Question

For parts a, b, c and d, use the following function: f(x) = e-5x a) (3 points) Write the Taylor polynomial of degree four for f(x) centered at 0. b) (2 points) Use the Taylor polynomial from part a to estimate the value of e-0.5. (Hint: let find x). c) (3 points) Write the series generated by f(x) at zero in sigma notation. d) (3 points) Find the radius of convergence and state the interval of convergence.

Answer 1

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Answer: - 5x fex)= polynomial af degree four for tex) centered (la) Taylor at o. given by fen) ~ (iv) fol txt'(o) + ? +10) + ht"(0) +x* f (62 31 21 [f'lo) = -5 4). S6 -50 -se f'(X) t"'(X) = -58 -58 25e = 25 (-5)× (-5) frog +'0) = -125 -5x - 125 e f(x)= 5x 2581-5) e 5x a 52 flli, 625e (0) = 625 fling (x) 12581-5) e 0 as e 7 625x4 + 2522 fex) 1 - 5+25x2-125X3 21 3! 4! ( 4. 125 x² 625 24 e + f(x). 1 - 5x + 25x² 6 - 5X OUI To estimate eois е so here x = 0.1 a 0.5 by earo 1 -5.0.1) + 25 (0.1)? - 125.60.137 6 f(0.1) e 625 4 (001) 24 =) 1 -0.5 + 0.125 -0.02083 + 0.012604 015 0.6067742 CS Scanned with CamScanner

f(x) -5X e ' Taylos series for & sta centered at o is given by. f(n) n f(x) (0) n! x n-0 dx lohed in baut (a) f'lol= -5 f"10) = f"10) = -125 25 =6512 653 ficos (-5) so, Maclaurian Series for fex) 5x Z ni - о ( d). Radius of convergence (R) by ear ② (5)n an = ni antl (-5)+1 n! ㅗ R х lim hyoo lin mo an (M+1)! (-5) 5 lin o las no nuo nti So, O =) Rua R Interval convergence is - RLXLR - - olas CS Scanned with CamScanner

(ej e -!+5X lin ko 2x2 by k (-5) 120 kl x 1 +50 = lin koo 2x2 K x E1-5) - 1 +5k + व lim 11 - 5x + 250 Ky 02x24 K-3k! ㅋ Ms (-5) x lim 25 ko + 4 k=3 K x (-5) as x 0 E -> 0 [ a as 25 25 k=3 ) CS Scanned with CamScanner