compare the average rates of change for f(x)=2x + 2, g(x)=2x² + 2, and h(x)=2^x over the interval x = 2 to 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, $a = 2$ and $b = 5$.

Step2: Calculate average rate of change for $f(x)=2x + 2$

First, find $f(5)$ and $f(2)$. $f(5)=2\times5 + 2=12$, $f(2)=2\times2+2 = 6$. Then $\frac{f(5)-f(2)}{5 - 2}=\frac{12 - 6}{3}=2$.

Step3: Calculate average rate of change for $g(x)=2x^{2}+2$

Find $g(5)$ and $g(2)$. $g(5)=2\times5^{2}+2=2\times25 + 2=52$, $g(2)=2\times2^{2}+2=2\times4 + 2 = 10$. Then $\frac{g(5)-g(2)}{5 - 2}=\frac{52 - 10}{3}=\frac{42}{3}=14$.

Step4: Calculate average rate of change for $h(x)=2^{x}$

Find $h(5)$ and $h(2)$. $h(5)=2^{5}=32$, $h(2)=2^{2}=4$. Then $\frac{h(5)-h(2)}{5 - 2}=\frac{32 - 4}{3}=\frac{28}{3}\approx9.3$.

Answer:

2,9.3,14, f(x)