compare the estimated avarage rates of the graph square root function over the interval [2,7] to the estimated average rates of the cube root functions y=3vx+2 over the same intervernal

Answer 1

To compare the estimated average rates of the square root function and the cube root function over the interval \([2, 7]\), we'll first define the functions and then calculate the average rates for both.

1. Square Root Function: \[ f(x) = \sqrt{x} \] We will calculate the values at the endpoints of the interval:

   • \(f(2) = \sqrt{2}\)
   • \(f(7) = \sqrt{7}\)

   The average rate of change over the interval \([2, 7]\) can be calculated using the formula: \[ \text{Average Rate} = \frac{f(b) - f(a)}{b - a} \] Substituting the values: \[ \text{Average Rate for } f(x) = \frac{\sqrt{7} - \sqrt{2}}{7 - 2} = \frac{\sqrt{7} - \sqrt{2}}{5} \]

2. Cube Root Function: \[ g(x) = 3\sqrt[3]{x} + 2 \] Again, we calculate the values at the endpoints of the interval:

   • \(g(2) = 3\sqrt[3]{2} + 2\)
   • \(g(7) = 3\sqrt[3]{7} + 2\)

   The average rate of change over the interval \([2, 7]\) is: \[ \text{Average Rate} = \frac{g(b) - g(a)}{b - a} \] Substituting the values: \[ \text{Average Rate for } g(x) = \frac{(3\sqrt[3]{7} + 2) - (3\sqrt[3]{2} + 2)}{7 - 2} = \frac{3\sqrt[3]{7} - 3\sqrt[3]{2}}{5} = \frac{3(\sqrt[3]{7} - \sqrt[3]{2})}{5} \]

Comparison

Now we can summarize the average rates of change over the interval \([2, 7]\):

• Average rate for \(f(x) = \sqrt{x}\): \[ \frac{\sqrt{7} - \sqrt{2}}{5} \]

• Average rate for \(g(x) = 3\sqrt[3]{x} + 2\): \[ \frac{3(\sqrt[3]{7} - \sqrt[3]{2})}{5} \]

We can now evaluate these expressions numerically:

1. Calculate square roots:

   • \(\sqrt{2} \approx 1.414\)
   • \(\sqrt{7} \approx 2.646\)
   • Therefore, \(\sqrt{7} - \sqrt{2} \approx 2.646 - 1.414 \approx 1.232\)
   • Average rate = \(\frac{1.232}{5} \approx 0.2464\)

2. Calculate cube roots:

   • \(\sqrt[3]{2} \approx 1.259\)
   • \(\sqrt[3]{7} \approx 1.913\)
   • Therefore, \(\sqrt[3]{7} - \sqrt[3]{2} \approx 1.913 - 1.259 \approx 0.654\)
   • Average rate = \(\frac{3 \times 0.654}{5} \approx \frac{1.962}{5} \approx 0.3924\)

Conclusion

• Average rate for \(f(x) = \sqrt{x}\) over \([2, 7]\): \(\approx 0.2464\)
• Average rate for \(g(x) = 3\sqrt[3]{x} + 2\) over \([2, 7]\): \(\approx 0.3924\)

Thus, the average rate for the cube root function is higher than that for the square root function on the specified interval.