In exercises 1 and 2, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers _

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1 Chapter 9 Curve I eercises ad, (a) write the repeatig decimal as a geometric series ad (b) write its sum as the ratio of two itegers _ Distace A ball is dropped from a height of 8 meters. Each time it drops height h meters it rebouds.7h meters/ Fid the total distace traveled by the ball. 4 Salary You accept a job that pays a salary of, the first year. Durig the et 9 years, you will receive.% raise each year. What would be your total compesatio over the 4-year period Compoud Iterest A deposit of $ is made at the ed of each moth for years i a accout that pays 6% iterest, compouded cotiuously. Determie the balace i the accout at the ed of years. 6 Compoud Iterest a deposit of $ is made at the ed of each moth for years i each accout that pays.% compouded mothly. Determie the balace i the accout at the ed of years. Determie the covergece or divergece of the series 7 l _ ( ) ( ) **...( ) *4*6...( ) 4 4 l 9 e

2 e! **...( ) **8...( ) I eercise ad 4, (a) verify that the series coverges, (b) use a graphical utility to fid the idicated partial sum S ad complete the table(c) use a graphig utility to graph the first terms of the sequece of partial sums ad (d) use the table to estimate the sum of the series. S ( ) 4 Writig Use a graphig utility to complete the table for (a) p= ad (b) p= The write a short paragraph describig ad comparig the etries i the table Writig You are told that the terms of a positive series appear to approach zero very slowly as approaches ifiity (i fact a7.7 ) If you are give o other iformatio ca you coclude that series diverges? Support your aswer with a eample. I Eercises 6 ad 7, fid the third-degree Taylor polyomial cetered at c 6 f()= e /, c 7 f()= ta, c 4 I eercises 8- use a Taylor polyomial to approimate the fuctio with a error of less tha. 8 si 9 9 cos.7 l(.7 ) e. A Taylor polyomial cetered at will be used to approimate the cosie fuctio. Fid the degree of the polyomial required to obtai the desired accuracy over the iterval Maimum error Error (a). [-.,.] (b). [-,] (c). [-.,.] (d). [-,]

3 I eercises -8, fid the iterval of covergece of the power series (be sure to iclude a check for covergece at edpoits of the iterval ( ) ( ) ( ) ( ) ( ) ( )!( ) ( ) Show that the fuctio represeted by the power series is a solutio of the differetial equatio 9 y y y 4 (!) y 4 ( ) y! y y y Fid a geometric power series cetered at for the fuctio 4 h( ) Fid a fuctio represeted by the series ad give the domai of the fuctio ( - ).( - ) ( )... Fid a power series for the fuctio cetered at c 44 f() = si, c= 4 4 f() = cos, c= 4 46 f() =, c 47 f() = csc, c= (first terms) 48 f() =, c 49 f() =, c 4 g() =, c h() =, c ( ) 8 Fid the sum of the coverget series by usig a well-kow fuctio. Idetify the fuctio ad eplai how you obtaied the sum 4

4 4!! 6 7 ( ) ( )!! () 8 Formig Maclauri Series Determie the first four terms of the Maclauri series for e (a) by usig the defiitio of Maclauri series ad the formula for the coefficiet of the th ( ) term a f () /! (b) by replacig by i the series for e (c) by multiplyig the series for e by itself because e = e * e 9 Follow the patter of eercise 9 to fid the first four terms of the series for si 6 si t e dt t 6 t cos dt e l( t ) e t 6 dt t e 6 e dt t Use a power series to fid the limit. Verify the result by usig L Hopital s Rule 64 lim arcta Write a epressio for the th term of the sequece,,,, , 4,, 7... Use a graphig utility to graph the first terms of the sequece. Use a graph to make a iferece about the covergece or divergece of the sequece. Verify your iferece aalytically ad, if the sequece coverges, fid its limit a 4 a si Determie the covergece or divergece of the sequece with the give th term. If the sequece coverges, fid its limit (b ad c are positive real umbers a 6 a 7 a

5 8 a l 9 a a ( ) si a a (b c )/ A deposit of $ is made i a accout that ears % iterest compouded quarterly. The balace i the accout after quarters is a (.4 ),,,... (a) compute the first eight terms of the sequece (b) fid the balace i the accout after years by computig the 4th term of the sequece. 4 A compay buys a machie for $,. durig the et years the machie will depreciate at a rate of % per year. (a) fid a formula for the th term of the sequece that gives the value V for the machie t full years after it was purchased. (b) fid the depreciated value of the machie at the ed of full years. (a) use a graphig utility to fid the idicated partial sum S k ad complete the table, ad (b) use a graphig utility to graph the first terms of the sequece of partial sums K Sk 6 ( ) 7 ( ) 8 ( ( ))! 9 ( ) Determie the covergece of divergece of the series (.8) (.8) ( l) Fid the sum of the covergig series 4 ( )

6 6 ( ) 7 [( ) ( )( ) ] 8 cosider the fuctio f defied by e /, f()=, (a) sketch a graph of the fuctio (b) use the alterative form of the defiitio of the derivative (.) ad L Hopital s rule to show that f ()= (c) usig the result of part (b), fid the Maclauri series for f. does the series coverge to f 9 (a)fid the power series cetered at for the fuctio f()= l( ) (b)use a graphig utility to graph f ad the eighth-degree Taylor polyomial P8 ( ) for f (c)complete the table where f()= X F() G(). z l( t ) t. dt ad g()=.7 z Ps (t ). (d) describe the relatioship betwee the graph of f ad P ad the results give i the table i part (c) prove that lim! = for ay real Fid the Maclauri series for f() = l ad determie its radius ad covergece. Use the first four terms of the series to approimate l prove that e is irratioal (e=!...! )

7 64 It is sometimes possible to covert a improper itegral ito a proper iteral havig the same value by makig a appropriate substitutio. Evaluate the followig itegral by makig the idicated substitutio, ad ivestigate what happes if you evaluate the itegral directly without usig a CAS d; u