find the average rate of change of f(x)=7x² -...

# Explanation: ## Step1: Recall average - rate - of - change formula The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 5$ and $f(x)=7x^{2}-2$. ## Step2: Calculate $f(b)$ and $f(5)$ $f(b)=7b^{2}-2$, $f(5)=7\times5^{2}-2=7\times25 - 2=175 - 2 = 173$. ## Step3: Substitute into the formula The average rate of change is $\frac{f(b)-f(5)}{b - 5}=\frac{(7b^{2}-2)-173}{b - 5}=\frac{7b^{2}-175}{b - 5}$. ## Step4: Factor and simplify Factor the numerator: $7b^{2}-175 = 7(b^{2}-25)=7(b + 5)(b - 5)$. Then $\frac{7(b + 5)(b - 5)}{b - 5}=7(b + 5)$. # Answer: $7(b + 5)$