Certainly! The Maclaurin Series is a representation of a function as an infinite sum of terms, calculated from the derivatives at a single point. We are interested in finding the representation up to the 5th degree for the function f(x) = x^3 * cos(x).
Here are the steps:
1. First, let's recall the general formula for a Maclaurin series which is:
f(x) ≈ f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ...
2. We start by finding the value of the function and its derivatives at x=0.
f(0) = (0)^3 * cos(0) = 0
f'(x) = 3x^2 * cos(x) - x^3 * sin(x), so f'(0) = 0
f''(x) = -2x^3*sin(x) - 6x*cos(x), so f''(0) = 0
f'''(x) = -9x^2*sin(x) - 6*cos(x), so f'''(0) = -6
f''''(x) = 18x*sin(x) + 6*x^2*cos(x), so f''''(0) = 0.
3. Now, we will substitute these values into the Maclaurin series formula.
4. But since we're only interested in the series upto the fifth degree, we should stop and truncate the series at x^5. This is denoted by the big O notation, O(x^5).
5. As you can see, the n-th derivative of our function at 0 is 0 for n not equal to 3, the coefficients for all the terms up to x^5 are zero except for the x^3 term. This leads to the fact that all terms except x^3 are zero.
So, the Maclaurin series for `x^3 * cos(x)` only includes the x^3 term and is `x^3 + O(x^5)`.