Question

Compare the estimated average rates of change of the functions f(x)=3x−4−−−−−−√
and g(x)=2x−4/3 (this is a fraction)−−−−−√3
over the interval [2,3]
. State the difference in the estimated average rates of change to the nearest tenth

Answer 1

To find the estimated average rates of change of the functions \( f(x) = \sqrt{3x - 4} \) and \( g(x) = \sqrt[3]{2x - 4} \) over the interval \([2, 3]\), we will use the formula for the average rate of change of a function over an interval \([a, b]\):

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

**For \( f(x) = \sqrt{3x - 4} \)**:

1. Calculate \( f(2) \):
\[
f(2) = \sqrt{3(2) - 4} = \sqrt{6 - 4} = \sqrt{2}
\]

2. Calculate \( f(3) \):
\[
f(3) = \sqrt{3(3) - 4} = \sqrt{9 - 4} = \sqrt{5}
\]

3. Calculate the average rate of change:
\[
\text{Average Rate of Change of } f = \frac{f(3) - f(2)}{3 - 2} = \frac{\sqrt{5} - \sqrt{2}}{1} = \sqrt{5} - \sqrt{2}
\]

**For \( g(x) = \sqrt[3]{2x - 4} \)**:

1. Calculate \( g(2) \):
\[
g(2) = \sqrt[3]{2(2) - 4} = \sqrt[3]{4 - 4} = \sqrt[3]{0} = 0
\]

2. Calculate \( g(3) \):
\[
g(3) = \sqrt[3]{2(3) - 4} = \sqrt[3]{6 - 4} = \sqrt[3]{2}
\]

3. Calculate the average rate of change:
\[
\text{Average Rate of Change of } g = \frac{g(3) - g(2)}{3 - 2} = \frac{\sqrt[3]{2} - 0}{1} = \sqrt[3]{2}
\]

Now we have:

- Average Rate of Change of \( f = \sqrt{5} - \sqrt{2} \)
- Average Rate of Change of \( g = \sqrt[3]{2} \)

Next, we will compute the numerical values of these average rates of change:

1. Calculate \( \sqrt{5} - \sqrt{2} \):
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{5} - \sqrt{2} \approx 2.236 - 1.414 \approx 0.822 \)

2. Calculate \( \sqrt[3]{2} \):
- \( \sqrt[3]{2} \approx 1.2599 \)

Finally, we find the difference between the two average rates of change:

\[
\text{Difference} = |\sqrt{5} - \sqrt{2} - \sqrt[3]{2}| \approx |0.822 - 1.2599| \approx | -0.4379| \approx 0.4379
\]

Rounding this to the nearest tenth gives:

\[
\text{Difference} \approx 0.4
\]

Thus, the difference in the estimated average rates of change to the nearest tenth is \( \boxed{0.4} \).