Calculus II - Problem Drill 21: Power Series, Taylor and Maclaurin Polynomial Series
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1 Calculus II - Problem Drill : Power Series, Taylor ad Maclauri Polyomial Series Questio No. of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as Fill i the blak: + is of l( ) +. Questio #0 (A) a Taylor polyomial approimatio (B) a Maclauri polyomial approimatio (C) a Taylor series represetatio (D) a Maclauri series represetatio (E) Noe of these A. Icorrect! Recall that if the polyomial is cetered aroud = 0, it is called Maclauri. Feedback o B. Correct! This is the correct aswer. The epressio is a Maclauri polyomial approimatio of the fuctio. C. Icorrect! It is ot a series because it stops at the fourth power. It is ot a series because it stops at the fourth power. Oe of the give aswers is correct. 3 4 l( + ) is approimated by the Maclauri polyomial give by
2 Questio No. of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as. The d order Taylor polyomial of f( ) = e aroud = is give by e + e ( ) + ( ). What is the missig factor? Questio #0 (A) e (B) e (C) e 6 (D) (E) A. Correct This is the correct aswer. The missig factor is e. B. Icorrect! Remember the formula for the coefficiets for a Taylor Polyomial. Feedback o C. Icorrect! Remember the formula for the coefficiets for a Taylor Polyomial. Remember the formula for the coefficiets for a Taylor Polyomial. Remember the formula for the coefficiets for a Taylor Polyomial. The formula for a Taylor polyomial aroud = c is give by: 3 f( ) f( c) + f '( c)( c) + f ''( c)( c) + f '''( c)( c) +...! 3! f ''( c) = e '' = 4e = e.! therefore, the d order term has a coefficiet of ( ) ( ) =
3 Questio No. 3 of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as The 5 th 3 5 order Maclauri polyomial of f( ) = si( ) is give by What is the missig factor? (A) Questio #03 (B) (C) 6 (D) 6 (E) A. Icorrect! Remember the formula for the coefficiets for a Maclauri Polyomial. B. Icorrect! Remember the formula for the coefficiets for a Maclauri Polyomial. Feedback o C. Correct! This is the correct aswer. The missig factor is -/6. Remember the formula for the coefficiets for a Maclauri Polyomial. Remember the formula for the coefficiets for a Maclauri Polyomial. The formula for a Maclauri polyomial is give by: f f + f + f + f + f +! 3! 4! 3 (4) 4 ( ) (0) '(0) ''(0) '''(0) (0)... therefore, the 3 rd order term has a coefficiet of '''(0) f =. 3! 6
4 Questio No. 4 of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as 4. Fid the radius of covergece for the power series. = 0 Questio #04 (A) 0 (B) (C) (D) 3 (E) A. Icorrect! B. Icorrect! Feedback o C. Correct! This is the correct aswer. Usig the Ratio Test, the radius of covergece is R =. Give = 0, we use the Ratio Test to fid a = = =. Recall that + + lim lim lim a a the Ratio Test states that the series coverges if lim + <. Therefore, sice we a have <, the radius of covergece is R =.
5 Questio No. 5 of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as 5. Fid the radius of covergece for the power series!. = 0 Questio #05 (A) (B) 0 (C) (D) 3 (E) A. Icorrect! B. Correct! This is the correct aswer. Usig the Ratio Test, the radius of covergece is R = 0. Feedback o C. Icorrect! Give = 0!, we use the Ratio Test to fid + a + ( + )! lim = lim = lim( + ) =. a! Thus, the series coverges oly whe =0. The radius of covergece is 0.
6 Questio No. 6 of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as 6. Fid the radius of covergece for the power series. = 0! Questio #06 (A) (B) 0 (C) (D) 3 (E) A. Correct! This is the correct aswer. Usig the Ratio Test, the radius of covergece is R is ifiity. B. Icorrect! Feedback o choice C. Icorrect! Give = 0!, we use the Ratio Test to fid a /( + )! + + lim = lim = lim = 0 a /! + Thus, by the Ratio Test the series coverges for all values of. The radius of covergece is R =..
7 Questio No. 7 of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as 7. Startig with the geometric power series, fid the power series of. Questio # (A) (B) (C) (D) 4 6 (E) A. Icorrect! You forgot to multiply by. B. Icorrect! This is the geometric power series. Feedback o C. Correct! This is the correct aswer, foud by usig mathematical operatios o the geometric power series. Substitute with. The sigs are icorrect. To fid the power series for... = multiply by to get = = , we start with. Now,. Lastly, substitute = to get
8 Questio No. 8 of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as 8. Startig with the geometric power series, fid the power series of. ( ) Questio # (A) (B) 4 6 (C) (D) (E) A. Icorrect! Try takig the derivative. B. Icorrect! This is the geometric power series. A Feedback o Each Aswer C. Icorrect! Try takig the derivative. D. Correct! This is the correct aswer, foud by takig the derivative of the geometric power series. Try takig the derivative. To fid the power series for ( )... = , we start with 3 the derivative o both sides to get ( ) = Now, take
9 Questio No. 9 of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as Questio # The power series represets which of the followig fuctios? (A) cos( ) (B) si( ) (C) e (D) l( ) (E) arcsi( ) A. Correct! This is the correct aswer. The series represets cos(). B. Icorrect! Try writig out the first few terms of the Maclauri series. A Feedback o Each Aswer C. Icorrect! Try writig out the first few terms of the Maclauri series. Try writig out the first few terms of the Maclauri series. Try writig out the first few terms of the Maclauri series. 3 (4) 4 Note that cos( ) cos(0) + (cos(0)) ' + (cos(0)) '' + (cos(0)) ''' + (cos(0)) +...! 3! 4! 4 =
10 Questio No. 0 of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as Give that cos( ) = , fid a power series for cos( ). Questio #0 4 6 (A) (B) 4 8 (C) (D) (E) 4 8 A. Icorrect! Try usig a substitutio. B. Correct! This is the correct aswer. It is foud by substitutig =. Feedback o C. Icorrect! Try usig a substitutio. Try usig a substitutio. Try usig a substitutio Give that cos( ) = , we let = so that 4 8 cos( ) =
e to approximate (using 4
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