Find the first, second and third derivatives of f (x) = xex. Evaluate these at x = 0 and use theseresults to write the first three non-zero terms in the Maclaurin series for this function.

Step 1: Find the first derivative of f(x) = xex

We use the product rule which states that the derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second.

f'(x) = 1ex + xex = ex(1 + x)

Step 2: Find the second derivative of f(x) = xex

Again, we use the product rule on the first derivative.

f''(x) = ex(1 + x)' = ex + ex = 2ex

Step 3: Find the third derivative of f(x) = xex

The third derivative is simply the derivative of the second derivative.

f'''(x) = (2ex)' = 2ex

Step 4: Evaluate these at x = 0

f'(0) = e^0(1 + 0) = 1 f''(0) = 2e^0 = 2 f'''(0) = 2e^0 = 2

Step 5: Write the first three non-zero terms in the Maclaurin series for this function.

The Maclaurin series for a function f(x) is given by:

f(x) = f(0) + f'(0)x/1! + f''(0)x^2/2! + f'''(0)x^3/3! + ...

So, the first three non-zero terms of the Maclaurin series for f(x) = xex are:

f(x) = 0 + 1x/1! + 2x^2/2! + 2*x^3/3! = x + x^2 + (2/3)x^3