Question

pls help a b and c

te the 5th degree aclaurin pol sin(x). Show your Taylkor polyia 1114 31 5 (a) Use Taylor's Remainder Theorem to express the remainder in using M) to appromate st the poin z That is, express R5 (끊) for some point c E (0, (b) Find a bound for the magnitude of the remainder when usingM toapproxamate sin (c) Graph the functions f and Ms on the same Cartesian plane. Also, plot the points . (5) to approximate sin( Compute (use your calculator )s2) and interpret this on you graph. compare to your answer in part (b)? Explain.

Answer 1

"A maclaurin series is given by f(x) = sigma_k = 0^infinity f^(k) (a)/k! x^k in our case, f(x) almostequalto p(x) = sigma_k = 0^n f^(k0 (a)/k! x^k = sigma_k = 0^5 f^(k) (a)/k1 x^k so, what we need to do to get the desired polynomial is to calculate the derivatives, evaluate them at the given point, and plug the results into the given formula f^(0) (x) = f(x) = sin (x) Evaluate the function at the point f(0) = 0 Find the 1^st derivative: f^(1) (x) = f^(0) (x))^' = (sin (x)^' = cos (x) Evaluate the 16st derivative at the given point: (f(0))^' = 1 Find the 26nd derivative: f^(2) (x) = (f^(1)(x))^' = (cos(x))^' = -sin (x) Evaluate the 2^nd derivative at the given point: (f(0))^"" = 0 Find the 3^rd derivative: f^(3) (x) = (f^(2) (x))^' = (- sin (x))^' = -cos (x)"