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,

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