Question 0809

Given that ${f(x)=\mathrm{e}^{\cos \frac{1}{3} x},}$ find ${f(0),}$ ${f'(0)}$ and ${f''(0).}$

Hence write down the first two non-zero terms in the Maclaurin series for ${f(x).}$ Give the coefficients in terms of ${\mathrm{e}.}$

Given that the first two non-zero terms in the Maclaurin series for ${f(x)}$ are equal to the first two non-zero terms in the series expansion of ${\displaystyle \frac{1}{a + bx^2},}$ where ${a}$ and ${b}$ are constants, find ${a}$ and ${b}$ in terms of ${\mathrm{e}.}$