find the difference quotient of f, that is, find (f(x + h)-f(x))/h, h≠0, for the following f f(x)=x² - 2x +…
find the difference quotient of f, that is, find (f(x + h)-f(x))/h, h≠0, for the following f f(x)=x² - 2x + 8 (f(x + h)-f(x))/h = □
Answer
Explanation:
Step1: Find f(x + h)
Substitute (x+h) into (f(x)): [ \begin{align*} f(x + h)&=(x + h)^2-2(x + h)+8\ &=x^{2}+2xh+h^{2}-2x-2h + 8 \end{align*} ]
Step2: Calculate f(x + h)-f(x)
[ \begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-2x-2h + 8)-(x^{2}-2x + 8)\ &=x^{2}+2xh+h^{2}-2x-2h + 8-x^{2}+2x - 8\ &=2xh+h^{2}-2h \end{align*} ]
Step3: Find the difference - quotient
[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-2h}{h}\ &=\frac{h(2x + h-2)}{h}\ &=2x+h - 2 \end{align*} ]
Answer:
(2x+h - 2)