Question

Compare the estimated average rates of the graphed square root function over the interval [2,7]
to the estimated average rates of the cube root function y=3^√x+2−−−−−
over the same interval. To the nearest tenth, what is the difference between the estimated average rates of the graphed square root function and the estimated average rates of the cube root function?

Answer 1

To compare the estimated average rates of the square root function and the cube root function over the interval \([2, 7]\), we need to calculate the average rate of change for each function over this interval.

1. **Calculating Average Rate of Change for the Square Root Function**:

The square root function generally takes the form \(y = \sqrt{x}\).

- Find the values at the endpoints:
\[
f(2) = \sqrt{2} \approx 1.414
\]
\[
f(7) = \sqrt{7} \approx 2.646
\]

- The average rate of change is given by:
\[
\text{Average Rate of Change} = \frac{f(7) - f(2)}{7 - 2} = \frac{\sqrt{7} - \sqrt{2}}{5}
\]
\[
\text{Average Rate of Change} \approx \frac{2.646 - 1.414}{5} \approx \frac{1.232}{5} \approx 0.2464
\]

2. **Calculating Average Rate of Change for the Cube Root Function**:

The cube root function is represented as \(y = 3^{\sqrt{x} + 2}\).

- Find the values at the endpoints:
\[
g(2) = 3^{\sqrt{2} + 2} \approx 3^{1.414 + 2} = 3^{3.414} \approx 40.29
\]
\[
g(7) = 3^{\sqrt{7} + 2} \approx 3^{2.646 + 2} = 3^{4.646} \approx 138.34
\]

- The average rate of change is given by:
\[
\text{Average Rate of Change} = \frac{g(7) - g(2)}{7 - 2} = \frac{3^{\sqrt{7} + 2} - 3^{\sqrt{2} + 2}}{5}
\]
\[
\text{Average Rate of Change} \approx \frac{138.34 - 40.29}{5} \approx \frac{98.05}{5} \approx 19.61
\]

3. **Finding the Difference**:

Now, we find the difference between the average rates of change:
\[
\text{Difference} = 19.61 - 0.2464 \approx 19.3636
\]

Rounding to the nearest tenth, the difference is:
\[
\approx 19.4
\]

**Final Answer**: The difference between the estimated average rates of the square root function and the cube root function over the interval \([2, 7]\) is approximately **19.4**.