Question

compare the average rate of change for f(x)=\sqrt3{x} and g(x)=\sqrt3{x}+5 for 0\leq x\leq4. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. (type integers or decimals rounded to the nearest thousandth as needed.) a. the average rates of change of f(x) and g(x) are the same. b. the average rate of change of f(x), , is greater than that of g(x), . c. the average rate of change of g(x), , is greater than that of f(x), .

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function y = h(x) over the interval [a,b] is $\frac{h(b)-h(a)}{b - a}$.

Step2: Calculate average rate of change for f(x)

For $f(x)=\sqrt[3]{x}$, $a = 0$, $b = 4$. Then $f(4)=\sqrt[3]{4}$ and $f(0)=\sqrt[3]{0}=0$. The average rate of change of f(x) is $\frac{f(4)-f(0)}{4 - 0}=\frac{\sqrt[3]{4}-0}{4}=\frac{\sqrt[3]{4}}{4}\approx\frac{1.587}{4}=0.397$.

Step3: Calculate average rate of change for g(x)

For $g(x)=\sqrt[3]{x}+5$, $a = 0$, $b = 4$. Then $g(4)=\sqrt[3]{4}+5$ and $g(0)=\sqrt[3]{0}+5 = 5$. The average rate of change of g(x) is $\frac{g(4)-g(0)}{4 - 0}=\frac{(\sqrt[3]{4}+5)-5}{4}=\frac{\sqrt[3]{4}}{4}\approx0.397$.
Since the average rate of change of f(x) and g(x) are both $\frac{\sqrt[3]{4}}{4}\approx0.397$, the average rates of change are the same.

Answer:

A. The average rates of change of f(x) and g(x) are the same,