given the following function, determine the difference quotient, $\frac{f(x + h)-f(x)}{h}$. $f(x)=3x^{2}-2x + 4$

Question

Accepted Answer

# Explanation:
## Step1: Find $f(x + h)$
Substitute $x+h$ into $f(x)$:
$$
\begin{align*}
f(x + h)&=3(x + h)^2-2(x + h)+4\
&=3(x^{2}+2xh+h^{2})-2x-2h + 4\
&=3x^{2}+6xh+3h^{2}-2x-2h + 4
\end{align*}
$$
## Step2: Calculate $f(x + h)-f(x)$
$$
\begin{align*}
f(x + h)-f(x)&=(3x^{2}+6xh+3h^{2}-2x-2h + 4)-(3x^{2}-2x + 4)\
&=3x^{2}+6xh+3h^{2}-2x-2h + 4-3x^{2}+2x - 4\
&=6xh+3h^{2}-2h
\end{align*}
$$
## Step3: Find the difference quotient $\frac{f(x + h)-f(x)}{h}$
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{6xh+3h^{2}-2h}{h}\
&=\frac{h(6x + 3h-2)}{h}\
&=6x+3h - 2
\end{align*}
$$
# Answer:
$6x+3h - 2$

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QUESTION IMAGE

Question

Explanation:

Step1: Find \(f(x + h)\)

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

Step3: Find the difference quotient \(\frac{f(x + h)-f(x)}{h}\)

Answer: