compare the average rate of change for f(x) = ∛x and g(x)=∛x + 5 for 0≤x≤4. select the correct choice below…

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.

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.