I cant figure out this taylor polynomial question. Could i get some help please? Thanks!

Question

Use an appropriate Taylor polynomial to approximate 245^(4/5) to within 0.001.

Clearly explain why you know that your answer has this accuracy.

Answer 1

Let f(x) = x^4/5

Since 243^4/5 = 3⁴ = 81, use the Taylor Series for f centered at 243:

81 + f'(243)(x - 243) + [f"(243) / 2!](x-243)² + [f'''(243)} / 3!](x - 243)³ + ...

245^4/5 = f(245) = 81 + (4/5)(1/3)(2) - (4/25)(243)^-6/5(1/2)(2)² + (24/125)(243)^-11/5(1/6)(2)³ + ...

On a calculator, 245^4/5 ≈ 81.53289581

Using the Taylor Polynomial of degree 2 for the Taylor Series defined above (I.e., using the sum of the first 3 terms of the Taylor Series), we get 245^4/5 ≈ 81.53289438.