Question

Write the first four terms of a Maclaurin series for 𝑓(𝑥)= 𝑒𝑥

Answer 1

$f(x) = {e^x} \Rightarrow f'(x) = {e^x} \Rightarrow f''(x) = {e^x} \Rightarrow f'''(x) = {e^x}$

Then

$f(x) = f(0) + \frac{{f'(0)}}{{1!}}x + \frac{{f''(0)}}{{2!}}{x^2} + \frac{{f'''(0)}}{{3!}}{x^3} + ... = {e^0} + \frac{{{e^0}}}{1}x + \frac{{{e^0}}}{2}{x^2} + \frac{{{e^0}}}{6}{x^3} + ... = 1 + x + \frac{1}{2}{x^2} + \frac{1}{6}{x^3} + ...$

Answer: $f(x) = 1 + x + \frac{1}{2}{x^2} + \frac{1}{6}{x^3} + ...$