Question

find the difference quotient of f, that is, find \\(\frac{f(x + h) - f(x)}{h}\\), \\(h \
eq 0\\), for the following function. be sure to simplify\\(f(x) = 2x^2 + x - 3\\)\\(\frac{f(x + h) - f(x)}{h} = \square\\) (simplify your answer.)

Explanation:

Step1: Compute $f(x+h)$

$f(x+h)=2(x+h)^2+(x+h)-3=2(x^2+2xh+h^2)+x+h-3=2x^2+4xh+2h^2+x+h-3$

Step2: Calculate $f(x+h)-f(x)$

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

Step3: Divide by $h$ and simplify

$$\begin{align*} \frac{f(x+h)-f(x)}{h}&=\frac{4xh+2h^2+h}{h}\\ &=\frac{h(4x+2h+1)}{h}\\ &=4x+2h+1 \end{align*}$$

Answer:

$4x + 2h + 1$