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Question

A curve labeled m of x and seven points on the curve are plotted on a coordinate plane. The curve begins on the x-axis sloping upward from left to right, passes through quadrant 2, and then rises through quadrant 1.

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=x+2−−−−√3
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?

(1 point)
The difference between the rates of change is
.
Remaining Attempts : 3
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Answer 1

To compare the estimated average rates of the graphed square root function over the interval \([2,7]\) with the estimated average rates of the cube root function \(y = \sqrt[3]{x + 2}\) over the same interval, we need to find the average rates of change for both functions.

1. Square Root Function: Let's assume the function is represented as \(f(x) = \sqrt{x}\).

   The average rate of change of \(f(x)\) over the interval \([2, 7]\) can be calculated using the formula: \[ \text{Average rate of change} = \frac{f(7) - f(2)}{7 - 2} \]

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

   Thus, the average rate of change is: \[ \frac{\sqrt{7} - \sqrt{2}}{5} \]

2. Cube Root Function: For the cube root function \(g(x) = \sqrt[3]{x + 2}\), we will perform a similar calculation:

   The average rate of change of \(g(x)\) over the interval \([2, 7]\) is given by: \[ \text{Average rate of change} = \frac{g(7) - g(2)}{7 - 2} \]

   • \(g(2) = \sqrt[3]{2 + 2} = \sqrt[3]{4}\)
   • \(g(7) = \sqrt[3]{7 + 2} = \sqrt[3]{9}\)

   Thus, the average rate of change is: \[ \frac{\sqrt[3]{9} - \sqrt[3]{4}}{5} \]

3. Estimations: We will now compute the numerical values for these requests using approximate values of square roots and cube roots:

   • \( \sqrt{2} \approx 1.414 \)
   • \( \sqrt{7} \approx 2.646 \)
   • So, \( f(7) - f(2) \approx 2.646 - 1.414 \approx 1.232 \)

   Thus, the average rate of change for the square root function is: \[ \frac{1.232}{5} \approx 0.2464 \]

   • \( \sqrt[3]{4} \approx 1.587 \) and \( \sqrt[3]{9} \approx 2.080 \)

   So: \[ g(7) - g(2) \approx 2.080 - 1.587 \approx 0.493 \]

   Thus, the average rate of change for the cube root function is: \[ \frac{0.493}{5} \approx 0.0986 \]

4. Calculate the Difference: The difference between the average rates of change: \[ \text{Difference} = 0.2464 - 0.0986 \approx 0.1478 \]

Rounding to the nearest tenth gives approximately: \[ \text{Difference} \approx 0.2 \]

Thus, the final answer to the nearest tenth for the difference between the estimated average rates of the square root function and the cube root function is: \[ \boxed{0.2} \]