We know that \(f'(2) = \lim_{h \to 0} \frac{f(2+h) - f(2)}{h}\text{.}\) But since we don’t have a graph or a formula for the function, we can neither sketch a tangent line nor evaluate the limit algebraically. We can’t even use smaller and smaller values of \(h\) to estimate the limit. Instead, we have just two choices: using \(h = -1\) or \(h = 1\text{,}\) depending on which point we pair with \((2,3.25)\text{.}\)

So, one estimate is

\begin{equation*}
f'(2) \approx \frac{f(1)-f(2)}{1-2} = \frac{2.5-3.25}{-1} = 0.75\text{.}
\end{equation*}

The other is

\begin{equation*}
f'(2) \approx \frac{f(3)-f(2)}{3-2} = \frac{3.625-3.25}{1} = 0.375\text{.}
\end{equation*}

Because the first approximation looks backward from the point \((2,3.25)\) and the second approximation looks forward, it makes sense to average these two estimates in order to account for behavior on both sides of \(x=2\text{.}\) Doing so, we find that

\begin{equation*}
f'(2) \approx \frac{0.75 + 0.375}{2} = 0.5625\text{.}
\end{equation*}