given the following function, determine the difference quotient, $\frac{f(x + h)-f(x)}{h}$. $f(x)=x^{2}-6x…
given the following function, determine the difference quotient, $\frac{f(x + h)-f(x)}{h}$. $f(x)=x^{2}-6x - 8$
Answer
Explanation:
Step1: Find $f(x + h)$
Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=(x + h)^2-6(x + h)-8\ &=x^{2}+2xh+h^{2}-6x-6h - 8 \end{align*} ]
Step2: Calculate $f(x + h)-f(x)$
[ \begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-6x-6h - 8)-(x^{2}-6x - 8)\ &=x^{2}+2xh+h^{2}-6x-6h - 8-x^{2}+6x + 8\ &=2xh+h^{2}-6h \end{align*} ]
Step3: Find the difference - quotient
[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-6h}{h}\ &=\frac{h(2x + h-6)}{h}\ &=2x+h - 6 \end{align*} ]
Answer:
$2x+h - 6$