for the function f(x) = x² + 7, construct and simplify the difference quotient $\frac{f(x + h)-f(x)}{h}$.
for the function f(x) = x² + 7, construct and simplify the difference quotient $\frac{f(x + h)-f(x)}{h}$.
Answer
Explanation:
Step1: Find f(x + h)
Given (f(x)=x^{2}+7), then (f(x + h)=(x + h)^{2}+7=x^{2}+2xh+h^{2}+7).
Step2: Substitute f(x + h) and f(x) into the difference - quotient formula
(\frac{f(x + h)-f(x)}{h}=\frac{(x^{2}+2xh + h^{2}+7)-(x^{2}+7)}{h}).
Step3: Simplify the numerator
((x^{2}+2xh + h^{2}+7)-(x^{2}+7)=x^{2}+2xh + h^{2}+7 - x^{2}-7=2xh+h^{2}). So the difference - quotient becomes (\frac{2xh + h^{2}}{h}).
Step4: Factor out h from the numerator and simplify
(\frac{h(2x + h)}{h}=2x + h) (assuming (h\neq0)).
Answer:
(2x + h)