find the average rate of change for the function over the given interval. y = 4x² between x = 0 and x =…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 0$, $b=\frac{7}{4}$, and $f(x)=4x^{2}$.

Step2: Calculate $f(a)$ and $f(b)$

When $x = a=0$, $f(0)=4\times0^{2}=0$. When $x = b = \frac{7}{4}$, $f(\frac{7}{4})=4\times(\frac{7}{4})^{2}=4\times\frac{49}{16}=\frac{49}{4}$.

Step3: Calculate the average rate of change

Substitute into the formula: $\frac{f(\frac{7}{4})-f(0)}{\frac{7}{4}-0}=\frac{\frac{49}{4}-0}{\frac{7}{4}}=\frac{\frac{49}{4}}{\frac{7}{4}}=\frac{49}{4}\times\frac{4}{7}=7$.

Answer:

A. 7