AP Calculus BC Homework - Chapter 8B Taylor, Maclauri, ad Power Series # Taylor & Maclauri Polyomials Critical Thikig Joural: (CTJ: 5 pts.) Discuss the followig questios i a paragraph: What does it mea to say a elemetary fuctio is approimated by a polyomial? Why would you wat to use a polyomial istead of the eact elemetary fuctio to approimate a fuctio value? For eample, why use P istead of f() = e to 6 approimate f() (compare values)? Where will a elemetary fuctio ad its Taylor polyomial have the eact same value? Discuss the accuracy of the Taylor polyomial approimatio based o the ceter ad the degree of the polyomial. # More Taylor & Maclauri Polyomials, Lagrage error p. 6 #- all, 5,, 7,, 5, 9,,, 5, 5, (A) The fuctio f has derivatives of all orders for all real umbers. Assume f() =-, f () = 5, f () =, ad f () = -8. (This problem was o the calculator portio of the AP test.) (a) Write the third-degree Taylor polyomial for f about = ad use it to approimate f(.5) (b) The fourth derivative of f satisfies the iequality f () () < for all i the closed iterval [.5, ]. Use the Lagrage error boud o the approimatio of f(.5) foud i part (a) to eplai why f(.5) -5. (c) Write the fourth-degree Taylor polyomial, P(), for g() = f( + ) about =. Use P to eplai why g must have a relative miimum at =. As: (A) (a) P () = - + 5(-) + / (-) 8/6 (-) (You would get pts for polyomial, for each error ad for + ) f(.5) P () = - + 5(.5) + / (-.5) 8/6 (.5) =.958.958 (You would get poit for.958 but lose a poit for misuse of equality) (b) Lagrage Error Boud =.5.785 so! f(.5) > -.958 -.785 = -.966 > -5. Therefore, f(.5) -5 ( pts for part (b), oe for umber ad oe for eplaatio) (c) P () = + 5 +. The coefficiet of i P() is g (). This coefficiet is, so g () =. The coefficiet of i P() is g ()/!. This coefficiet is 5, so g () = which is greater tha. Therefore, g has a relative miimum at =. ( pts. for correct polyomial but - for each icorrect or missig term, ad - for etra terms, or +, pt. for the eplaatio.) (B) Let f be the fuctio give by f si 5, ad let P() be the third degree Taylor polyomial for f about =. (a) Fid P(). (b) Fid the coefficiet of i the Taylor series for f about = (c) Use the Lagrage error boud to show that f P. (d) Let G be the fuctio give by about =. G f t dt. Write the third-degree Taylor polyomial for G
AP Calculus BC - Ch. 8B (As: (B) (a) (b) (c) (d) 5 5 5!! P (You would get pts for part (a)) 5!! f P ma f c c 5 5 ( pts for part (b) oe for umber ad oe for egative sig) 65! 8 ( pt for error boud i appropriate iequality) ( pts. for correct polyomial but - for each icorrect or missig term, ad- for etra terms, or +, or misuse of equality) # Power Series p. 6 #a,, 7,, 5, 7,, 5, 7, 9- all, 7, 59-6 all (Eplai both T ad F) # Represetig Fuctio as Power Series p. 6 #,, (Hit: use partial fractios), 9,, 5, 7 Bous Critical Thikig Joural: (CTJ: + pts.) p. 6 # ad 6 - Eplai how to fid the series AND fid the series. #5 Taylor ad Maclauri Series p. 6 #, 6, 9 (develop the formulas, do t just fill i) Use the formula chart for the followig #, 9,,,, 5, 5, 55 Test I: Fill i the chart of series. I will check your homework o the day I give back the test papers for Test I for the bous 9% poits (ot higher tha 5%) o Test I. Remember - it s all or othig. Test II: (You may use your calculator o the whole test.) Fidig a geometric power series cetered o ad o, composite fuctios of si ad cos, a fuctio defied as a itegral, fidig value of e with a series, MacLauri polyomial for cos or si ad Lagrage Formula for error. I will check your homework agai with all problem sets o the day I give back the test papers for Test II. It s all or othig. ad it will raise your Test II grade 9% poits (ot higher tha 5%).
AP Calculus BC - Ch. 8B Problem Set Taylor Polyomial, Lagrage Error, Iterval of Covergece of Power Series Do ot work o this sheet. Show all your work ad eplaatios i your homework otebook. () The power series coverges if ad oly if (A) < < (B) < < (C) < < (D) < < (E) = () The power series diverges!!! (A) for o real (B) if < < (C) if < or > (D) if < < (E) if () The series ad oly if (A) = (B) < < (C) = (D) < < (E) < or >! coverges if () The iterval of covergece of the series obtaied by differetiatig term by term the series (A) (B) < (C) < (D) 9 6 (E) oe of the precedig is (5) Let f() = covergece of (A) = oly (B) (C) < < (D) < (E) < <. The iterval of f t dt is (6) The first four terms of the power series A C E i for f() = are B 8 8 6 D 8 6 8 8 8 (7) Cosider the power series a, 7 where a = ad a a for. (a) Fid the first four terms ad the geeral term of the series. (b) For what values of does the series coverge? (c) If f() = a, fid the value of f (). (8) Determie all values of for which the series k k k l k coverges. Justify your aswer.
AP Calculus BC - Ch. 8B (9) Let f be the fuctio give by f t ad G be the fuctio give by t (a) (b) (c) G f t dt. Fid the first four ozero terms ad the geeral term for the power series epasio of f(t) about t =. Fid the first four ozero terms ad the geeral term for the power series epasio of G() about =. Fid the iterval of covergece of the power series i part (b). (Your solutio must iclude a aalysis that justifies your aswer.) Aswers: Problem Set Ch 8B () C, () A, () C, () B, (5) D, (6) B 7 7 7 (7) (a) 7!!! (b) for all d 7 d 7 7 7 7 7 7 7 ( c) 7 7 7 d! d!!!!!!! 7 7 f ' 7 7e!! 7 (8) By the Ratio test the limit = which must be < ½ < < ½. Check edpoit = ½: First prove diverges by the Itegral test k k l k ad the prove k l k Check edpoit = - ½: Prove ANSWER: - ½ < < ½ diverges by direct compariso to k k l k l k k k coverges by the alteratig series test. (9) (a) t + t t 6 + + (-) (t) 5 7 (b) 5 7 (c) - < <
AP Calculus BC - Ch. 8B 5 Problem Set : Power Series for Commo Fuctios, Taylor ad Maclauri Series () si () = 5 ( A )! 5!! 5 ( B )! 5!! ( C )!!! 6 ( D )!! 6!! 5 ( E )! 5!! () For < < if f() =, the f () = () Which of the followig epressios is impossible? (A) i powers of (B) i powers of (C) l i powers of ( ) (D) ta i powers of (E) l ( ) i powers of (5) The coefficiet for i the Maclauri series of f() = e / is (A) / (B) / (C) /96 (D) /8 (E) /8 (6) the Taylor series epasio for e about = is A B C D A! B e C e! E D e!! () The coefficiet of i the Taylor series for e about = is (A) /6 (B) / (C) ½ (D) / (E) 9/ E e!!
AP Calculus BC - Ch. 8B 6 (7) The coefficiet of i the Taylor series about of f() = cos is A B C D E 6 (8) Which of the followig series ca be used to compute l.8? (A) l ( ) epaded about = (B) l about = (C) l i powers of ( ) (D) l ( ) i powers of ( ) (E) oe of the precedig () Let f a, g( ) b, ad be a umber for which both these series coverge. Which of the followig statemets is false? D a coverges to f ' A a b coverges to f g B a b coverges to f g C f a is cotiuous at E oe of the precedig (9) If e. is computed usig series, the, correct to three decimal places, it equals (A).95 (B).95 (C).9 (D).9 (E).99 () If the approimate formula si = - is used ad < (radia),! the the error is umerically less tha (A). (B). (C).5 (D).8 (E).9 () The coefficiet of i the Maclauri series for e si is (A) (B) (C)! (D) (E) ¼ () If the approimatio =.75 (radia) is used, the the value of si correct to four decimal places is (A). (B).5 (C).9 (D).5 (E).5 () The coefficiet of ( ) 5 i the Taylor series for l about = is (A) / (B) /5! (C) /5! (D)! (E)! (5) If a appropriate series is used to. evaluate e d, correct to three decimal places, the defiite itegral equals (A).9 (B).8 (C).98 (D).8 (E).9 (6) The fuctio f a ad f () = f() for all. If f() =, the f(.), correct to three decimal places, is (A).95 (B). (C).89 (D).8 (E).
(7) (a) Write the Taylor series epasio about = for f() = l ( + ). Iclude a epressio for the geeral term. (b) For what values of does the series i part (a) coverge? (c) Estimate the error i evaluatig l by usig oly the first five ozero terms of the series i part (a). Justify the aswer. (d) Use the result foud i part (a) to determie the logarithmic fuctio whose Taylor series is. (8) (a) Fid the first four ozero terms i the Taylor series epasio about = for (b) (c) f Use the results foud i part (a) to fid the first four ozero terms i the Taylor series epasio about = for g() =. Fid the first four ozero terms i the Taylor series epasio about = for the fuctio h such that h () = ad h() =. (9) Let f be the fuctio defied by f (a) Write the first four terms ad the geeral term of the Taylor series epasio for f() about =. (b) use the result from part (a) to fid the first four terms ad the geeral term of the series epasio about = for l. (c) Use the series i part (b) to compute a umber that differs from l / by less tha.5. Justify. () (a) Fid the first five terms i the Taylor series about = for f (b) Fid the iterval of covergece for the series i part (a). (c) Use partial fractios ad the result from part (a) to fid the first five terms i the Taylor series about = for g.
AP Calculus BC - Ch. 8B 8 Aswers Problem Set : () B, () A, () E, () A, (5) E, (6) C, (7) D, (8) C, (9) A, () C, () C, () A, () E, ()E, (5) A, (6) C (7) (a) l (b) - < < (c) It is a alteratig series R5 a 6 8 6 8 (d) If l (+) = l the l l l (8) (a) 8 6 (If you write more tha terms, you lose poits) 6 9 (b) 8 6 6 9 7 (c) h() = d C 8 6 8 56 7 Sice h, first terms of h 8 56 (9) (a) (-) + (-) (-) + +(-) (-) + (b) (c) By the alteratig series test R <.5. Fidig.5 l implies that (Whe =, C from atidiff. = ) 5. whe =, the remaider would be <.5 5 5 5 5 l.5 () (a) + + + () + () = + + + 8 (b) ½ < < ½ (c) + + 7 + 5 +
AP Calculus BC - Ch. 8B 9 Little Gree Book of Calculus BC Properties Ch. 8B Power Series 8B- Taylor polyomial, Maclauri Polyomial Approimatio (why used, epasio formula, otatio, work out a eample) 8B- Lagrage Error Approimatio (eplaatio ad formula ad meaig of all variables, how to determie the bouds for a Lagrage Error, work out a eample) 8B- Power Series (defiitio, use graphs to eplai i words, geeral formula ad epaded formula cetered o ad o c) 8B- Guidelies for Fidig a Taylor Series Formula (Discuss ad work a eample) 8B-5 Covergece of Power Series ( cases ad e of each, eplai radius ad iterval of covergece, how to fid the iterval of covergece usig the Ratio Test, how to fid the iterval of covergece usig the Geometric Series Test) 8B-6 Elemetary fuctios represeted as a power series (recreate the chart, whole page) 8B-7 Usig basic series to fid series for Composite Fuctios (e. power series for si, geeral ad epaded form ad eplai how you got it ad whe you ca do this, e. power series for si, geeral ad epaded form ad eplai how you got it) 8B-8 e defied as a series (use power series to evaluate e, geeral ad epaded form ad eplai how you got it e: Fid!! 8B-9 defied as a series (use power series to evaluate, geeral ad epaded form ad eplai how you got it ) 8B- Operatios o Power Series (additio of series, term by term differetiatio ad cos itegratio, e. use a power series to fid d through the poit (, )) 8A- Disher Philosophy Our greatest glory is ot i ever fallig, but risig each time we fall. (Commet o how you see it i my life ad how you ca use it i your life.)