Question

compare the average rate of change for f(x) = ∛x and g(x)=∛x + 5 for 0≤x≤4. 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 formula for the average rate of change of a function \(y = h(x)\) over \([a,b]\) is \(\frac{h(b)-h(a)}{b - a}\)

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

\(f(0) = 0\), \(f(4)=\sqrt[3]{4}\), \(\text{Average rate}=\frac{\sqrt[3]{4}-0}{4}\)

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

\(g(0)=5\), \(g(4)=\sqrt[3]{4}+5\), \(\text{Average rate}=\frac{(\sqrt[3]{4}+5)-5}{4}=\frac{\sqrt[3]{4}}{4}\)

Step4: Compare the results

Since the average rate of change of \(f(x)\) and \(g(x)\) is \(\frac{\sqrt[3]{4}}{4}\), they are the same.

Answer:

We need to first find the average rate of change formula for a function \(y = f(x)\) over the interval \([a,b]\) which is \(\frac{f(b)-f(a)}{b - a}\). For \(f(x)=\sqrt[3]{x}\) and \(g(x)=\sqrt[3]{x}+5\) over the interval \([0,4]\):

For \(f(x)=\sqrt[3]{x}\), when \(a = 0\) and \(b=4\), \(f(0)=\sqrt[3]{0}=0\) and \(f(4)=\sqrt[3]{4}\approx1.587\). The average rate of change of \(f(x)\) is \(\frac{f(4)-f(0)}{4 - 0}=\frac{\sqrt[3]{4}-0}{4}=\frac{\sqrt[3]{4}}{4}\approx\frac{1.587}{4}=0.397\)

For \(g(x)=\sqrt[3]{x}+5\), when \(a = 0\) and \(b = 4\), \(g(0)=\sqrt[3]{0}+5=5\) and \(g(4)=\sqrt[3]{4}+5\approx1.587 + 5=6.587\). The average rate of change of \(g(x)\) is \(\frac{g(4)-g(0)}{4 - 0}=\frac{(\sqrt[3]{4}+5)-(5)}{4}=\frac{\sqrt[3]{4}}{4}\approx0.397\)

The average rates of change of \(f(x)\) and \(g(x)\) are the same. So the answer is A. The average rates of change of \(f(x)\) and \(g(x)\) are the same.