Question

follow the step by step process to determine the difference quotient, $\frac{f(x + h)-f(x)}{h}$. f(x)=x² - 15x 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.) c. determine the difference quotient $\frac{f(x + h)-f(x)}{h}$. $\frac{f(x + h)-f(x)}{h}=$

Explanation:

Step1: Find f(x + h)

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

$$\begin{align*} f(x + h)&=(x + h)^2-15(x + h)\\ &=x^{2}+2xh+h^{2}-15x-15h \end{align*}$$

\]

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

\[

$$\begin{align*} f(x + h)-f(x)&=(x^{2}+2xh + h^{2}-15x-15h)-(x^{2}-15x)\\ &=x^{2}+2xh+h^{2}-15x-15h - x^{2}+15x\\ &=2xh+h^{2}-15h \end{align*}$$

\]

Step3: Determine the difference quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-15h}{h}\\ &=\frac{h(2x + h-15)}{h}\\ &=2x+h - 15 \end{align*}$$

\]

Answer:

a. \(x^{2}+2xh+h^{2}-15x-15h\)
b. \(2xh+h^{2}-15h\)
c. \(2x+h - 15\)