Use P4(x) to approximate e^x when x = 1.5.

Question

Use P₄(x) to approximate e^x when x = 1.5.

Please give a detailed answer, thank you.

Answer 1

Macloren polynomial P₄(x) = f(0) +f'(0)x/1! + f''(0)x²/2! + f'''(0)x³/3! + f⁽⁴⁾(0)x⁴/4!

f(x) = e^x, f'(0) = f''(0) = f'''(0) = f⁽⁴⁾(0) = e⁰ = 1;

So f(1.5) = e^1.5≈ P₄(1.5) = 1 + 1.5/1! + 1.5²/2! + 1.5³/3! + 1.5⁴/4! = 4.3984375