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

To find the estimated average rates of change of the functions \( f(x) = \sqrt{3x - 4} \) and \( g(x) = \sqrt[3]{2x - 43} \) over the interval \([2, 3]\), we first need to evaluate each function at the endpoints of the interval.

### Step 1: Calculate \( f(2) \) and \( f(3) \)

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

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

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

The average rate of change of \( f(x) \) over \([2, 3]\) is given by:
\[
\text{Average rate of change of } f = \frac{f(3) - f(2)}{3 - 2} = f(3) - f(2) = \sqrt{5} - \sqrt{2}
\]

### Step 3: Calculate \( g(2) \) and \( g(3) \)

1. **Calculate \( g(2) \)**:
\[
g(2) = \sqrt[3]{2(2) - 43} = \sqrt[3]{4 - 43} = \sqrt[3]{-39}
\]

2. **Calculate \( g(3) \)**:
\[
g(3) = \sqrt[3]{2(3) - 43} = \sqrt[3]{6 - 43} = \sqrt[3]{-37}
\]

### Step 4: Calculate the average rate of change of \( g(x) \)

The average rate of change of \( g(x) \) over \([2, 3]\) is:
\[
\text{Average rate of change of } g = \frac{g(3) - g(2)}{3 - 2} = g(3) - g(2) = \sqrt[3]{-37} - \sqrt[3]{-39}
\]

### Step 5: Estimate and Compare the Rates

To compute the actual numerical values of the averages:

1. **Estimate \( f(3) - f(2) \)**:
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
\[
\sqrt{5} - \sqrt{2} \approx 2.236 - 1.414 \approx 0.822
\]

2. **Estimate \( g(3) - g(2) \)**:
- \( \sqrt[3]{-37} \approx -3.332 \)
- \( \sqrt[3]{-39} \approx -3.389 \)
\[
\sqrt[3]{-37} - \sqrt[3]{-39} \approx -3.332 + 3.389 \approx 0.057
\]

### Step 6: Find the Difference in Average Rates

Finally, we calculate the difference:
\[
\text{Difference} = (\sqrt{5} - \sqrt{2}) - (\sqrt[3]{-37} - \sqrt[3]{-39}) \approx 0.822 - 0.057 \approx 0.765
\]

### Step 7: Round to the nearest tenth

The difference between the estimated average rates of change is approximately:
\[
\boxed{0.8}
\]