given the function f(x)=9x² - 1/2x, find the difference quotient (f(-3 + h)-f(-3))/h. enter exact answers…

Explanation:

Step1: Find $f(-3 + h)$

Substitute $x=-3 + h$ into $f(x)=9x^{2}-\frac{1}{2}x$. [ \begin{align*} f(-3 + h)&=9(-3 + h)^{2}-\frac{1}{2}(-3 + h)\ &=9(9 - 6h+h^{2})+\frac{3}{2}-\frac{1}{2}h\ &=81-54h + 9h^{2}+\frac{3}{2}-\frac{1}{2}h\ &=9h^{2}-54h-\frac{1}{2}h + 81+\frac{3}{2}\ &=9h^{2}-\frac{109}{2}h+\frac{162 + 3}{2}\ &=9h^{2}-\frac{109}{2}h+\frac{165}{2} \end{align*} ]

Step2: Find $f(-3)$

Substitute $x = - 3$ into $f(x)=9x^{2}-\frac{1}{2}x$. [ \begin{align*} f(-3)&=9(-3)^{2}-\frac{1}{2}(-3)\ &=9\times9+\frac{3}{2}\ &=81+\frac{3}{2}\ &=\frac{162 + 3}{2}\ &=\frac{165}{2} \end{align*} ]

Step3: Calculate $f(-3 + h)-f(-3)$

[ \begin{align*} f(-3 + h)-f(-3)&=(9h^{2}-\frac{109}{2}h+\frac{165}{2})-\frac{165}{2}\ &=9h^{2}-\frac{109}{2}h \end{align*} ]

Step4: Calculate the difference - quotient

[ \begin{align*} \frac{f(-3 + h)-f(-3)}{h}&=\frac{9h^{2}-\frac{109}{2}h}{h}\ &=\frac{h(9h-\frac{109}{2})}{h}\ &=9h-\frac{109}{2} \end{align*} ]

Answer:

$9h-\frac{109}{2}$