Find \(f(0)\).
\[ f(0) = \answer {40} \]
Write out a few terms of \(f(x)\) and evaluate the expression at \(x=0\).
Find \(f''(0)\).
\[ f''(0) = \answer {40} \]
Is it easier to compute the series represented by \(f''(x)\) explicitly or use the relationship
between the coefficients of the power series and the derivatives of the function \(f(x)\)?
Find \(f^{(54)}(0)\).
\[ f^{(54)}(0) = \answer {\frac {10}{7} \cdot 54!} \]
Is it easier to compute the series represented by \(f^{(54)}(x)\) explicitly or use the relationship
between the coefficients of the power series and the derivatives of the function
\(f(x)\)?
If you want to use the formula:
\[ a_n = \frac {f^{(n)}(c)}{n!} \]
then \(c=\answer {0}\) and \(n=\answer {54}\).
In order to find \(a_{54}\), note that \(a_{54}\) is the coefficient of \(x^{54}\).
What can you take away from this exercise?