\h^gu_]bi_j[hebq_kdbobh[jZlguo]bi_j[hebq_kdbonmgdpbc 44. Hij_^_e_gby ijhba\h^guo \ukrbo ihjy^dh\ Ijhba\h^gu_ \ukrbo ihjy^dh\ kmffubijhba\_^_gbynmgdpbcNhjfmeZE_c[gbpZ 45. >bnn_j_gpbZeu \ukrbo ihjy^dh\ Hlkmlkl\b_ bg\ZjbZglghklb nhjfu ^bnn_j_gpbZeh\\ukrboihjy^dh\ 46. EhdZevguc wdklj_fmf nmgdpbb L_hj_fZ N_jfZ h g_h[oh^bfhf mkeh\bb ehdZevgh]hwdklj_fmfZ 47. L_hj_fZJheey 48. L_hj_fZEZ]jZg`Zhdhg_qguoijbjZs_gbyo

L_hj_fZDhrbhdhg_qguoijbjZs_gbyo JZkdjulb_g_hij_^_e_gghkl_cihijZ\bemEhiblZey Fgh]hqe_gL_cehjZNhjfmeZL_cehjZkhklZlhqgufqe_ghf\nhjf_I_Zgh NhjfmeZL_cehjZkhklZlhqgufqe_ghf\nhjf_EZ]jZg`Z NhjfmeZL_cehjZkhklZlhqgufqe_ghf\nhjf_Dhrb JZaeh`_gb_ihnhjfme_L_cehjZFZdehj_gZ we_f_glZjguonmgdpbc m x y = e , y = sin x, y = cos x, y = (1 + x ) (m ≠ 1,2 ), y = ln (1 + x ) hklZlhqgh_mkeh\b_kljh]hcfhghlhgghklb 56. L_hj_fZ h kf_g_ agZdZ ijhba\h^ghc \ hdj_klghklb lhqdb >hklZlhqgu_ mkeh\by fhghlhgghklb\hdj_klghklblhqdbehdZevgh]hwdklj_fmfZ 57. IjbagZdbwdklj_fmfZnmgdpbbbf_xs_cijhba\h^gu_\ukrboihjy^dh\ 58. hklZlhqgh_mkeh\b_\uimdehklb 59. LhqdZi_j_]b[Z>hklZlhqgh_mkeh\b_i_j_]b[Z 60. :kbfilhluMjZ\g_gb_Zkbfilhl 61. Bkke_^h\Zgb_nmgdpbc 62. NmgdpbbaZ^Zggu_iZjZf_ljbq_kdb
Ijbf_juj_r_gbyaZ^Zqij_^eZ]Z\rbokygZwdaZf_g_ihfZl_fZlbq_kdhfmZgZebam x n _keb x n = Ijbf_jGZclb lim n →∞

n n

8 −1 16 − 1

.

n n a = 1 _keb a > 1 H[hagZqbf 0 < a n = n a − 1, ⇒ a = (1 + a n ) ≥ na n J_r_gb_: IhdZ`_fqlh lim n →∞

a n

ih g_jZ\_gkl\m ;_jgmeeb Hlkx^Z ke_^m_l 0 < a n ≤ AgZqbl ih l_hj_f_ h lj_o a n = 0 , ⇒ lim n a = lim(1 + a n ) = 1 Imklv z n = n 2 , lim z n = 1 ih ihke_^h\Zl_evghklyo lim n →∞ n →∞ n→ ∞ n →∞ n

^hdZaZgghfm lim n n →∞

Hl\_l

lim x n = n →∞

( 2 ) − 1 = lim z = lim z 16 − 1 ( 2) −1 8 −1

n

n →∞ n

3 4

n →∞

3 n 4 n

(

)

(z − 1) z n2 + z n + 1 3 −1 = lim n = . − 1 n→∞ (z n − 1) z n2 + 1 (z n + 1) 4

(

)

3 . 4

sin (k!) . k =1 k ⋅ (k + 1) n

Ijbf_jBkke_^h\ZlvgZkoh^bfhklvdjbl_jbcDhrb _keb x n = ∑ J_r_gb_>ey ∀ ε > 0 b ∀ p ∈ N bf__f xn+ p − xn =

sin((n + p )!) + (n + 1)(n + 2)

sin ((n + 2)!) sin ((n + p )!) 1 1 ++ ≤ + + (n + 2)(n + 3) (n + p )(n + p + 1) (n + 1)(n + 2) (n + 2)(n + 3) 1 1 1 1 1 1 1 + = − + − ++ − = (n + p )(n + p + 1) n + 1 n + 2 n + 2 n + 3 n + p n + p +1 1 1 1 1 = − < < ε ijb Ke_^h\Zl_evgh x n + p − x n < n +1 n + p +1 n +1 n +1 +

1 1 − 1 = N (ε ) bex[hfgZlmjZevghf p BlZd ∀ ε > 0 ∃N = N (ε ) = − 1 ε ε ∀ n ≥ N ∀ p ∈ N : x n + p − x n < ε Mkeh\b_ Dhrb \uiheg_gh b ihwlhfm ^ZggZy qbkeh\Zy ∀n >

ihke_^h\Zl_evghklvkoh^blky 1

 sin x  x 2 Ijbf_jIjbf_gyyijZ\behEhiblZeygZclb lim   . x →0  x  J_r_gb_ A^_kv g_hij_^_e_gghklv \b^Z 1∞ Ijbf_gyy nhjfmem u v = e v ln u

(u > 0, v > 0) b

ihevamykv \hafh`ghklvx i_j_oh^Z d ij_^_em \ ihdZaZl_e_ nmgdpbb e x gZoh^bf 1

1

sin x

ln lim z ( x ) 2  sin x  x 2 lim  = lim e x x = e x →0 , x →0 x →0  x  x  x cos x − sin x  sin x ⋅  ln sin x  x2  = lim 1 ⋅ x ⋅ x cos x − sin x  = x = lim z (x ) = lim lim   2 0 → x →0 x →0 x x → 0 2 x sin x 2x x x2   x cos x − sin x 1 1 1 sin x 1  − x sin x  = lim = lim = − lim =− .  3 2 2 x →0 2 x →0  2 x 6 x →0 x 6 x  1

1

−  sin x  x 2 =e 6. Ke_^h\Zl_evgh lim   x →0  x  1

1

−  sin x  x 2 =e 6. Hl\_l lim   x →0  x 

Ijbf_jFh`gheb^hhij_^_eblvnmgdpbx y = f (x )\lhqd_jZaju\Z x0 lZdqlh[uhgZklZeZ 1

sin x  x 2 g_ij_ju\ghc\wlhclhqd__keb y =   , x0 = 0 ?  x  1

1

− sin x  x 2 = e 6 kf j_r_gb_ ijbf_jZ lh nmgdpby y = f (x ) [m^_l J_r_gb_ LZd dZd lim0   x→  x  1  x sin x   2 1    , x≠0 − g_ij_ju\ghc \ lhqd_ x0 = 0 _keb iheh`blv f (0) = e 6 BlZd nmgdpby y =  x   −1 e 6 , x=0

g_ij_ju\gZ\lhqd_ x0 = 0 .

 x 2 ⋅ sin x , x ≠ 0 ′ Ijbf_j GZclb ijhba\h^gmx y (x ) nmgdpbb y =  bkke_^h\Zlv y\ey_lky eb x=0 0 , y ′(x ) g_ij_ju\ghc\lhqd_ x0 = 0 ?

J_r_gb_ ?keb x ≠ 0 lh y ′(x ) = 2 x sin − cos AgZq_gb_ y ′(0) \uqbkebf ih hij_^_e_gbx 1 x

1 x

1 ∆y = lim ∆x ⋅ sin = 0 BlZd nmgdpby y (x ) ^bnn_j_gpbjm_fZ gZ \k_c qbkeh\hc ∆x → 0 ∆x ∆x → 0 ∆x 1 1  2 x sin − cos , x ≠ 0 ijyfhcb y ′(x ) =  . x x 0 , x=0 y ′(0 ) = lim

y ′(x ) Hq_\b^gh qlh >ey bkke_^h\Zgby g_ij_ju\ghklb y′(x ) \ lhqd_ x0 = 0 jZkkfhljbf lim x →0

1 = 0 ijhba\_^_gb_ [_kdhg_qgh fZehc nmgdpbb gZ h]jZgbq_ggmx _klv [_kdhg_qgh x 1 cos g_kms_kl\m_l^ey^hdZaZl_evkl\Z\hkihevah\Zlvkyhij_^_e_gb_f fZeZynmgdpby Z lim x →0 x ij_^_eZnmgdpbbih=_cg_gZyaud_ihke_^h\Zl_evghkl_c LZdbfh[jZahf y′(x ) jZaju\gZ\ lim 2 x ⋅ sin x →0

lhqd_ x0 = 0 dhlhjZyy\ey_lkylhqdhcjZaju\Zjh^Znmgdpbb y′(x ) . Ijbf_jBkke_^h\ZlvgZ^bnn_j_gpbjm_fhklvnmgdpbx y = arcsin (cos x ) . J_r_gb_ Nmgdpby y (x ) = arcsin (cos x ) hij_^_e_gZ gZ \k_c qbkeh\hc ijyfhc i_jbh^bq_kdZy k

i_jbh^hf 2π h[eZklvagZq_gbc y ≤

π ]jZnbdgZjbk 2

y π 2

y = arcsin(cos x ) −π −

π 2

π 2

−

Ijbf_gyy

ijZ\beh

y ′(x ) =

=

− sin x

1 − cos 2 x

π

0

3π 2

π 2

jbk

^bnn_j_gpbjh\Zgby

− sin x = − sgn (sin x ) , sin x

x

(x ≠ kπ ,

keh`ghc

nmgdpbb

ihemqbf

k ∈ Z ).

< lhqdZo [ = N π , N ∈Ζ ih hij_^_e_gbx h^ghklhjhggbo ijhba\h^guo bf__f \(N π + ∆[ ) − \(N π ) \ +′ = OLP

∆[ →+

∆[

=

= OLP

∆[ →+

= OLP

DUFVLQ [FRV(N π + ∆[ )] − DUFVLQ[FRV(N π )] = ∆[

∆[ →+

= OLP

[

∆[

]

[

DUFVLQ (− ) FRV(∆[ ) − DUFVLQ (− )

∆[ →+

(− ) N +

= OLP

[

DUFVLQ (FRV(N π ) ⋅ FRV(∆[ ) − VLQ (N π ) ⋅ VLQ (∆[ )) − DUFVLQ (− )

∆[ →+

N

N

]=

N

]=

(− ) N + DUFVLQ VLQ (∆[ )

OLP ∆[ →+ ∆[ DUFVLQ VLQ (∆[ ) VLQ (∆[ ) N+ ⋅ = (− ) . ∆[ VLQ (∆[ )

∆[

=

DUFVLQ[FRV (N π + ∆[ )] − DUFVLQ [FRV(N π )] N+ = − (− ) . ∆[ LZd dZd \ +′ (N π ) ≠ \ −′ (N π ) N ∈ Ζ lh \ lhqdZo [ = N π nmgdpby \([ ) = DUFVLQ (FRV [ ) g_

:gZeh]bqgh \ −′ (N π ) = ∆OLP [ →−

^bnn_j_gpbjm_fZ AZf_qZgb_ AZ^Zqm fh`gh j_rblv b ^jm]bf kihkh[hf Z bf_ggh ihdZaZlv qlh \ +′ () ≠ \ −′ () \ +′ (π ) ≠ \ −′ (π ) b\hkihevah\Zlvkyk\hckl\hfi_jbh^bqghklbnmgdpbb \([ ) . Hl\_lNmgdpby \([ ) = DUFVLQ (FRV [ ) ^bnn_j_gpbjm_fZgZ\k_cqbkeh\hcijyfhcdjhf_lhq_d [ = N π N ∈Ζ . Ijbf_jGZclb \ ( Q) ([ ) ^eyke_^mxsbonmgdpbc Z \([ ) = VLQ [ + FRV [ ; [ \([ ) = VLQ [ ⋅ VLQ [ ; \ \([ ) = [ ⋅ FRV [ . J_r_gb_Z Ihke_lh`^_kl\_gguoij_h[jZah\Zgbcihemqbf

− FRV [ \([ ) = (VLQ [ + FRV [ ) − VLQ [ ⋅ FRV [ = − VLQ [ = =− = + FRV [ π (FRV α [ )( Q) = α Q FRV α [ + πQ  lh \ ( Q) ([ ) = Q− FRV  [ + Q  . − FRV [ VLQ [ = VLQ [ − VLQ [ . [ Bf__f \([ ) = VLQ [ ⋅ VLQ [ = π Q π Q π Q ( Q)    LZddZd (VLQ α [ ) = α Q VLQ  α [ +  lh \ ( Q) ([ ) = Q− VLQ  [ +  − Q− VLQ  [ +  .      

LZd

dZd

\ Ijbf_gbf nhjfmem E_c[gbpZ (X Y) \(

)

([ ) = & Q [ (FRV [ ) + & Q ([

( Q)

)′ (FRV [ )(

Q −)

( Q)

+ & Q ([

Q

= ∑ FQN X ( Q− N ) Y( N ) iheh`b\ \ g_c X = FRV [ Y = [ N =

)″ (FRV [ )(

Q−)

=

  π Q π (Q − ) π (Q − )  = Q ⋅ [ ⋅ FRV  [ +  + [ Q ⋅ Q − FRV  [ +  + + Q (Q − ) ⋅ FRV  [ + =      

π Q π Q   πn  = Q ⋅ [ FRV [ +  + Q ⋅ Q[ ⋅ VLQ  [ +  − n(n − 1)2 n − 2 cos  2 x + =     2    π Q π Q Q(Q − )   = Q  [ −  FRV [ +  + Q ⋅ Q[ ⋅ VLQ  [ +  .      

π Q  ;  π Q π Q   VLQ  [ +  − Q− VLQ  [ +  ;      

Hl\_lZ \ ( Q) ([ ) = Q− FRV  [ +

[ \ ( Q) ([ ) = Q−

 

\ \ ( Q) ([ ) = Q  [ −

Q(Q − ) π Q π Q    FRV [ +  + Q ⋅ Q[ ⋅ VLQ  [ + .     

AZ^Zqb^eykZfhklhyl_evgh]hj_r_gby I.Ij_^_eqbkeh\hcihke_^h\Zl_evghklb [ Q _keb 1. GZclb OLP Q→∞ Q + Q Z [ Q = 0,7+0,29+...+ [ [ Q = Q

\ [ Q = ^ [ Q = ` [ Q = b

−

Q Q

−

Q

Q

−

−

Q

Q − Q +

Q

Q

Q+ Q+ Q

[Q =

Q +

;

Q

Q + Q Q −

Q

 Q +Q     Q + Q + 

] [ Q =

−

Q + + Q + ; Q +

+ Q _ [ Q = Q

Q

;

;

a [ Q = Q + Q − Q − Q ;

;

d [ Q =

(Q) QQ

GZclb\_jogbcbgb`gbcij_^_eu

;

πQ πQ [ [ Q = + VLQ ; Q πQ Q Q \ [ Q = Q − + ; ] [ Q = + ; FRV Q + Q πQ Q π Q   FRV _ [ Q =  +  ⋅ (− ) Q + VLQ ; ^ [ Q =  Q + Q

Z [ Q = DUFWJ Q ⋅ FRV

Bkke_^h\ZlvgZkoh^bfhklvdjbl_jbcDhrb Q Q Q FRV (N ) ; \ [ Q = ∑ ; Z [ Q = ∑ [ [ Q = ∑ N N= N N = N ⋅ (N + ) N=

Q

] [ Q = ∑ N = Q

` [ Q = ∑ N =

Q

VLQ N FRV N ^ [ Q = ∑ N N N=

(− ) N ⋅ N − FRV π N

Q

; a [ Q = ∑ N =

_ [ Q = ∑

(− ) N ⋅ N N

Q

N =

;

,,Ij_^_enmgdpbb

; N

FRV [ g_kms_kl\m_l >hdZaZlvqlh [OLP →+∞ I ([ ) _keb Kms_kl\m_leb OLP [ →D [

Z I ([ ) = ([ + ) ⋅ VLQ Z [ I ([ ) = VLQ [ ⋅ VLQ

, Z [

 [  [ + [ [ >   ( ) \ I [ =  Z ] I ([ ) = DUFFWJ   + [ ⋅ VLQ Z [ [  − FRV [ [ <  [   ^ I ([ ) = DUFWJ   + [ ⋅ FRV [ ⋅ FRV Z [ [

GZclbij_^_eunmgdpbc [



 [

[

π [



Z OLP  VLQ + FRV  [ OLP [ ⋅ OQ  FRV  \ OLP (FRV [ ) ( + [→ [ →∞  [ →∞    H[ − H−

] OLP [→

[

[ + VLQ ( [

  ( + [ ) [ ` OLP [→  H 

)

^ OLP [→

[

[

)

;

− FRV [ _ OLP [ FRV [ ;

[ [ ( + [ ) [ − H    OLP OLP DUFWJ [ a b   ;  [ →+∞  π [→  [ 

[ →

 DUFVLQ [  [ d OLP   e OLP + [ − [ [→  [→  [

g OLP [→

OQ VLQ

(

)

[

  [ ⋅  [ − [ +  ; f OLP [ →∞  

[ [ − ( + WJ [ ) ( + h [OLP ([ + [ ) [ i OLP →+∞ [→ OQ [

OQ [

) ;

VLQ (H [ − − ) [ k OLP l OLP ; j [OLP →+ [ → OQ ( + FRV [ ) [→ OQ [ OQ FRV [

[

m OLP [→

[

+ [ VLQ [ − FRV [

n OLP [ →

+ WJ [ − + VLQ [ [

.

,,,G_ij_ju\ghklvnmgdpbb IjbdZdhfagZq_gbbiZjZf_ljZanmgdpby f(x) , [m^_lg_ij_ju\gZ_keb (( + [ ) Q − ) [ [ ≠ Q ∈ Ν [ OQ ( + [ ) [ ≠ Z I ([ ) =  [ I ([ ) =  ; D

[ =

D

[ =

 [ [ ( − ) [ ≠ (VK [ ) [ + [ VLQ [ ≠ [ \ I ([ ) =  ] I ([ ) =  ; [ = D D [ =

GZclb lhqdb jZaju\Z nmgdpbb mklZgh\blv bo jh^ ^hhij_^_eblv nmgdpbx ih g_ij_ju\ghklb\lhqdZomkljZgbfh]hjZaju\Z

 [ − ([ + ) [ VLQ [ ≠ Z \ = [ \ =  [ \ \ = ; ([ − ) − [ VLQ π [  [ = ] \ = VJQ (VLQ [ ) ^ \ = VJQ (FRV [ ) _ \ = ; [ −[ − H  VLQ [ − [ ≠ [⋅  [ a \ = H [ b \ = H ; ` \ =  [  [ = 

VLQ

VLQ

d \ = H g \ =

[

[

e \ = [

−

⋅H

−

f \ =

[

π [  FRV    

VLQ [ h \ = VLQ [ [ −[

FRV

DUFVLQ [ ; VLQ [

;

Fh`gh eb ^hhij_^_eblv nmgdpbx y=f(x) \ lhqd_ jZaju\Z [ g_ij_ju\ghc\wlhclhqd_ Z \ = [ + [ [ = [ \ = [

−

 VLQ [    [ 

\ \ = 

[

[ = ] \ = [

⋅H

−

lZd qlh[u hgZ klZeZ

[

[ = ;

− [

[ = ;

Ijb dZdbo agZq_gbyo iZjZf_ljh\ a b b nmgdpby f(x) g_ij_ju\gZ gZ k\h_c h[eZklb hij_^_e_gby_keb ([ − ) [ ≤ [ [ ≤  Z I ([ ) = D[ + E < [ < [ I ([ ) =  ;

[ + D[ + E [ >  [ ≥  [ [ FRV([ ) VLQ [ [ ∈[− π π ] [ ≠ [ ≠ π  . \ I ([ ) = D [ = E [ = π 

IVIjhba\h^gZynmgdpbb IjZ\beZ^bnn_j_gpbjh\Zgby GZclb h^ghklhjhggb_ ijhba\h^gu_ nmgdpbb I ([ ) = [ − [ ⋅ J([ ) \ lhqd_ [ ]^_ J([ ) g_ij_ju\gZy\lhqd_ [ nmgdpbyBf__lebnmgdpby I ([ ) ijhba\h^gmx\lhqd_ [ ? 2GZclbijhba\h^gmx \ ′([ ) nmgdpbb

 [ ⋅ VLQ \([ ) =  [ 

ij b [ ≠ ij b [ =

bbkke_^h\Zlvy\ey_lkyeb \ ′([ ) g_ij_ju\ghc\lhqd_ [ =0.

GZclb qbkeZ D E D E lZd qlh[u nmgdpby y=f(x) [ueZ ^bnn_j_gpbjm_fZ gZ \k_c qbkeh\hcijyfhc

[>

π D [ + E [ >  π π  [ ∈ −  [ \ = [ ⋅ VLQ π [ [ ∈[− ] ;    D [ + E [ < − π [ <− [< H − H ≤[ ≤H ; [ >H

 D [ + E   Z \ = FRV [   D [ + E   D [ + E  \ \ = [ OQ [ D [ + E  

Bkke_^h\ZlvgZ^bnn_j_gpbjm_fhklvke_^mxsb_nmgdpbb Z \ = π − [ ⋅ VLQ [ [ \ = [ ⋅ [ \ \ = VLQ [ ; ] \ = DUFVLQ ( VLQ [ ) ^ \ = DUFVLQ ( FRV [ ) _ \ = DUFFRV ( FRV [ ) ;  π [ ⋅ FRV _keb [ ≠ [ a \ = ([ + ) ` \ =   _keb [ = 

b \ = DUFFRV

− [ d \ = + [

([ + )

− ([ − )

;

;

GZclbh^ghklhjhggb_ijhba\h^gu_\mdZaZgguolhqdZo^eyke_^mxsbonmgdpbc π [= N ∈ Ζ [ I ([ ) = VLQ [ [ = ; [ N + [ \ I ([ ) = VLQ [ [ = [ = π ] I ([ ) = DUFVLQ [ = ± . + [

Z I ([ ) = [ ⋅ FRV

Bkke_^h\ZlvgZ^bnn_j_gpbjm_fhklv\lhqd_x=0 nmgdpbx   [ − H [ − _keb [ ≠ I ([ ) =   _keb [ = 

GZclbijhba\h^gu_nmgdpbc

[

 VLQ [   + ORJ [ ;  [ 

Z \ = [ + [ [ + [ [ [ \ = FK [ H + [ [ \ \ =  [

[

] \ = ORJ [ ⋅ ORJ [ H + OQ ⋅ ORJ [ ⋅ [ ^ \ = [ H

;

DUFWJ [

_ \ = ( FRV[ )

H[

;

` \ = [ H + ( [ + ) ; 8. Bkoh^ybahij_^_e_gbyijhba\h^ghc, gZclb I ′ ( ) , _keb FWJ [

WJ [

    [ [ ⋅ FRV VLQ  [ ⋅ VLQ  [ ≠ + [ ≠ [ [ I [ =  Z I ([ ) =   [   [ = [ =  

;

 H [ − FRV [  OQ FRV [  [ ≠  [ ≠ \ I [ =  [ ] I [ =  ; [   [ = [ =  GZclb G \ ^eynmgdpbb \ = VLQ [ _keb

Z [ -g_aZ\bkbfZyi_j_f_ggZy[ [ -nmgdpbyg_dhlhjhci_j_f_gghc GZclb \ ( Q) ([ ) ^eyke_^mxsbonmgdpbc [ \ = H [ ⋅ VLQ [ \ \ = ([ + ) ⋅ H [ ; Z \ = VLQ [ + FRV [ ; ] \ = [ ⋅ (VLQ [ + FRV [ ) ^ \ = [ ORJ ( − [ ) _ \ = ([ − )⋅ [ ⋅ [ ;

[ a \ = ( + [ ) − [ b \ = VLQ [ ⋅ VLQ [ ; d \ = ([ + [ ) ⋅ FRV [ e \ = ([ + [ )(VLQ [ − FRV [ ) f \ = [ − [ .

` \ = [ FRV

V.NhjfmeZL_cehjZ Ijbf_g_gb_ijhba\h^ghcdbkke_^h\Zgbxnmgdpbc 1. JZaeh`blvihnhjfme_FZdehj_gZke_^mxsb_nmgdpbb Z I ([ ) = FRV [ + VLQ [ ^h R([ Q+ ) [ I ([ ) = H [ [ ^h R([

\ I ([ ) = OQ

+ [ −[

VLQ

^h R([ Q ) ] I ([ ) =

−

)

;

^h R( [ Q ) ;

− [

^ I ([ ) = OQ ( − [ ) ^h R([ ) _ I ([ ) = ([ +) Bkihevamyhkgh\gu_jZaeh`_gbygZclbij_^_eu Q

^h R([

[

)

.

( + [ ) − H H [ VLQ [ − [ ( + [ ) FRV [ − H [ OLP \ OLP ; Z OLP [→ [→ [→ [ [ [ π  FRV  FRV [  [ H [ + OQ ( − [ )   H [ − + [ OLP OLP ] OLP ^ _ . [→ [→ [→ [ FRV [ − VLQ [ OQ FRV [ VLQ (VLQ [ )

[

Bkke_^h\ZlvgZwdklj_fmfnmgdpbb Z \ = DUFVLQ (VLQ [ ) [ \ = ([ + ) − ([ − ) \ \ = VLQ [ + FRV [ ;

[ + [ − [ ` \ = [ OQ [ a \ = [ OQ [ b \ = DUFFRV + [ − [− d \ = [ ⋅ H e \ = VLQ [ + FRV [ f \ = [ − ⋅ H [ .

] \ = ([ + ) H [ ^ \ = ([ − ) H − [ _ \ = DUFVLQ

GZclbbgl_j\Zeu\uimdehklbblhqdbi_j_]b[Z]jZnbdZnmgdpbb Z

;

\ = [ + [ \ =

[

[ −

\ \ = H − [ ;

] \ = [ ⋅VLQ (OQ [ ) ^ \ = H DUFWJ [ _ \ = [ OQ [ ; ` \ = H − [ ⋅ VLQ [ a \ = [ ( + [ ) .

;

EBL?J:LMJ:

>_fb^h\bq;IK[hjgbdaZ^ZqbmijZ`g_gbcihfZl_fZlbq_kdhfm ZgZebam-FGZmdZ- 527 c. 2. K[hjgbdaZ^ZqihfZl_fZlbq_kdhfmZgZebamij_^_eg_ij_ju\ghklv^bnn_j_gpbjm_fhklvE> Dm^jy\p_\:>DmlZkh\<BQ_oeh\ FBRZ[mgbg-FGZmdZ- 592 c. FZl_fZlbq_kdbcZgZeba\\hijhkZobaZ^ZqZoMq_[ihkh[b_^ey \mah\<N;mlmah\GQDjmlbpdZy=GF_^\_^_\::Rbrdbg Ih^j_^<N;mlmah\Z-FDmjkfZl_fZlbq_kdh]hZgZebaZ-F