UNIT 1: ANALYTICAL METHODS FOR ENGINEERS

Transcription

1 UNIT : ANALYTICAL METHODS FOR ENGINEERS Ui code: A// QCF Level: Credi vale: OUTCOME TUTORIAL SERIES Ui coe Be able o aalyse ad model egieerig siaios ad solve problems sig algebraic mehods Algebraic mehods: polyomial divisio; qoies ad remaiders; se of facor ad remaider heorem; rles of order for parial fracios (icldig liear, repeaed ad qadraic facors); redcio of algebraic fracios o parial fracios Epoeial, rigoomeric ad hyperbolic fcios: he are of algebraic fcios; relaioship bewee epoeial ad logarihmic fcios; redcio of epoeial laws o liear form; solio of eqaios ivolvig epoeial ad logarihmic epressios; relaioship bewee rigoomeric ad hyperbolic ideiies; solio of eqaios ivolvig hyperbolic fcios Arihmeic ad geomeric: oaio for seqeces; arihmeic ad geomeric progressios; he limi of a seqece; sigma oaio; he sm of a series; arihmeic ad geomeric series; Pascal s riagle ad he biomial heorem Power series: epressig variables as power series fcios ad se series o fid approimae vales eg epoeial series, Maclari s series, biomial series Yo shold jdge yor progress by compleig he self assessme eercises. D.J.D

2 FACTORIALS Yo shold already kow ha a facorial mber is deoed wih! so! meas facorial.! = ()()()() =!=()(-)(-)(-)... Wiho proof,! Is always ake as SUMMATION of a SERIES May epressios ca be represeed by a series of mbers added ogeher. This migh be a series of mbers like... I migh be a power series of like y... I geeral a series may be wrie as = where is he erm i he series. Cosider he followig calclaio.... The vale of is he sm of a series of fracios wih he meraors formig a descedig series of iegers (whole mbers). We cold wrie his more simply as: or The symbol is a capial leer Sigma ad meas he sm of The limis of he variable bewee he smmaio akes place is show. WORKED EXAMPLE No. Wrie o he series represeed by for he firs erms. = = 8 = 8 9 hece WORKED EXAMPLE No. Wrie o he series represeed by!! = =! ()() for he firs erms.!!! = =! Hece ()()() D.J.D

3 LIMITING VALUES Cosider he followig eqaio. Sppose we wish o kow if his has a vale 7 whe =. A simple way o fid o is o make se of he fac ha / = If we rearrage he eqaio by dividig hrogh by he highes order of ( i his case) we ge / / 7 / / Now p = We wrie his as so here is a limiig vale whe = 7 7 L 7 WORKED EXAMPLE No. Fid he limiig vale of he followig epressios whe i) ii) iii) ( )( 7) i) / / ii) ( )( 7) / / 7/ iii) / / / SELF ASSESSMENT EXERCISE No.. Wrie o he series represeed by - Aswer... 7 for he firs erms. -! - -!. Fid he limiig vale of he followig epressios whe - i. ii. iii. Aswers,. ad D.J.D

4 CONVERGENCE ad DIVERGENCE I he series... we migh hik ha sice each erm is smaller ha he oe before i, he he vale of wold ed o coverge o some figre as we add more ad more erms. I fac he vale of will go o geig bigger so i he limi as he vale of will also ed o ifiiy. This series has o limiig vale. We ms be very carefl dealig wih series becase he vale of each erm may ge bigger ad bigger (divergece) or i migh ge smaller ad smaller (covergece) ad if i coverges here is a limiig vale. I order o fid he vale of a series whe = we wrie he series i he form: = or If he erm a = is o zero he i seems likely ha series has o limiig vale ad is diverge. If L he series migh coverge o a limiig vale b his is o cerai. For eample cosider agai he followig series i. The series is... L I was show earlier ha his series i diverge so havig a zero vale does o prove he series is coverge. WORKED EXAMPLE No. Deermie if he followig series is diverge. The series is 8... / L ad sice his is o zero he series is diverge. / D'ALEMBERT'S RATIO For ay series L heseries coverges. L heseries diverges. This does o ell s wha happes if he resl is iy. D.J.D

5 D.J.D WORKED EXAMPLE No. Deermie if he followig series is diverge. The series is... 7 We oe ha ad ) ( / / - - L This is less ha so he series is coverge SELF ASSESSMENT EXERCISE No. Fid he limiig vale of he followig epressios whe. Deermie if he followig are diverge or coverge. i. 9 ii) Aswers i. L hece diverge. ii. L hece ideermiae b L ad sice his is o zero i ms be diverge.

6 ABSOLUTE CONVERGENCE Wiho proof i ca be show ha if we deermie he sm of he modlli of each erm i a series sch ha S... ad if his is coverge, he is also coverge ad has a defiie vale hece he se of he words absole covergece. I follows ha if all he erms are posiive ayway, he if he series is coverge i is absolely coverge WORKED EXAMPLE No. 7 Tes he followig series o see if i has absole covergece This is ow he same as eample whece L 7 This is less ha so he series is coverge ad he series... is absolely coverge. WORKED EXAMPLE No. 7 Deermie if he followig power series is coverge or diverge. -...!!! -!!!!! L for all vales of This is less ha so he series is absolely coverge for all vales of. D.J.D

7 SELF ASSESSMENT EXERCISE No. Tes he followig series for covergece. 7...!! 7!..... Aswers - -!. L so if < he series is absolely coverge b if >i is diverge.. L for all vales of hece he series is absolely coverge for all. PASCAL'S TRIANGLE ( -)( - )( -)...( -{r }) A impora facorial epressio is C r r! The op lie is he firs r facors of ad he boom lie is facorial r If we evalaed all he vales of C r from r = o r = we wold fid he vales are symmerical. For eample ake he case = ()() ()()() ()()()() C C C C C ()() ()()() ()()()() ()()()()() C ()()()()() Pascal's Triagle is made of rows as show. The h row is made of all he mbers C r for r = o r = The zero row is C = The s row is C = C = The d row is C = C = C = The rd row is C = C = C = C = D.J.D 7

8 BINOMIAL THEOREM Cosider a polyomial made p of repeaed facors of (y + ) If = he (y+) = y +y + If = he (y+) = y + y + y + If = he (y+) = y + y + y + y + If = he (y+) = y + y + y + y + y + Le y =. If = he ( + ) = + + If = he ( + ) = If = he ( + ) = If = he ( + ) = This is a series wih each erm coaiig o a power i ascedig order from o. Eamie his ad yo will see ha he coefficies mach he rows of Pascal's Triagle. I follows ha he coefficies are C r. Sice o = ad we ormally wrie as simply ad oig ha C = C = ( + ) = C + C + C + C + C C ( + ) = + C + C + C + C ( ) ( ) r C r r r...!! Rerig o (y + ) we ca see ha his is a series sch ha : (y + ) = C y + C - y - + C - y - + C - y C (y + ) = y + C - y - + C - y - + C - y This resl ca be obaied epadig ( + ) as show i he e eample. WORKED EXAMPLE No. 8 Usig he biomial heorem epad S = (z + a) Firs we ms rearrage he epressio io a form ha ca be epaded. a (z a) z le a/z = (z a) z z Epadig we ge S = z ( + ) = z [ + C + C + C + C ] S = z ( + ) = z [ + C (a/z) + C (a/z) + C (a/z) + C (a/z) (a/z) ] S = z ( + ) = z [ + C z - a + C z - a + C z - a + C z - a z - a ] S = z ( + ) = z + C z - a + C z - a + C z - a + C z - a a Noe if we chage z o y ad a o we ge he resl for epadig ( + y) y + C y - + C y - + C y - + C y D.J.D 8

9 WORKED EXAMPLE No. 9 Epad ( + ) wih he biomial heorem. r ( ) C we kow ha r ( ) C C C C C C C C... as solved earlier C C C C C ( ) Check his o by pig i ay vale of say = (+) = = 79 ( ) ()() () () () () 79 = = 79 WORKED EXAMPLE No. Usig he biomial heorem epad aswer is y = - ( ) C C C C... ( ) ( )( ) ()() - C C ad show ha if is very small he he ( )( )( ) ()()()... ( )... The resl is a ifiie series ad i is oly sefl for evalaio whe is small sch ha higher powers are egligible. Check if =. he y =. - =.9 =. =.9 WORKED EXAMPLE No. Usig he biomial heorem epad y ad evalae whe =. y pig = - we ca epad. y C C C... ( )( ) ( )( )( ) y... ()() ()()() y... y Noe for small vales of y is qie accraely give by - D.J.D 9

10 SELF ASSESSMENT EXERCISE No.. Epad y sig he biomial heorem.. Epad y = ( q) - o for erms sig he biomial heorem.. Epad y ad show ha for small mbers y. Epad y ad show ha for small mbers y 9 MACLAURIN'S SERIES This is a series ha demads special aeio ad he sde migh also look p he Taylor Series o which i is based. The series is based o sccessive differeials. f''() f'''() f f() f() f'()...!! WORKED EXAMPLE No. ()!... Epad io a series sig Maclari's mehod. f() f() f () f() Hece...!!... f () f()... b clearly if = he resl is ifiiy f () f() D.J.D

11 SELF ASSESSMENT EXERCISE No. Epad he followig Maclari's mehod heorem. i. cos() ii. si() iii. e iv. sih() e e Noe ha sih() = Check yor aswers i he able below. TABLE OF FUNCTIONS AND THEIR SERIES cos() si() cosh() sih() e l( + ) ( )! ( )!!!! D.J.D