The analysis of the method on the one variable function s limit Ke Wu

Transcription

1 Ieraioal Coferece o Advaces i Mechaical Egieerig ad Idusrial Iformaics (AMEII 5) The aalysis of he mehod o he oe variable fucio s i Ke Wu Deparme of Mahemaics ad Saisics Zaozhuag Uiversiy Zaozhuag 776 Chia Keywords: Fucio of oe variable; he i; Evaluaio mehod Absrac The i of oe variable fucio is a impora problem i Higher Mahemaics i is a basic mehod o sudy he relaioship bewee he fucio of variables This emplae describes some impora mehod abou couig he i ad gives he solvig skills Iroducio The cocep of i is produced i order o solve some pracical problems of precise I he hird Ceury AD Chia's well-kow mahemaicia Liu Hui used he mehod of cyclosome Surgery ad used he regular polygo iscribed i a circle o approimaed he area of a circle his is he esimaig hikig Liu hui sars from he heago iscribed i a circle does he Iscribed regular welve sided i shape wey-four polygo ec doubles he umber of edges every ime So we ge a series of iscribed regular polygo areas A A A A Imagie ifiie icrease ad he umber of edges iscribes regular ifiie icrease I his process iscribed regular ifiie closes o a circle A he same ime A ihibiig closes o a deermied value he deermied values ca be udersood as he area of a circle This is he origi of he cocepio of i The i of oe variable fucio is a impora problem i Higher Mahemaics Limi mehod is a basic mehod o sudy he relaioship bewee he fucio of variables i is a impora basis for he i of fucio of may variables he derivaive differeial ad iegral calculus The i is a impora foudaio of calculus i is ecessary o make a summary ad aalysis for he mehod of solvig he i prepare for Calculus Learig The various mehod of fidig he i of various Calculaio of i by usig usig he cocep Defiiio assumig he fucio f( ) a he poi of a deleed eighborhood has a defiiio If here eiss a cosa A for ay posiive umber here is always a posiive umber whe < < δ here have bee f( ) A < ε he he cosa A is called he i fucio whe wrie as: f( ) = A Eample Verificaio: whe > Prove ε > as: I order o make As log as So make = = < ε < ε 5 The auhors - Published by Alais Press 79

2 Whe The value of correspodig fucios δ = ε < < δ mee < ε Accordig o he defiiio of i of fucio: = Calculaio of i by defiiio is he mos direced mehod he key is o deermie suiable Posiive values δ we ca use he scalig mehod o ideify he posiive value I addiio usig he defiiio of i is direced bu i is o simple mehod we eed oher mehods o i he auiliary Calculaio of i by usig properies of ifiiesimal Theorem A produc of bouded fucio ad a ifiiesimal is a ifiiesimal si Eample Couig Aalysis: si is he produc a si ad Because is he ifiiesimal whe ad si is bouded fucio Accordig Theorem si = Whe usig Theorem be sure o see he chage process of variables judge clearly which is Bouded fucio which is he ifiiesimal Oly he Muliplicaio abou a bouded fucio ad a ifiiesimal is a ifiiesimal muliplicaio As i eample if he is o ifiiesimal bu ifiie I fac we kow ha: si = Calculaio of i by usig squeezig rule Theorem If whe U ( r) g () f() h () Ad g ( ) = A h ( ) = A The f( ) = A Eample Couig [ ] Cou: As [] Whe () because ( ) = Ad = he [ ] = Aalysis: Calculaio of i by usig squeezig rule he key is o deermie g () h () ad g ( ) = h ( ) =A Calculaio of i by he equivale ifiiesimal Theorem Whe: β β 79

3 Ad β α eis so β β β β = = = α α α α The commoly used equivale ifiiesimal: Whe si a arcsi arca cos ( ) Eample 4 Couig ( ) cos Cou: Whe ( ) So ( ) = = cos cos Aalysis: Whe calculae he i by he equivale ifiiesimal we mus pay aeio o ha he idepede variable eds o zero whe usig he equivale ifiiesimal Calculaio of i by he coiuiy of a fucio Defiiio he fucio y = f() is defied i a eighborhood of he poi If f( ) ( ) he The fucio a his poi is coiued Theorem 4 every elemeary fucio i he defiiio rage is coiued Eample 5 Couig Aalysis: The fucio Is he elemeary fucio ad = f() = is a poi i he defiiio so = f = f () = Calculaio of i by he coiuiy of a fucio we mus esure ha his fucio is defied a his poi ca direcly Subsiue o he fucio Calculaio of i by he L Hospial rule Theorem 5 Whe a () The fucio f() ad F () all ed o zero; ()I a deleed ceer eighborhood abou he poi a f () ad F () all eis ad F () ; f() f () () = a F () a F () si Eample 6 Couig Cou: Whe we ca fid ha si ad he si cos si = = = 6 6 I he eample 6 he umeraor ad deomiaor ed o zero we ca cou he i afer couig he derivaives If molecular ad deomiaor all ed o zero i he same chagig process we ca use he Hospial rule 79

4 Calculaio of i by he wo impora is si The firs impora i = The secod impora i ( ) = e (or ( ) = e) The secod impora i has hree characerisics: Whe we ca see ( ) as ; There mus be i he fucio ad mus be reciprocal i ( ) a Eample 7 Couig = a si si Cou: = ( ) = ( ) ( ) = cos cos Eample 8 Couig ( ) Cou: ( ) = [ ( )] {[ ( )] = } = e Calculaio of i by derivaives of he fucios abou he idefiie iegral upper i Theorem 6 If he fucio f( ) is coiuous o he secio [ abhe ] fucio abou he iegral upper i f ( ) = f () d a has derivaive ad is derivaive is d f ( ) = f ( ) d f ( ) d = a This heorem pois o a impora coclusio: he coiuous fucios f( ) calculae he defiie iegral abou he chagig upper i The resul reducive iself f( )Associae he couig i we ca use he L Hospial rule o couig he i If he molecular or he deomiaor of he fracioal appears his fucio of he idefiie iegral upper i we ca use he idea of he heorem 6 o cou he derivaio e d cos Eample 9 Couig Aalysis: This is a ucerai formula clearly we ca cou i by he L Hospial rulethe fracio ca be wrieed o cos e d = e d cos This iegral s upper i is cos i is a fucio by I ca be see a compoud fucio by he middle variable u = cos Accordig o he heorem 6we ca calculae: d cos d e d e d d = cos d d u = e d (cos ) u= cos du cos = e ( si ) So = si e cos e d cos cos si e = = e 79

5 Calculaio of i by Taylor formula Taylor formula: If he fucio f( ) has ( ) duplicaes derivaives he ( ab ) f ( ) f( ) = f( ) f ( )( ) ( )! f! ( ) ( ) ( ) R ( ) ( ) f ( ) R ( ) = ( ) his ξ is a value bewee ad ( )! Eample Couig he i by he McLaughli formula who has he peao remaider: si( ) cos si Cou: As he deomiaor of he fracioal si ~ ( ) we ca oly use he McLaughli formula si = o( ) cos = o( )!! Subsiue: si cos = o( ) o( ) = o( )!! I he above formula we ca hik he wo ο( ) as oeο ( ) The coclusio The above mehods are jus some mai mehod for couig i some specific problems have specific aalysis For eample some is eed o simplify ad he subsiue; some is eed o chage eleme; some is eve eed raioal he molecular deomiaor ec For he i of piecewise fucio a he ierruped poi We eed cou he i o he lef ad righ side of his poi he do he judgme If you wa o be familiar wih he mehod of couig he i you eed do he various ypes of eercises he lay he good foudaio o he Calculus Ackowledgemes This work is suppored by Naioal Naure Sciece Foudaio of Shadog educaio provice plaig projec o (GG) This work is suppored by Naioal Naure Sciece Foudaio of he iformaioal ad elecrical egieerig professioal plaform s course o Zaozhuag College (4C) Refereces [] Deparme of Mahemaics i og ji Uiversiy Higher Mahemaics (Referece of Higher Educaio Chia 7) [] Jiafu Wag The Sychroous Tuorship ad The Eercise s Quesio o Higher Mahemaics(Referece of Chiese Miig ad Techology Uiversiy Chia 8) [] Weli Li A iroducio o he mahemaics hisory (Referece of Higher Educaio Chia ) 794