Taylor Series Mixed Exercise 6

Transcription

1 Taylor Series Mied Eercise Let f() cot ad a f(a) f () ( cos ec ) + cot f (a) f''() cotcosec + ( cosec ) f''(a) f'''() ( cosec cot cosec ) + cotcosec Substitutig ito the Taylor series epasio gives f() !! + + as required f'''(a) a f( ) l(+ e ) so f() l e ( + e ) f ( ) + e f () + e So e f ( ) ( + e ) or use the quotiet rule f () b ( + e ) e e + ( e )e f ( ) ( + e ) ( + e )e {( + e ) e } e ( e ) f ( ) ( + e ) ( + e ) Use the quotiet rule ad chai rule. c Usig Maclauri s epasio: l(+ e ) l The epasio is valid for < e, e so for a b ( ) ( ) ( ) cos + +!!! cos si, 5 so si cos si Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.

2 Usig e ad cos + + cos e e e e e e o other terms required e e e a dy + + siy ad, y () so d d Differetiatig () gives d y cos d y + d y () d Substitutig, y, ito() gives d d Differetiatig () gives d y cos d y y siy dy () d d d Substitutig y,, ito () gives d d d Substitutig foud values ito yy d! d! d y b At., y (. ) + (. ) + (. ) 55. l [( + ) ( )] l( + ) + l ( ) ( ) ( ) ( ) ( ) Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.

3 7 d y dy ( + ) + y () d d Differetiatig () gives d d y y ( ) d y d y + + d d d d So that d ( ) d y y d y + + () d d d Substitutig iitial data i () gives d Substitutig kow data i () gives So y + + +!! + + d 8 a f( ) l(sec+ ta ) f() l sec ta + sec sec (ta + sec ) f ( ) sec sec+ ta sec+ ta f () f ( ) secta f () f ( ) sec sec + sec ta ta f () Substitutig ito Maclauri s epasio gives y + + b We use the epasios: l(sec+ ta ) + si! to see that: si l(sec+ ta ) (cos) (! ) ( + )... + ( si l(sec+ ta ) lim (cos) lim... + ) Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.

4 9 We first make ote of the fact that: l l sih(l) ( e e ) ( ) l l cosh(l) ( e + e ) 5 ( + ) ad that d cosh sih, d sih cosh, which implies that: d d k k+ d d cosh cosh, cosh sih k k+ d d k k+ d 5 d cosh, cosh k k+ d d l The, the Taylor series about l is: l l d cosh ( l ) cosh l l l! d l Thus we deduce that: th a The term whe is eve is: 5 ( l)! th b The term whe is odd is: ( l)! l Cosider the first two terms i the Taylor series of cos aroud : cos + ( ) ( ) The the limit is give by: ( ) ( ) lim lim + cos ( ) + ( ) lim + + ( )... Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.

5 Cosider the first two terms i the Taylor series: 5 arcta arcta si +! 5! 5 si +! 5! The we ca evaluate the limit: 5 arcta + 5 lim lim si ! +! 5!... lim... a We differetiate the respective Taylor series term by term ad match that up with the derivative. Firstly: r e !! r! d e d!!! r r r+ r r! ( r+ )! d e d!! + + ( r)! r! r r + e b c r d ( ) (r+ ) r si d! (r+ )! r d ( ) si d! (r )! r d ( ) ( r) cos d!! ( r)! cos r ( r ) d ( ) cos + d! (r)! r ( r ) ( )! (r)! r ( r ) si Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free. 5

6 d y d y y d + d () Differetiatig d y d y y, d + d gives d d y y d + y + d d d Substitutig iitial values ito () gives d Substitutig ad ito () gives d d Usig Taylor s epasio i the form with ( ) ( ) y +( ) + ( ) + ( ) +!! ( ) + ( ) ( ) + d. () a You ca write cos + ; it is ot ecessary to have higher powers: sec + cos ( + ) Usig the biomial epasio but oly requirig powers up to ( )( ) sec + ( ) + +! higher powers of b si ta si sec cos ! 5! ! (!) 5! Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.

7 c Usig the series epasios: ta + e + + e +!! we ca evalaute the limit: + + ta lim lim lim e !! ( ) 5 a Usig e ad cos +!!! e cos b Usig the series epasio i the first part we ca deduce that: e cos + + si cos 7 5 e cossi cos The we ca evaulate the limit: e cossicos lim lim lim + + Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free. 7

8 d y dy a Differetiatig + + y () with respect to, gives: d d d y dy d y dy () d d d d Substitutig give data, y ad ito () gives d d d Substitutig, ad ito () gives d d d So usig Taylor series y y d! d! d y + + b Differetiatig () with respect to gives: d y d y dy d y d y d y d + d + d + d + d + d () Substitutig,, ad ito () gives, d d d d y d y at, () ( ), so d + + d 7 a f( ) ( + ) l( + ) f ( ) ( + ) + ( + ) l ( + )(+ )(+ l ( + )) + f ( ) ( + ) + (+ l( + )) + l( + ) + f ( ) + f(),f (), f (), f () b Usig Maclauri s epasio ( ) l( ) ()!! Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free. 8

9 8 a l(+ si ) l + +! !!!! ( + ) ( + ) ootherterms ecessary + + b 9 a b l( + si ) d + d ( d.p.) ta f( ) e e e e (As oly terms up to are required, oly first two terms of ta are eeded.) o other terms required.!! ta ta( ) e e, e !! so replacig by i a gives ta + + c Usig the series epasios: ta e e ta e e s i! We ca evaluate the limit: ta e e lim lim si! lim Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free. 9

10 We use the followig series epasios: 5 si...! 5! 8 8 ( si ) (!) cos +! co s! The we ca evaluate the limit: 8 ( si ) lim lim cos 8 + lim a Differetiatig the give differetial equatio with respect to gives: d y dy d y dy d y dy y d d d d d d d y dy d y So + d y d d b Give that y, at, d ( ) ( ), so, + + d d Ad ( )( ( ) ), so + 5 d ( ) d ( ) 5 5 So y+ ( ) !!. c The approimatio is best for small values of (closed to ):., therefore, would be acceptable, but ot 5 a f( ) lcos f() si f ( ) ta f () cos f ( ) sec f () f ( ) sec ta f () f ( ) sec sec ta f () Substitutig ito Maclauri: l cos ( ) + ( ) +!! Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.

11 b Usig cos cos +, l (+ cos ) lcos l+ lcos so l( cos ) l + + l 9 c Usig the series epasios derived above, we deduce that: l(cos ) l+ lcos l l(+ cos ) l(cos ) ( l ) (l ) cos The we ca calculate the limit: l(+ cos ) l(cos ) lim lim cos lim a Let y, the l y y I l l e so e l ( l) ( l) !! (l) (l) + l+ + + l e ( l) b Put l (l) (l) :.7 (s.f.) a f( ) cosec f ( ) coseccot i ii + f ( ) cosec ( cosec ) cot (cosec cot ) cosec + cosec ( cot ) + cosec {cosec (cosec )} cosec {cosec } f ( ) cosec ( cosec cot ) coseccot (cosec ) cosec cot (cosec ) Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.

12 b f, f, f, f ( ) Substitutig all values ito y y+ ( ) + with + d! d ( ) ( ) cosec + ( ) + + +!! a We take f( ) l ( cos ( )) f ( ) ( ) ( si ( )) +, the differetiatig: si + cos + cos ( ) ( ) + ) ( + os ) ( + cos( )) ( + os( )) ( ) ( ) cos( ) si f ( ) + ( cos( ) c c b Evaluatig the above at, we fid: f() l,f (),f () Hece the Taylor epasio about is: f( ) ( ) + ( )( )! l( + cos ( )) ( ) ( ) c We use the Taylor epasio of l( ) about : l( ) ( ) ( )... + The we ca evaluate the limit: l( + cos ( )) ( ) ( ) lim lim l( ) ( ( ) ( ) ) ( ) lim + ( ) Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.

13 Challege a We have d + ()! l ( ), so holds for d Assume true for k where k The: k+ d d k+ k l ( ) ( k)! k+ d d k+ k ( k+ ) + (( k+ ) )! k( ) ( k)! ( ) k+ So true for k+ The result the follows by iductio. b Hece the Taylor series about a, a> is: + ( ) ( )! l l a+ ( a)! a + ( ) l a+ ( a) a c I our case we have a + ( ) ( a), so: a a+ a ( a) a a( + ) a + a lim a + a a a where we have used lim This is strictly less tha if ad oly if < < a So the ratio test shows that the Taylor series epasio coverges for such that < < a d We wat to eted the rage to iclude a Settig a, we have a alteratig series with b Clearly, b for all, ad limb lim + Fially, bb + + which is true for all Hece, by the alteratig series test, the Taylor series coverges at a Hece, the Taylor series coverges for all < a as required. Pearso Educatio Ltd 8. Copyig permitted for purchasig istitutio oly. This material is ot copyright free.