Error for power series (Day 2) YOU MAY USE YOUR CALCULATOR TO COMPUTE FRACTIONS AND OTHER SIMPLE OPERATIONS

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Elmer Thompson
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1 AP Calculus BC CHAPTE B WOKSHEET INFINITE SEQUENCES AND SEIES Name Seat # Date Error or power series (Day ) YOU MAY USE YOU CALCULATO TO COMPUTE FACTIONS AND OTHE SIMPLE OPEATIONS a) Approimate si usig the Maclauri polyomial o degree three or y si b) Determie the maimum error o your approimatio rom part (a) Is there aother method to determie this maimum error? I so, which oe is easier to use? c) Usig your aswer to part (b), id a closed iterval where the eact value o si must be a) Approimate e usig the Maclauri polyomial o degree three or y e b) Determie the maimum error o your approimatio rom part (a) Is there aother method to determie this maimum error? I so, which oe is easier to use? c) Usig your aswer to part (b), id a closed iterval where the eact value o e must be Let be a uctio that has derivatives o all orders or all real umbers ad or which ad ' 0 For >, the th derivative o at = is give by! a) Determie whether has a local maimum, a local miimum, or either at = Justiy your aswer b) Write the third-degree Taylor polyomial or about = c) For >, write a epressio or the th term o the Taylor series or about = d) Fid the radius o covergece o the Taylor series or about = Show the work that leads to your aswer e) Show that the sith-degree Taylor polyomial or about = approimates with a error less tha 00 (Hit: determie the maimum error o the approimatio suggested by the problem) Let be a uctio with derivatives o all orders or all real umbers The third-degree Taylor polyomial or the uctio about is give by T a) Fid ' ad '' b) Determie whether has a local maimum, a local miimum, or either at = Justiy your aswer 0 c) Use T to id a approimatio or d) The ourth derivative o satisies the iequality 0 8 Use Lagrage error boud to demostrate that 0 where the eact value or (0) must be) e) Let G be a uctio such that G' ad G polyomial or G about = or all values o such that (Hit: id a closed iterval Write the third degree Taylor SEE OTHE SIDE

Let e cos Graph o The graph o y y is show above a) Use your kowledge o power series o basic uctios to write the irst our ozero terms ad a epressio or the th term o the Maclauri series epasio or e b)

2 Let e cos Graph o The graph o y y is show above a) Use your kowledge o power series o basic uctios to write the irst our ozero terms ad a epressio or the th term o the Maclauri series epasio or e b) Use your kowledge o power series o basic uctios to write the irst our ozero terms ad a epressio or the th term o the Maclauri series epasio or cos c) Use your aswers oud i parts (a) ad (b) to write the irst our ozero terms o the Maclauri series epasio or e cos d) Fid the value o 0 without dieretiatig e) Let P be the ourth-degree Maclauri polyomial or e cos iormatio rom the graph o y show above, show that (Hit: determie the maimum error o the approimatio suggested by the problem) Usig P 000

3 AP Calculus BC CHAPTE B WOKSHEET INFINITE SEQUENCES AND SEIES Error or power series (Day ) ANSWE KEY a) si 0 (this is S, the sum o the series cosiderig oly the! irst two terms) b) The Maclauri series or si is a coverget alteratig series, so the error made i the approimatio is less tha the absolute value o the irst term o cosidered: c) si! 9,0 This problem ca also be doe usig Taylor s Theorem To do so, we would have: z 0, where represets the remaider whe a polyomial o! degree is used i the approimatio Sice si ad si z So: 0!,9 9,9 9, si or 0 si 0 9 9,0 9,0 I Taylor s Theorem was used or part (b), we would have: si or 0 si 0,9,9 9 a) e 88!! 8 z b) Usig Taylor s Theorem: 0, where represets the! remaider whe a polyomial o degree is used i the approimatio Sice e which icreases, ad 0 z z e So: 0!,0 The series or e rom part (a) is NOT alteratig, so Taylor s theorem is the oly method we ca use to estimate the error made i the approimatio

4 c),9, e or 8 e 8,0,0 Let be a uctio that has derivatives o all orders or all real umbers ad or which For >, the th derivative o at = is give by!! " has a local miimum at = The uctio has a horizotal taget lie ad it is cocave up (secod derivative test) " ad P 8!!! a lim lim lim : adius o a covergece is P We ca see that the resultig series is a alteratig series 8 We ca estimate the error usig the remaider o the alteratig series a ad ' 0 a) Sice ' 0 ad 0 b) c) d) e) Let be a uctio with derivatives o all orders or all real umbers The third-degree Taylor polyomial or the uctio about is give by T a) ' ad ''! 8 b) Neither Sice ' 0 is icreasig at = c) 0 T0 d) We ca calculate the error committed i our approimatio i part c): z 0 0 8! 0! Sice 8 or all values o such that 0 : 0 Sice the error is smaller tha 0 e) G C Sice G C G Ad so

5 a) e b)!!! or e!!! cos!!!! or cos!!! 90 c) e cos d) a 0!, so 0 8 e) Usig Lagrage s orm o the remaider: where 0 z The graph above clearly shows that the error as z! 0 0!! z P 0! 0,09 o the iterval, which shows that 0, Thereore we ca boud P 000

e to approximate (using 4

e to approximate (using 4 Review: Taylor Polyomials ad Power Series Fid the iterval of covergece for the series Fid a series for f ( ) d ad fid its iterval of covergece Let f( ) Let f arcta a) Fid the rd degree Maclauri polyomial