Calculus and Its Applications, Chapter 1, Review Exercises, Section Review Exercises, Problem 32

Below is the graph of the function $y = g(x)$. Determine the simplified difference quotient for
$f(x) = 2x^2 - 3$.



We have,
If $f(x) = 2x^2 - 3$, so

$
\begin{equation}
\begin{aligned}
f(x + h) &= 2(x + h)^2 -3 \\
\\
&= 2(x^2 + 2xh + h^2) - 3\\
\\
&= 2x^2 + 4xh + 2h^2 - 3
\end{aligned}
\end{equation}
$


Then,

$
\begin{equation}
\begin{aligned}
f(x + h) - f(x) &= 2x^2 + 4xh + 2h^2 - 3 - (2x^2 - 3) \\
\\
&= 2x^2 + 4xh + 2h^2 - 3 - 2x^2 + 3\\
\\
&= 4xh + 2h^2
\end{aligned}
\end{equation}
$


Thus,

$
\begin{equation}
\begin{aligned}
\frac{f(x + h) - f(x)}{h} &= \frac{4xh + 2h^2}{h}\\
\\
&= \frac{2h (2x + h)}{h}\\
\\
&= 2(2x + h)
\end{aligned}
\end{equation}
$