To find the 3rd Maclaurin polynomial for the function f(x) = sin(x), we start by recalling that the Maclaurin series is a Taylor series expansion around 0. The general formula for the Maclaurin series is:
f(x) = f(0) + f'(0)x + rac{f”(0)}{2!}x^2 + rac{f”'(0)}{3!}x^3 + ext{higher order terms}
First, we need to compute the function and its derivatives at 0:
- f(0) = sin(0) = 0
- f'(x) = cos(x) → f'(0) = cos(0) = 1
- f”(x) = -sin(x) → f”(0) = -sin(0) = 0
- f”'(x) = -cos(x) → f”'(0) = -cos(0) = -1
Now, substituting these values into the Maclaurin series formula up to the third degree, we get:
f(x) = 0 + 1*x + rac{0}{2!}x^2 + rac{-1}{3!}x^3
This simplifies to:
f(x) = x – rac{x^3}{6}
So, the 3rd Maclaurin polynomial for the function f(x) = sin(x) is:
P_3(x) = x – rac{x^3}{6}