Write out the Maclaurin Expansion for the function f(x) = ln(x 2 + 3x + 2).

Answer 1

We can approximate the Maclaurin expansion for the function by using 4 terms as shown below:

f(x) = f(0) + f'(0)x/1! + f''(0)x²/2! + f'''(0)x³/3! + ... + fⁿ(0)xⁿ/n! (remember Maclaurin series are centered at a=0)

f(x) = ln(x²+3x+2) --> f(0) = ln(2)

f'(x) = (2x+3)/(x²+3x+2) --> f'(0) = 3/2

f''(x) = [2(x²+3x+2)-(2x+3)²]/(x²+3x+2)² --> f''(0) = -5/4

f'''(0) = 9/4

f(x) = f(0) + f'(0)x/1! + f''(0)x²/2! + f'''(0)x³/3! + ... + fⁿ(0)xⁿ/n!

ln(x²+3x+2) ≈ ln(2) + (3/2)x + (-5/4)x²/2 + (9/4)x³/6 = ln(2) + (3/2)x - (5/8)x² + (3/8)x³

Hope this helped! Let me know if I need to clear anything up!