Question

Estimate the average rate of change of the functions (cubic function) f(x)= 3√2x-8 and (square root function) g(x)=√2x-8 over the given intervals. Select the interval over which f(x) has an estimated average rate of change greater than that of g(x)
[4,4.7]
[4,4.5]
[4,4.9]
[4,4.3]

Answer 1

To estimate the average rate of change of the functions \( f(x) = \sqrt[3]{2x - 8} \) and \( g(x) = \sqrt{2x - 8} \) over the given intervals, we can use the formula for average rate of change:

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

for function \( f \) over the interval \([a, b]\).

For each interval, we will compute the average rate of change for both \( f(x) \) and \( g(x) \), and compare them.

**1. First Interval: \([4, 4.7]\)**

- For \( f(x) \):
- \( f(4) = \sqrt[3]{2(4) - 8} = \sqrt[3]{0} = 0 \)
- \( f(4.7) = \sqrt[3]{2(4.7) - 8} = \sqrt[3]{1.4} \approx 1.12 \)
- Average Rate of Change for \( f \):
\[
\frac{f(4.7) - f(4)}{4.7 - 4} = \frac{1.12 - 0}{0.7} \approx 1.60
\]

- For \( g(x) \):
- \( g(4) = \sqrt{2(4) - 8} = \sqrt{0} = 0 \)
- \( g(4.7) = \sqrt{2(4.7) - 8} = \sqrt{1.4} \approx 1.18 \)
- Average Rate of Change for \( g \):
\[
\frac{g(4.7) - g(4)}{4.7 - 4} = \frac{1.18 - 0}{0.7} \approx 1.68
\]

**2. Second Interval: \([4, 4.5]\)**

- For \( f(x) \):
- \( f(4.5) = \sqrt[3]{2(4.5) - 8} = \sqrt[3]{1} = 1 \)
- Average Rate of Change for \( f \):
\[
\frac{f(4.5) - f(4)}{4.5 - 4} = \frac{1 - 0}{0.5} = 2
\]

- For \( g(x) \):
- \( g(4.5) = \sqrt{2(4.5) - 8} = \sqrt{1} = 1 \)
- Average Rate of Change for \( g \):
\[
\frac{g(4.5) - g(4)}{4.5 - 4} = \frac{1 - 0}{0.5} = 2
\]

**3. Third Interval: \([4, 4.9]\)**

- For \( f(x) \):
- \( f(4.9) = \sqrt[3]{2(4.9) - 8} = \sqrt[3]{1.8} \approx 1.22 \)
- Average Rate of Change for \( f \):
\[
\frac{f(4.9) - f(4)}{4.9 - 4} = \frac{1.22 - 0}{0.9} \approx 1.36
\]

- For \( g(x) \):
- \( g(4.9) = \sqrt{2(4.9) - 8} = \sqrt{1.8} \approx 1.34 \)
- Average Rate of Change for \( g \):
\[
\frac{g(4.9) - g(4)}{4.9 - 4} = \frac{1.34 - 0}{0.9} \approx 1.49
\]

**4. Fourth Interval: \([4, 4.3]\)**

- For \( f(x) \):
- \( f(4.3) = \sqrt[3]{2(4.3) - 8} = \sqrt[3]{0.6} \approx 0.84 \)
- Average Rate of Change for \( f \):
\[
\frac{f(4.3) - f(4)}{4.3 - 4} = \frac{0.84 - 0}{0.3} \approx 2.8
\]

- For \( g(x) \):
- \( g(4.3) = \sqrt{2(4.3) - 8} = \sqrt{0.6} \approx 0.775 \)
- Average Rate of Change for \( g \):
\[
\frac{g(4.3) - g(4)}{4.3 - 4} = \frac{0.775 - 0}{0.3} \approx 2.58
\]

### Summary of Average Rates of Change:

- On \([4, 4.7]\):
- \( f: \approx 1.60 \); \( g: \approx 1.68 \)
- On \([4, 4.5]\):
- \( f: 2 \); \( g: 2 \)
- On \([4, 4.9]\):
- \( f: \approx 1.36 \); \( g: \approx 1.49 \)
- On \([4, 4.3]\):
- \( f: \approx 2.8 \); \( g: \approx 2.58 \)

### Conclusion:

The only interval where \( f(x) \) has a greater average rate of change than \( g(x) \) is \([4, 4.3]\). Thus, the answer is \([4, 4.3]\).