Example 2. Find the upper bound for the remainder for the approximation from Example 1.

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1 Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute value of the st term (this is the et term or the first uused term). Eample. Approimate the sum of by usig the first 6 terms.! Eample. Fid the upper boud for the remaider for the approimatio from Eample. Eample. Fid upper ad lower bouds for the actual sum of the series i Eample. Eample. Approimate 0 with a error of less tha.00.! Eample 5. Use a elemetary series to fid the actual value of the series i Eample. This remaider ca be writte as Alteratig Series Remaider First uused term!! f c c If a oalteratig series is approimated the method is slightly differet ad slightly harder. It is called the Lagrage Remaider or Taylor s Theorem Remaider. Lagrage Remaider! f z c Where z is the -value betwee ad c iclusive which makes f z a maimum. As i a alteratig series remaider the st term of the Taylor series is used however, the derivative factor is carefully chose. st

2 Choose a value of z which makes the f z 05 factor a maimum. This may be at the ceter, at the - value where f is to be evaluated, or you may kow the maimum value i advace (sie ad cosie fuctios have a maimum value of ). Eample 6. Estimate e usig a Maclauri polyomial of degree 0 for e. Eample 7. Use the Lagrage form of the remaider (error) to estimate the accuracy of usig this partial sum. Eample 8. If 5 f 700si ad if.7 is i the covergece iterval for the power series of f cetered at 0, fid a upper limit for the error whe the fourth-degree Taylor f.7. polyomial is used to approimate Eample 9. If f 6 is a decreasig fuctio, fid the error boud whe a 5th degree Taylor polyomial cetered at = is used to approimate f.. Assume the series coverges for =.. Assigmet 8-. Approimate the sum of the alteratig series. If the first four terms are used to approimate the series remaider. e with a error less tha or equal to fid a upper boud for the. Approimate with a sith degree Maclauri Polyomial ad fid a upper limit of the Alteratig Series Remaider.

3 . How may terms of a Maclauri Polyomial are eeded to approimate si with a error of less tha 0.00? 5. How may terms of a Maclauri Polyomial are eeded to approimate si with a error of less tha 0.00? 6. If a Taylor Polyomial cetered at is used to approimate l with a error of less tha 0.00, how may terms are eeded? 7. If f fid the Lagrage error boud if a third degree Taylor Polyomial cetered at is used to approimate f. Assume the series coverges for =. 8. If P 5 for the fuctio from problem 7, fid the rage of possible values for 9. If 6 f. f 00si ad.5 is i the iterval of covergece of the power series for f, the fid the error whe a fifth-degree Taylor polyomial, cetered at 0 is used to f.5. approimate 0. If a sith degree Taylor Polyomial cetered at 0 is used to approimate f, fid the Lagrage error boud for each of the followig if the graph show is a portio of the graph of 7 f. Assume the series coverges for =. 06 a. y b. y c. y. Assumig the fuctio from problem 0 is represeted by a alteratig series, which of the three aswers would be the same usig a alteratig series error boud?. The fuctio f e is approimated by the polyomial f. For what -values will this approimatio have a error of less tha 0.00?. For f l, c : a. Write a Taylor Polyomial P. b. Write a power series for f usig otatio. c. Approimate f. usig P.. d. Fid the actual value of f.. e. Fid the Lagrage error (remaider) boud, R.. f. Fid the umber of terms from the Taylor Polyomial eeded to approimate f. with a error (remaider) less tha.00.. Fid a upper limit for the error whe the Taylor polyomial T approimate f si at 0.5. is used to!

4 07 5. Let f be a fuctio whose Taylor series coverges for all. If f what is the miimum umber of terms of the Taylor series, cetered at, ecessary to approimate f. with a error less tha ? Assume the series has o zero terms. 6. (calculator allowed) h h h h h Let h be a fuctio havig derivatives of all orders for 0, selected values of h ad its first four derivatives are idicated i the table above. h.9. a. Write the first-degree Taylor polyomial for h about = ad use it to approimate b. Write the third-degree Taylor polyomial for h about = ad use it to approimate h.9. c. Assumig the fourth derivative of h is a icreasig fuctio, use the Lagrage error boud h.9 to show that the third-degree Taylor polyomial for h about = approimates with error less tha If d y y, fid. d 8. Fid the poit(s) where the lie(s) taget to the graph of to the graph of y 5. f( ) is/are parallel Itegrate: t 9. dt 0. t 9 arcsi d. d hit: let u. Fid a geeral solutio of the differetial equatio t y. Solve for y.. The rate of growth of bacteria i a culture is proportioal to the umber of bacteria preset at ay time t. If there were 000 bacteria preset days after the itroductio of bacteria ito the culture, ad 5000 preset days later, fid: a. the growth rate for the bacteria i the culture. b. the umber of bacteria iitially itroduced ito the culture. (Roud to the earest whole umber.) c. the estimated umber of bacteria i the culture days after the bacteria were iitially itroduced. (Roud to the earest hudred.) dy dt dy dt

5 . A small dog keel with 8 idividual rectagular holdig pes of equal size is to be costructed usig ft of chai lik fecig material. Oe side of the keel is to be placed agaist a buildig ad requires o fecig, as show i the figure below. a. Fid the dimesios (for each holdig pe) that produce a maimum area for each pe. b. What is that maimum area for each holdig pe? 5. A block of ice is eposed to heat i such a way that the block maitais a similar shape as it melts. The block of ice is iitially feet wide, feet high, ad feet log, as show at right. If the chage i the width of the ice is ft/hr, fid: a. the rate of chage i the volume of the block of ice whe the width is ft. b. the amout of time it will take for the block of ice to completely melt. 6. Fid cos lim e 08 Selected Aswers:..90. R 9..68, R. three terms 5. five terms terms f 6 6 R R.00 0a. R.867 or.868 b. R.75 or.76 c. R.0 or.0. c..97 or or.97 a. P, b. f c. f. P..6975, d. e. R , f. four terms f. l..66,,. Alt Series Error.0006 or Lagrage Error five terms h h.9.6, 6 (a), 7. 5 (b) h, (c) R y 6 0. arcsi h.9.6 0,,, 9. t arcta l t 9 C. arcta C C. yc t

6 09 More Selected Aswers: a. k.58 b. 505 or 506 bacteria c.,500 bacteria a. 8 ft. by 9 ft. b. 7 ft 5a. ft hr b. 6 hours 6. Lesso 8- Itegral Test ad p-series Itegral Test: If the fuctio a ad f f is positive, cotiuous, ad decreasig for ad d either both coverge or both diverge. a f the Note: a f d Eamples: Use the itegral test to determie covergece or divergece... l Eamples: Use the itegral test to determie covergece or divergece of these series... 5.

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