Question

3. Answer the following questions. Justify your answers. A. (&pts) Find the first three nonzero terms of the Taylor series of the function f(x) = px? centered at a = -1. B. (&pts) What is the Maclaurin series for f(x) = sin x? Use it to obtain the Maclaurin series for the function g(x) = x. sin 2x. C. (4pts) Suppose that f(2)= 1, f'(2) = 3, and f" (2) = -2. How can you use this information to estimate the value of f(2.1)? Find the best estimate you can for f(2.1).

Answer 1

A) To Find out 1st three non-zero terms of the taylor series of the function flor) - centered at a = -1. x² a We know that taylor series expansion is of function fra) centered at given by atbox F Cath) = f(x) = 2 bepka) KI h=x-a kao heat! F(t)= e Kao Ź (2+1) FT) k! (+1)°F64), (XH)'F'(U), Carls?f"44) 이 Ist three terms 2! era FO) (2x) F"(x) = 872 2 + (2x)². ex2 f(x)= e ex? » F (H) = e putting all there is FH) = -2 e F" () = 2e + 4.2 = 60

Eet ()-e) + (2+1)? (60) U (**){-20 2! U ē -2e (x+1) +32 (x+i)? B We know that, Me clauris series of F(1) sing + xs a 7 71 t X-23 3 * 51 to find meclaurin series of g(1) - a sinza put a=2x ve get F(2x) = sin2x = (2x) - {x)3 31 + (2x) (2x)*+- 7! -Ő 51 mutiply above by X → gid= XF(2x) = q sin2x g(x) 2x? + 2. X SI 8.74 27. 28 31 7!

ue know that By taylor series Formule. f(a+b)# kº Fla) + h' f'(a) O ! !! + h? F"(a) 2! take a = 2 në 0.1 putting in above ath=2.1 → F/2.1) F12) + (0:1)F(2) + (0:1).25"(2) 1 + (0:1) 3 + (0:01) I + 03 0:01 1.29 F(2-1) =