Сборник задач по высшей математике 2 том
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2007
YAK 517(075.8)
BBK 22.1a73-4
C23
ABTOPLI:
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2007. - 592 c.: HJI. - (BLIcmee 06pa30BaHHe).
ISBN 978-5-8112-2948-2
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BBK 22.1H73-4
Y,l(K 517(075.8)
© |
000 .M3.11aTeJILCTBO |
ISBN 978-5-8112-2948-2 |
• .A:a:PMC-npecct, 2004 |
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rnaaa 2. ALilCI>CI>EPEH4Li1AnbHbiE YPABHEHLilH nEPBOrO nOPHAKA |
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JIHHeitHhIe ypaBHeHHz. YpaBHeHHZ BepHYJIJIH ......................... |
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YpaBHeHHZ B rrOJIHbIX ,D;Hq,q,epeHD;HaJIax. .. . |
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YpaBHeHHZ JIarpaHJKa H KJIepo ............................... |
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HHTerpHpOBaHHe ,D;Hq,q,epeHD;HaJIbHhIX YPaBHeHHit BhICIlIHX rrOpZ,D;KOB. . |
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§ 7. |
JIHHeitHhIe ,D;Hq,q,epeHD;HaJIbHhIe ypaBHeHHZ BbICiliero rrOpZ,D;Ka. . . . . . |
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HHTerpHpOBaHHe CHCTeM ,D;Hq,q,epeHD;HaJIbHhIX YPaBHeHHit . . . . |
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KOHTPOJIbHa:II pa6oTa. . . . . . . . . . . . . . . . . . |
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rnaaa 3. KPATHblE LilHTErPAnbl |
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,I1;BOitHOit HHTerpaJI. CBoitcTBa U MeTO)l;hI BhI'IHCJIeHHZ. |
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3aMeHa rrepeMeHHbIx B ,D;BOitHOM HHTerpaJIe.................. |
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TIpHMeHeHHz ,D;BoitHoro HHTerpaJIa. |
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TPOitHOit HHTerpaJI. CBoitcTBa, BhI'IHCJIeHHe,rrpHMeHeHHe. . . . . |
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rnaaa 4. KPLilBOnLilHEMHblE LiI nOBEPXHOCTHblE LilHTErPAnbl |
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KPHBOJIHHeitHhIit HHTerpaJI nepBoro pO,D;a . . . |
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KpHBOJIHHeitHhIit HHTerpaJI BToporo pO,D;a . . . |
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TIoBepxHocTHbIit HHTerpaJI . . . . . . . . . |
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rnaaa 5. TEOPLilH nonH |
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§ 1. CKaJIZpHhIe H BeKTopHhIe rrOJIz. TIoBepxHocTb ypOBHZ. |
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BeKTopHhIe JIHHHH . . . . . . . . . . . . . . . . . |
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)J;HBepreHD;Hz H POTOP BeKTopHoro rrOJIz. OrrepaTop raMHJIbToHa. ...... |
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TIOTOK BeKTopHoro rrOJIz............................................... |
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IJ;HPKYJIZD;HZ BeKTopHoro rrOJIZ . . . . |
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rl1aaa 6. TEOPLilH BEPOHTHOCTEM |
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CJIY'IaitHhIeC06hITHli. )J;eitcTBHlI HM C06hITHlIMH . .. . . . |
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§ 4. |
YCJIOBHaH BepoHTHocTb . . . |
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<D0pMYJIa nOJIHoil: BepOHTHOCTH. <D0pMYJIa Beil:eca. . |
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CxeMa HCnblTaHHil: BepHYJIJIH • . . . |
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IIpH6JIH:lKeHHble <P0PMYJIbI B cxeMe BepHYJIJIH .. . . . |
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KOHTPOJIbHaH pa6oTa. . . . . . . . . . |
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)J;HcKpeTHble CJIY'lail:HbleBeJIH'lHHbI. . |
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HenpepbIBHble CJIY'lail:HbleBeJIH'lHHbI.. . |
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§ 10. qHCJIOBble XapaKTepHCTHKH CJIY'lail:HblXBeJIH'lHH.• |
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§ 11. BalKHeil:wHe pacnpe)J;eJIeHHH CJIY'lail:HblXBeJIH'lHH. |
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§ 12. CHcTeMbI CJIY'lail:HblXBeJIH'lHH.. |
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§ 13. <DYHKIT;HH CJIY'lail:HblXBeJIH'lHH. . |
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§ 14. IIpe)J;eJIbHbIe TeopeMbI TeopHH BepoHTHocTeil:. |
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rnasa 7. TEOP&IIH CPYHKU.&IIIII KOMnJ1EKcHoro nEPEMEHHOro |
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§ 1. OCHoBHble 9JIeMeHTapHble <PYHKIT;HH KOMllJIeKCHOro nepeMeHHoro. . |
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AHaJIHTH'leCKHe<PYHKIT;HH |
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MHTerpHpoBaHHe <PYHKn;Hil: KOMnJIeKCHOro nepeMeHHoro. • . |
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§4. |
PH)J;bI JIopaHa. M30JIHpOBaHHble OC06bIe TO'lKH.• . . . |
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BbI'leTbI............................................................... |
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KOHTPOJIbHaH pa6oTa. • . • . • . • . . |
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rnasa 8. onEPAu.&IIOHHOE &IIC'"I&IICJ1EH&IIE |
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§ 1. |
OpHrHHaJI H306palKeHHH. IIpe06pa3oBaHHe JIaITJIaca. |
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HaxO:lK)J;eHHe H306palKeHHil: .... |
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§ 2. CBepTKa <PYHKIT;Hil:. OTblcKaHHe OpHrHHaJIa no H306palKeHHIO........... |
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§3. |
IIpHJIo:lKeHHH OnepaIT;HOHHoro HC'lHCJIeHHH. . . |
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KOHTPOJIbHas: pa6oTa. . . . . . . . . . |
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OTBeThI.................................................................... |
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IIpHJIolKeHIUI . . . . . . . . . . . . . . . . |
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5
KHHra HanHCaHa npenO,!l;aBaTeJIHMH HeCKOJIbKHX Pa3JIJI'IHhIXMOCKOBCKHX BY-
30B, HMeIOID;HMH MHOrOJIeTHHit OnblT JIeKn;HOHHOit H CeMHHapcKoit pa60TbI co CTY- ,!l;eHTaMH. TIPH 3TOM rJIaBbI 3, 4 H §§ 6-8 rJIaBbI 2 HanHCaHbI JIYHry K. H.j rJIaBbI 5
H |
8 - |
HopHHblM B. TI.j rJIaBa 6 H §§ 1-5 rJIaBbI 2 - TIHcbMeHHblM ,r:t;. T.j rJIaBbI 1 |
H |
7 - |
llieB'IeHKO10. A.j KYJIaHHH E.,r:t;. HanHCaJI § 4 rJIaBbI l. |
ABTOPbl 6Y,!l;JT npH3HaTeJIbHbI 3a JII06b1e OT3b1BbI, nOlKeJIaHHH H KpHTH'IeCKHe 3aMe'laHHH,KOTopble MOlKHO npHCbIJIaTb no Mpecy: 141103, MocK. 06JI., r. II1;eJI- KOBo-3, a/H 140j HJIH no MpeCaM 3JIeKTpoHHoit nO'ITbI: chislovo@yandex.ru HJIH editor@airis.ru (o6H3aTeJIbHO YKaJaTb TeMY: «3Ma'lHHK»).
ABTOPhI
ABTOPbl H H3,!l;aTeJIbCTBO 6JIarO,!l;apHT npenO,!l;aBaTeJIH MaTeMaTHKH TIaitKOBY JI. M.
H3 ,r:t;HenponeTpOBCKa (YKpaHHa) 3a n;eHHble 3aMe'laHHH,KOTopble 6b1JIH Y'lTeHbIB ,!l;aHHOM H3,!l;aHHH.
npMHHTbie 0603Ha'-leHMH
=? |
onpe,!l;eJIeHHe |
|
Q |
Ha'laJIOpeweHHH 3Ma'iH |
|
•N |
KOHen; peweHHH 3Ma'iH |
|
MHOlKeCTBO HaTypaJIbHhIX 'IHCeJI |
||
Z |
MHOlKeCTBO n;eJIhIX 'IHCeJI |
|
IR |
MHOlKeCTBO ,!l;eitcTBHTeJIbHhIX 'IHCeJI |
|
IR2 |
,!l;eitcTBHTeJIbHaH nJIOCKOCTb |
|
IR3 |
,!l;eitcTBHTeJIbHOe TpexMepHoe npOCTpaHCTBO |
|
IC |
MHOlKeCTBO KOMnJIeKCHblX 'IHCeJI |
|
U |
06'be,!l;HHeHHeMHOlKeCTB |
|
n |
nepeCe'leHHeMHOlKeCTB |
|
ACB |
A - |
nO,!l;MHOlKeCTBO MHOlKeCTBa B (A "I B) |
A~B |
A - |
nO,!l;MHOlKeCTBO MHOlKeCTBa B . |
V |
JII06oit, ,!l;JIH JII06oro |
|
|
TOlK,!l;eCTBeHHO paBeH |
|
sign(x) |
3HaK 'IHCJIax |
|
rnaBa 1. PflAbl
CI
§ 1. nOHSlT~E PSlAA. PSlAbl C n01l0>K~TEllbHbIM~
'-IllEHAM~
PJlA. CXOAl1MOCTb pJlAa. CYMMa pJlAa
~ TIYCTb 3a,Il;aHa 6eCKOHe'lHaHrrOCJIe,!l;OBaTeJIbHOCTb 'lHCeJI(,!l;eil:cTBHTeJIbHbIX HJIH KOMIlJleKCHblx)
al + a2 + a3 + ... + an + ...
00
COKparn;eHHO pX,!l; 0603Ha'laIOTCJIe,!l;yIOm;HM 06pa30M: 2: an. TIPH 3TOM 'lHCJIa n=l
al, a2, a3, .• . , an, . .. Ha3bIBaIOTCX "I.JleHaMU pX,!l;a, a 'lHCJIOan - 06tqUM "I.JleHOM pX,!l;a. CYMMbI BH,!l;a
Ha3bIBaIOTCX "I.aCmU"I.H'bt.MU CyMMaMU pX,!l;a. qHCJIOBOil: pX,!l; Ha3bIBaeTCX CXOOJltqUM-
CJI, eCJIH cym;ecTByeT KOHe'lHbIiI:rrpe,!l;eJI rrOCJIe,!l;OBaTeJIbHOCTH {Sn} ero 'lacTH'lHbIX CyMM:
S = lim Sn.
n-+oo
B 3TOM CJIy'laeYKa3aHHbIiI: rrpe,!l;eJI Ha3bIBaeTCX CY.M.Mo{J, pJloa.
ECJIH lim Sn He cym;ecTByeT HJlH paseH 6eCKOHe'lHOCTH,TO 'lHCJIOBOil:pX,!l; Ha- n-+oo
3bIBaeTCX paCXOOJltqUMCJI H CYMMbI He HMeeT. $
Ecm1 Y CXOA~l.I.IerOC~ p~Aa oT6poCIIITb KOHe'iHOe'iIilCIlOero 'illeHOB, TO nOIlY'ieHHbliii P~A TaK>Ke 6YAeT CXOAIIITbC~. BepHO III 06paTHoe: eC1l1ll CXOAIIITC~ P~A, nOIlY'ieHHbliiioT6paCblBaHilleM KOHe'iHOrO'iIilClla 'illeHOBY AaHHoro p~Aa, TO III AaHHbliii P~A TaK>Ke CXOAIIITC~.
TaKHM 06Pa30M, CXO,!l;HMOCTb pX,!l;a He MeHxeTCX rrpH oT6pacbIBaHHH JII06oro i{QHe'lHOrO'lHCJIaero 'lJIeHOB.
7
|
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00 |
|
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TeopeMa 1.2. nYCTb |
pl'lA 1: an CXOA~TCl'I, ~ |
era CYMMa paBHa |
8. TorAa pl'lA |
|
00 |
|
n=l |
|
|
= (ral + aa2 |
+ ... + aan + ..., rAe a |
|
|
|
1: (ran |
- npO~3BOJ1bHOe |
"'~CJ10, TaK>Ke |
||
n=l
CXOA~TCl'I, np~...eM ero CYMMa paBHa 08.
|
|
00 |
00 |
|
|
|
TeopeMa 1.3. nYCTb Pl'lAbl 1: |
an ~ 1: bn CXOAl'ITCl'I,~ ~x CYMMbl, COOTBeTCTBeH- |
|||
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n=l |
n=l |
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|
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|
00 |
+ b2) |
+ ... TaK>Ke |
|
HO, paBHbl 8 1 ~ 8 2 • TOrAa pl'lA 1: (an + bn ) = (al + bd + (a2 |
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n=l |
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CXOA~TCl'I, np~... eM ero cYMMa paBHa 8 1 + 8 2 . |
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Heo6xOAlilMbliii nplll3HaK CXOAIilMOCTIil |
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00 |
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ECJIH pH,D; 1: an CXO,D;HTCH, TO 06W;Hit 'LJIeHpx,n;a an CTpeMHTCH K HyJIIO npH |
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n |
-t 00, T. e.: |
n=l |
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lim an = O. |
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n~oo |
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00 |
|
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TaKHM 06pa30M, eCJIH lim an "10, TO pH,D; 1: an pacXO,D;HTCH. |
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|||
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n~oo |
n=l |
|
|
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PH,D; a+aq+ aq2+... + aqn-l +...,COCTaBJIeHHblit H3 'UleHOB6eCKOHe'lHoitreo- |
||||
MeTpH'IeCKoitnporpeCCHH CO 3HaMeHaTeJIeM q H nepBblM 'UleHOMa "I 0, Ha3blBaeTCH zeO.MempU"I.eCfC'U.M p"i}O.M. ECJIH Iql ~ 1, TO I'OOMeTpH'IeCKHitpH,D; pacXO,D;HTCH, eCJIH
Iql < 1 - CXO,D;HTCH (npH aTOM ero cyMMa S HaXO,D;HTCH no <p0pMYJIe 8 = -a1 ). |
||||||
1 |
1 |
1 |
|
1 |
|
-q |
00 |
|
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||||
PH,D; 1 + -2 |
+ -3 |
+ ... + n + ..., RJIH, 'ITOTO lKe caMoe, |
1: |
n' Ha3blBaeTCH zap- |
||
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n=l |
1 |
1 |
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1 |
|
+ ..., |
||
.MO~U"I.eCfCU.M. rapMOHH'IeCKHitpH,D; pacXO,D;HTCH. PH,D; 1 + p |
+ 3P + ... + p |
|||||
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2 |
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n |
|
r,D;e p > 0, Ha3blBaeTCH p"i}O.M j(upux.II.e. 9TOT pH,D; CXO,D;HTCH npH p |
> 1 H pacxo- |
|||||
,D;HTCH npH 0 < p |
~ 1. qacTHblM CJIY'laeMpH,D;a ,r:t;HpHXJ1e (npH p = 1) HBJIHeTCH |
|||||
rapMOHH'IeCKHitpH,D;.
npM3HaKM CXOAMMOCTM PRAOB C nOnO)l(MTenbHblMM "IneHaMM
I-iii nplll3HaK CpaBHeHIIIR
00 |
|
00 |
TIYCTb 1: an H |
1: bn - pH,D;bl C nOJIOlKHTeJIbHblMH 'UleHaMH,npH'IeMan ~ bn |
|
n=l |
n=l |
|
,D;J1H Bcex HOMepOB n, Ha'lHHaHC HeKOToporo. Tor,D;a: |
||
|
00 |
00 |
1) eCJIH pH,D; |
1: bn CXO,D;HTCH, TO CXO,D;HTCH H pH,D; 1: an; |
|
|
n=l |
n=l |
|
00 |
00 |
2) eCJIH pH,D; |
1: |
an pacXO,D;HTCH, TO pacXO,D;HTCH H pH,D; 1: bn . |
|
n=l |
n=l |
8
2-iii nplll3HaK CpaBHeHIIIR
00 |
00 |
|
|
TIYCTb L: an H |
L: bn |
- p:il,W>! C nOJIOlKHTeJIbHblMH qJIeHaMH, npHqeM cyW;e- |
|
n=l |
n=l |
|
|
cTByeT KOHeqHblit H OTJIHqHblit OT HyJI:iI npe,!l;eJI |
|||
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I. |
an |
|
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1m -b' |
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n-too |
n |
00 |
|
00 |
|
Tor,!l;a pH,W>! L: |
an H |
L: bn CXO,!l;HTCH HJ1H pacXO,!l;HTCH O,!l;HOBpeMeHHO. |
|
n=l n=l
TIPH HCnOJ1b30BaHHH 1-ro HJ1H 2-ro npH3HaKa CpaBHeHHH, KaK npaBHJ10, CPaBHHBaIOT HCXO,!l;Hblit pH,!l; C COOTBeTCTBYIOW;HM PH,!l;OM )J;HpHXJ1e. TIPH aTOM qacTO HCnOJIb3YIOT aKBHBaJIeHTHOCTb CJ1e,!l;yIOW;HX 6eCKOHeqHO MaJIblX nOCJ1e,!l;OBaTeJIbHOcTeit (npH n --+ 00):
|
• |
1 |
t 1 |
• 1 |
tIl |
(1 |
|
1 ) |
|
1 |
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|
|
n" |
|||||||
|
|
n |
n |
n |
n |
|
|
n |
f"OoJ |
|
|
SIn - '" g - '" arCSIn - |
'" arc g - '" n |
|
+ - |
|
|||||
nplll3HaK AanaM6epa |
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00 |
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TIYCTb |
L: an - |
pH,!l; C nOJIOlKHTeJIbHbIMH qJIeHaMH, H cyw;ecTByeT KOHeqHblit |
||||||||
npe,!l;eJI |
n=l |
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an +1 = l. |
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lim |
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n-+oo |
an |
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|
Tor,!l;a, eCJ1H l |
< 1, TO ,!l;aHHblit pH,!l; CXO,!l;HTCHj eCJ1H lKe l |
> 1, TO - pacXO,!l;HTCH. |
||||||||
ECJ1H l |
= 1, TO pH,!l; MOlKeT CXO,!l;HTbCH HJ1H pacXO,!l;HTbC:ilj B aTOM CJ1yqae Tpe6y- |
|||||||||
eTCH HCCJ1e,!l;OBaTb pH,!l; C nOMODI,bIO ,!l;pyrHx MeTO,!l;OB. |
|
|
|
|
|
|||||
00
TIYCTb L: an - p:il,!l; C nOJIOlKHTeJIbHblMH 'lJIeHaMH,H cyw;ecTByeT KOHeqHblit n=l
lim va; = l.
n~oo
Tor,!l;a, eCJ1H l < 1, TO ,!l;aHHblit pH,!l; CXO,!l;HTCHj eCJ1H lKe l > 1, TO - pacXO,!l;HTCH. ECJ1H l = 1, TO p:il,!l; MOlKeT CXO,!l;HTbCH HJ1H pacXO,!l;HTbCHj B aTOM CJ1yqae Tpe6y-
eTC:iI HCCJ1e,!l;OBaTb pH,!l; C nOMODI,bIO ,!l;pyrHx MeTO,!l;OB.
I.1HTerpanbHbliii nplll3HaK CXQAIIIMOCTIil
00
TIYCTb L: an - p:il,!l; C nOJIOlKHTeJIbHblMH qJIeHaMH, ,!l;JlH KOTOPOro cyw;ecTByeT n=l
nOJIOlKHTeJIbHaH, HenpepblBHaH H MOHOTOHHO y6b1BaIOW;aH Ha npoMelKYTKe [1, +00) <PYHKD;HH f(x) TaKaH, 'ITOf(n) = an, n = 1,2, •••
I+00
Tor,!l;a pH,!l; n~lan H Heco6cTBeHHblit HHTerpaJI f{x) dx CXO,!l;HTCH HJ1H pac-
1
XO,!l;HTCH O,!l;HOBpeMeHHO.
9