Question

Compare the estimated average rates of change of the functions f(x)=3x−4−−−−−√
and g(x)=2x−43−−−−−√3
over the interval [2,3]
. State the difference in the estimated average rates of change to the nearest tenth.(1 point)
The difference between the estimated average rates of change is approximately

Answer 1

so whats the answer

Answer 2

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 can use the formula for average rate of change over an interval \([a, b]\):

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

### Step 1: Calculate the average rate of change for \( f(x) \)

First, we need to evaluate \( f(2) \) and \( f(3) \):

\[
f(2) = \sqrt{3(2) - 4} = \sqrt{6 - 4} = \sqrt{2} \approx 1.414
\]

\[
f(3) = \sqrt{3(3) - 4} = \sqrt{9 - 4} = \sqrt{5} \approx 2.236
\]

Now we can calculate the average rate of change for \( f \):

\[
\text{Average rate of change of } f = \frac{f(3) - f(2)}{3 - 2} = \frac{\sqrt{5} - \sqrt{2}}{1} \approx 2.236 - 1.414 \approx 0.822
\]

### Step 2: Calculate the average rate of change for \( g(x) \)

Now evaluate \( g(2) \) and \( g(3) \):

\[
g(2) = \sqrt[3]{2(2) - 4} = \sqrt[3]{4 - 4} = \sqrt[3]{0} = 0
\]

\[
g(3) = \sqrt[3]{2(3) - 4} = \sqrt[3]{6 - 4} = \sqrt[3]{2} \approx 1.260
\]

Now we can calculate the average rate of change for \( g \):

\[
\text{Average rate of change of } g = \frac{g(3) - g(2)}{3 - 2} = \frac{\sqrt[3]{2} - 0}{1} \approx \sqrt[3]{2} \approx 1.260
\]

### Step 3: Compare the average rates of change

Now we compare the average rates of change:

- Average rate of change of \( f \): \( \approx 0.822 \)
- Average rate of change of \( g \): \( \approx 1.260 \)

### Difference in average rates of change:

\[
\text{Difference} = g_{\text{avg}} - f_{\text{avg}} \approx 1.260 - 0.822 \approx 0.438
\]

### Final result:

Rounding to the nearest tenth, the difference between the estimated average rates of change is approximately:

\[
\boxed{0.4}
\]

Answer 3

The difference between the estimated average rates of change of the functions \( f(x) = \sqrt{3x - 4} \) and \( g(x) = \sqrt[3]{2x - 4} \) over the interval \([2, 3]\), rounded to the nearest tenth, is approximately:

\[
\boxed{0.4}
\]