Question

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 compare 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 compute the average rate of change for both functions over that interval.

The average rate of change of a function \( h(x) \) over the interval \([a, b]\) is given by:

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

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

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

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

### Average rate of change for \( f(x) \):

\[
\text{Average Rate of Change for } f = \frac{\sqrt{5} - \sqrt{2}}{3 - 2} = \sqrt{5} - \sqrt{2}
\]

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

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

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

### Average rate of change for \( g(x) \):

\[
\text{Average Rate of Change for } g = \frac{\sqrt[3]{-37} - \sqrt[3]{-39}}{3 - 2} = \sqrt[3]{-37} - \sqrt[3]{-39}
\]

### Step 3: Calculate the difference

We need to find the difference in the average rates of change:

\[
\text{Difference} = \left( \sqrt{5} - \sqrt{2} \right) - \left( \sqrt[3]{-37} - \sqrt[3]{-39} \right)
\]

Let's calculate the values numerically:

1. \( \sqrt{5} \approx 2.236 \)
2. \( \sqrt{2} \approx 1.414 \)

So,
\[
\sqrt{5} - \sqrt{2} \approx 2.236 - 1.414 \approx 0.822
\]

Now, for the cube roots:

1. \( \sqrt[3]{-37} \approx -3.332 \)
2. \( \sqrt[3]{-39} \approx -3.386 \)

So,
\[
\sqrt[3]{-37} - \sqrt[3]{-39} \approx -3.332 + 3.386 \approx 0.054
\]

### Final Calculation of the Difference

Now, let's compute the difference:

\[
\text{Difference} \approx 0.822 - 0.054 = 0.768
\]

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

\[
\boxed{0.8}
\]