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.

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=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
.

Answer 1

To determine the difference between 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 rates of change for both functions over that interval.

1. Square Root Function: Let's define the square root function as \( f(x) = \sqrt{x} \).

   • We calculate the values at the endpoints:

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

   • The average rate of change over the interval \([2, 7]\) is given by: \[ \text{Average rate of change} = \frac{f(7) - f(2)}{7 - 2} = \frac{\sqrt{7} - \sqrt{2}}{5} \]

2. Cube Root Function: Let's define the cube root function as \( g(x) = \sqrt[3]{x} + 2 \).

   • We calculate the values at the endpoints:

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

   • The average rate of change over the interval \([2, 7]\) is given by: \[ \text{Average rate of change} = \frac{g(7) - g(2)}{7 - 2} = \frac{\left(\sqrt[3]{7} + 2\right) - \left(\sqrt[3]{2} + 2\right)}{5} = \frac{\sqrt[3]{7} - \sqrt[3]{2}}{5} \]

3. Calculating Values:

   • We will use approximate values for calculations,

     • \( \sqrt{2} \approx 1.414 \)
     • \( \sqrt{7} \approx 2.646 \)
     • \( \sqrt[3]{2} \approx 1.260 \)
     • \( \sqrt[3]{7} \approx 1.913 \)

   • Average rate of change for square root function: \[ \frac{f(7) - f(2)}{5} = \frac{2.646 - 1.414}{5} \approx \frac{1.232}{5} \approx 0.2464 \]

   • Average rate of change for cube root function: \[ \frac{g(7) - g(2)}{5} = \frac{(1.913 + 2) - (1.260 + 2)}{5} = \frac{1.913 - 1.260}{5} \approx \frac{0.653}{5} \approx 0.1306 \]

4. Finding the Difference:

   • Now, we calculate the difference between the average rates of change: \[ \text{Difference} = 0.2464 - 0.1306 \approx 0.1158 \]

   • Rounding to the nearest tenth: \[ \text{Difference} \approx 0.1 \]

Thus, to the nearest tenth, the difference between the average rates of change is approximately \( \boxed{0.1} \).