Question

Consider the cubic function: d (x) = ^3√2x-1, and the square root function: g (x) = √2x+1. . Select the interval where the square root function has the lower estimated average rate of change.

Answer 1

but what do I put as the input

Answer 2

To compare the average rate of change of the functions \(d(x) = \sqrt[3]{2x - 1}\) and \(g(x) = \sqrt{2x + 1}\) over some intervals, we need to calculate the average rate of change for both functions over specific intervals.

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

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

Let's choose intervals for comparison. A common choice for intervals could be \([0, 1]\) and \([1, 2]\), but you can choose any interval you prefer.

### Step 1: Calculate for the interval \([0, 1]\)

**For \(d(x)\):**
- \(d(0) = \sqrt[3]{2(0) - 1} = \sqrt[3]{-1} = -1\)
- \(d(1) = \sqrt[3]{2(1) - 1} = \sqrt[3]{1} = 1\)

Average rate of change for \(d(x)\) from \(0\) to \(1\):

\[
\text{Average Rate of Change}_d = \frac{d(1) - d(0)}{1 - 0} = \frac{1 - (-1)}{1 - 0} = \frac{2}{1} = 2
\]

**For \(g(x)\):**
- \(g(0) = \sqrt{2(0) + 1} = \sqrt{1} = 1\)
- \(g(1) = \sqrt{2(1) + 1} = \sqrt{3}\)

Average rate of change for \(g(x)\) from \(0\) to \(1\):

\[
\text{Average Rate of Change}_g = \frac{g(1) - g(0)}{1 - 0} = \frac{\sqrt{3} - 1}{1} \approx 0.732 - 1 = -0.268
\]

### Step 2: Calculate for the interval \([1, 2]\)

**For \(d(x)\):**
- \(d(1) = 1\) (calculated before)
- \(d(2) = \sqrt[3]{2(2) - 1} = \sqrt[3]{3}\)

Average rate of change for \(d(x)\) from \(1\) to \(2\):

\[
\text{Average Rate of Change}_d = \frac{d(2) - d(1)}{2 - 1} = \frac{\sqrt[3]{3} - 1}{1} \approx 0.442
\]

**For \(g(x)\):**
- \(g(1) = \sqrt{3}\) (calculated before)
- \(g(2) = \sqrt{2(2) + 1} = \sqrt{5}\)

Average rate of change for \(g(x)\) from \(1\) to \(2\):

\[
\text{Average Rate of Change}_g = \frac{g(2) - g(1)}{2 - 1} = \frac{\sqrt{5} - \sqrt{3}}{1} \approx 0.236
\]

### Summary of Average Rates of Change

- From \([0, 1]\):
- Average rate of change of \(d(x) \approx 2\)
- Average rate of change of \(g(x) \approx 0.732\)

- From \([1, 2]\):
- Average rate of change of \(d(x) \approx 0.442\)
- Average rate of change of \(g(x) \approx 0.236\)

### Conclusion

The interval where the square root function \(g(x)\) has the lower estimated average rate of change is \([1, 2]\) since \(0.236 < 0.442\). Thus, in this interval, \(g(x)\) has the lower average rate of change compared to \(d(x)\).