Question

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

Answer 1

To compare the estimated average rates of change of the functions \( f(x) = \sqrt{3x - 4} \) and \( g(x) = 2\sqrt[3]{x - \frac{4}{3}} \) over the interval \([2,3]\), we will calculate the average rate of change for both functions on this 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: Calculate 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. **Average Rate of Change for \( f(x) \)**:
\[
\text{Average Rate of Change of } f = \frac{f(3) - f(2)}{3 - 2} = \frac{\sqrt{5} - \sqrt{2}}{1} = \sqrt{5} - \sqrt{2}
\]

### Step 2: Calculate for \( g(x) = 2\sqrt[3]{x - \frac{4}{3}} \)

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

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

3. **Average Rate of Change for \( g(x) \)**:
\[
\text{Average Rate of Change of } g = \frac{g(3) - g(2)}{3 - 2} = g(3) - g(2) = \frac{2\sqrt[3]{5}}{\sqrt[3]{3}} - \frac{2\sqrt[3]{2}}{\sqrt[3]{3}} = \frac{2(\sqrt[3]{5} - \sqrt[3]{2})}{\sqrt[3]{3}}
\]

### Step 3: Numerical Evaluation

Let’s evaluate \( \sqrt{5} \), \( \sqrt{2} \), \( \sqrt[3]{5} \), \( \sqrt[3]{2} \), and \( \sqrt[3]{3} \) to approximate the difference in average rates of change.

1. **Calculate Approximations**:
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt[3]{2} \approx 1.260 \)
- \( \sqrt[3]{5} \approx 1.710 \)
- \( \sqrt[3]{3} \approx 1.442 \)

2. **Evaluate average rates of change**:
- \( \text{Average Rate of Change of } f = \sqrt{5} - \sqrt{2} \approx 2.236 - 1.414 \approx 0.822 \)
- \( \text{Average Rate of Change of } g = \frac{2(\sqrt[3]{5} - \sqrt[3]{2})}{\sqrt[3]{3}} \approx \frac{2(1.710 - 1.260)}{1.442} = \frac{2(0.450)}{1.442} \approx \frac{0.900}{1.442} \approx 0.624 \)

### Step 4: Find the Difference

\[
\text{Difference} = \text{Average Rate of Change of } f - \text{Average Rate of Change of } g \approx 0.822 - 0.624 = 0.198
\]

### Round to the Nearest Tenth

The difference in the estimated average rates of change to the nearest tenth is:

\[
\text{Difference} \approx 0.2
\]

Thus, the final answer is:

**0.2**