Find the Taylor or Maclaurin polynomial Pn(x) for the function f with the given values of c and n.

Question

Find the Taylor or Maclaurin polynomial P_n(x) for the function f with the given values of c and n. Then give a bound on the error that is incurred if P_n(x) is used to approximate f(x) on the given interval. (Round your answer to six decimal places.)

f(x) = e⁴x, c = 1, n = 4, [1, 1.1]

P₄(x)=

|R₄(x)| ≤

Answer 1

Well first we need 5 derivatives.

f'(x) = e^4

f''(x) = f'''(x) = f''''(x) = f'''''(x) = 0

f(1) = e^4

f'(1) = e^4

f''(1) = f'''(1) = f''''(1)=f'''''(1) = 0




P4(x) = f(1)/0!(x-1)^0 + f'(1)/1!(x-1)^1 + f''(1)/2!(x-1)^2 +
f'''(1)/3!(x-1)^3 + f''''(1)/4!(x-1)^4

= e^4 + e^4(x-1)

= e^4 * ( 1 + x - 1)

= e^4 * x

The error term is zero