Question

The prompt for this question is f(x) =sin(x^2)

A)Write the first four terms of the Maclaurin series for f(x)

B)Use the Maclaurin series found in Part A to approximate the integral from 0 to 1 of sin(x^2) dx

C)How many terms are needed to find the value of the integral given in Part B, correct to three decimal places? What is that value?

Answer 1

f' = 2x cos(x^2)
f" = 2cos(x^2) - 4x^2 sin(x^2)
f"' = -12x sin(x^2) - 8x^3 cos(x^2)

so the series is

x^2 -x^6/3! + x^10/5! - x^14/7!

The rest should now present no difficulty

Answer 2

I actually do not know how to do the rest, I am completely lost on this. For part A, when writing the first terms, are they just: x^2 -x^6/3! + x^10/5! - x^14/7! or do I have to plug in numbers 1 through 4, and then solve to get the first four terms?

Answer 3

I think you should look this problem up online or ask a teacher.