Question

follow the step by step process to determine the difference quotient, $\frac{f(x + h)-f(x)}{h}$. f(x)=4x^2 - 3 a. determine f(x + h). f(x + h)= (simplify your answer.) b. simplify the expression f(x + h)-f(x). f(x + h)-f(x)= (simplify your answer. do not factor.) c. determine the difference quotient $\frac{f(x + h)-f(x)}{h}$. $\frac{f(x + h)-f(x)}{h}=$ (simplify your answer. do not factor.)

Explanation:

Step1: Find f(x + h)

Substitute \(x+h\) into \(f(x)\):
\[

$$\begin{align*} f(x + h)&=4(x + h)^2-3\\ &=4(x^{2}+2xh+h^{2})-3\\ &=4x^{2}+8xh + 4h^{2}-3 \end{align*}$$

\]

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

\[

$$\begin{align*} f(x + h)-f(x)&=(4x^{2}+8xh + 4h^{2}-3)-(4x^{2}-3)\\ &=4x^{2}+8xh + 4h^{2}-3 - 4x^{2}+3\\ &=8xh+4h^{2} \end{align*}$$

\]

Step3: Determine the difference quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{8xh + 4h^{2}}{h}\\ &=\frac{h(8x + 4h)}{h}\\ &=8x+4h \end{align*}$$

\]

Answer:

a. \(4x^{2}+8xh + 4h^{2}-3\)
b. \(8xh+4h^{2}\)
c. \(8x + 4h\)