Calculus - D Yue Fial Eam Review (Versio //7 Please report ay possible typos) NOTE: The review otes are oly o topics ot covered o previous eams See previous review sheets for summary of previous topics Also study all homework problems, all worksheet problems, all quiz problems, ad all previous eams, ad all previous commo fial eams Power series c ( a) o Theorem: Every power series has a radius of covergece R, where R The power series coverges absolutely for a R ad diverges for a R Ad whe R, the power series may or may ot coverge at the edpoits where a R ; that is, other tests eed to be doe at the edpoits The power series could coverge coditioally or absolutely or diverge at such edpoits The eact iterval of values of where the power series coverges is call the iterval of covergece o To compute the radius of covergece, perform the ratio test or root test: Compute c ( a) c lim a lim c ( a) c or lim c ( a) a lim c ad solve the equatio for to get the bulk of where the power series coverges; you still eed to check ay edpoits for possible covergece If, the the series coverges absolutely everywhere If, the the series coverges (absolutely) oly at a o You ca take derivatives ad itegrals withi the radius of covergece This allows you to create ew power series from old power series Taylor series ad Maclauri series A Maclauri series is a Taylor series with base poit o For a fuctio f () to have a Taylor series epasio at base poit a, the fuctio f () must be ifiitely differetiable at a The formula for the Taylor series ( ) f ( a) to f () at a is ( a)! ( ) o Make a table of the derivatives f ( ), the evaluate at a, ad the put together i a series usig the formula above If there is a patter to the values, the write the series i summatio form o The Taylor series coverges to the origial fuctio i some radius of covergece Provig this requires usig the Taylor remaider formula, which is ( ) f ( c)( a) R ( ) for some c betwee a ad So ( )! M a ( R ( ) where f ) ( c) M for all c betwee a ad ( )! o Nice fuctios such as e, si, cos have a ifiite radius of covergece for their Maclauri series Ad the fuctios, l( ), have a radius of
covergece of for their Maclauri series The Maclauri series of these ice fuctios coverge to the origial fuctio withi the radius of covergece Differetial Equatios o Kow the meaig of a slope field correspodig to a first order differetial equatio Kow how to sketch a solutio curve through a slope field Kow how to match a first order differetial equatio to a slope field o Kow how to verify whether a give fuctio is a solutio to a give differetial equatio o Kow how to solve st order separable differetial equatios by separatig ad itegratig o Kow how to solve st order liear differetial equatios by multiplyig by a Pd itegratig factor v e where y' Py Q is i stadard form; this yields ( vy)' vq Note if a differetial equatio is both liear ad separable, the separable techique is usually easier o Kow how to solve d order liear homogeeous differetial equatios with costat coefficiets by fidig roots of the characteristic polyomial ar br c where the differetial equatio is ay '' by' cy Kow how to hadle the special cases of comple roots ad repeated roots o Kow how to solve for the arbitrary costats if give iitial values Additioal Practice Problems (your homeworks, worksheets, quizzes, previous eams, previous review sheets are also ecellet review problems) Fid a formula for the iverse fuctio of h if h ( ) Suppose you are give f has a iverse fuctio, ad that f ( ), f ( ) 7, f '(), f '(), f '(7) What is ( f )'() (the derivative of the iverse fuctio at ) Fid the derivative of the followig si ta (a) y (arcta()) y y (l( )) (d) y ( e ) Fid the followig itegrals d d d (a) d 9d (d) (e) (f) 9 9d d d (g) ( ) e d (h) (i) (j) e sec 6 () d (k) e cos() d (l) d (m) d () d
Fid the sum of the followig series (a) ( ) ( )( ) (d) (e) ( ) (f) ( )! 6 Determie which of the followig series are coverget Give full reasoig (a) ( ) si (d) (e)! 9 ()! ()! 7 (a) If you were told that cos diverges, ca you coclude aythig about whether cos coverges or diverges? If a coverges, ca you coclude aythig about whether ( a ) coverges or diverges? If a ad b both diverge, ca you coclude aythig about whether ( a b ) coverges or diverges? 8 Fid the Taylor series of the give fuctio at the give base poit, to the give umber of terms (a) f ( ) e at a, the whole series (d) f ( ) 7 at a, the whole series f ( ) 7 at a, the whole series f ( ) ta at a, order Taylor polyomial (e) f ( ) at a, order Taylor polyomial (f) f ( ) si at a, order Taylor polyomial Bous for the whole series (g) f ( ) arcsi at a, order Taylor polyomial 9 For the followig power series, fid the radius of covergece, the iterval of covergece State where the series coverges absolutely, coverges coditioally, ad diverges (a) ( )! ( ) ( ) (d) ( ) (e) ( )! If the fuctio f ( ) is replaced by its d order Taylor polyomial cetered at a Use a techique from the course to estimate the error R ( ) whe d Cosider the itegral (a) Use the Trapezoid Rule with to make a estimate of this itegral What is a estimate of the error i (a)? Use the Trapezoid Rule error boud from the formula sheet
Use Simpsos Rule with to make a estimate of this itegral (d) What is a estimate of the error i (a)? Use the Simpsos Rule error boud from the formula sheet Solve the followig differetial equatios or iitial value problems Give your aswer as y a eplicit formula i dy (a) y d y' y dy y, y ( ) d (d) y ' y ta (e) y '' y' y (f) y '' y' y (g) y '' y' y, y ( ), y '() (a) I the followig slope fields, draw the solutio curve through the poit (, ) Which of these slope fields matches the differetial equatio dy y d
Some Hits or Solutios h ( ) ( f )'() f '( f ()) f '() si (a) y' (arcta( )) (cos l(arcta( )) si arcta( )( 9 ) ta y' (l )sec y ' (l( )) (d) e (l ) y' ( e ) (l( e ) e d d (a) the sub w ( ) the sub w ( ) d the partial fractios (d) Trig sub si, get ( )( ) 7 si d the do sub w cos (e) Trig sub sec, get ta d (sec ) d ta, get ta sec d, the sub w sec (f) Tri sub wdw (g) Itegratio by parts u ( ), dv e d (h) Sub w, get, a easy w w dw itegral (i) Sub w e, get the partial fractios (j) Sub w ta (k) w(w ) Itegratio by parts cosistetly twice ad the abstractly solve for e cos() d (l) Improper itegral (m) Improper itegral, will require lim arcta () Improper itegral, will require combiig two logs ito oe usig l A l B l( A / B) Fid the sum of the followig series (a) Two geometric series: Two geometric series: Telescopic (f) Taylor series l( / ) l(/ ) (d) Taylor series e (e) Taylor series l( ) l 6 (a) Multiply by ozero costat to get positive series which the ca be limit compared to Series diverges Alteratig series Show is decreasig 9 by showig its derivative is evetually all egative, ad Series coverges 9 Show series is absolutely coverget by comparig the series of absolute values
si with (d) Ratio test yields lim Series is ( )( ) ( ) coverget (e) Ratio test yields lim Series is coverget ( )( ) 7 (a) No Yes, ( a ) must diverge or else ( ) ( a ) ( a ) would coverge No For eample, ( ) coverges but ad both diverge 8 (a) 7 6 ( ) ( ) ( ) (d)! 8 8 ( ) ( ) ( ) ( ) 7 77 (e) (f) 6 (g) 6 9 For the followig power series, fid the radius of covergece, the iterval of covergece State where the series coverges absolutely, coverges coditioally, ad diverges (a) Radius=, Iterval of covergece = ( 7,) Coverges absolutely for 7, diverges everywhere else Radius= /, Iterval of covergece = [ /, 9 /) Coverges absolutely for 9, coverges coditioally at, diverges everywhere else Radius=, Iterval of covergece = { } (sigle poit) Coverges absolutely for, diverges everywhere else (d) Radius=, Iterval of covergece = (, ) Coverges absolutely everywhere (e) Radius=, Iterval of covergece = [,] Coverges absolutely for, diverges everywhere else / / f '''( ) ( ) Hece we may choose M (9) 6 6 f ( ), a, b,, (a) T f ''( ) so may use / ( ) M () S (d) f ( ) so may use / ( ) M (a) y Ce y C y C (d) y l sec ta cos C cos (e) y Ce / (f) y e / 7 ( C cos C si ) (g) y e e The last choice / / C e /