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?
To find the estimated average rate of a function over an interval, we need to find the slope of the secant line that passes through the endpoints of the interval.
For the square root function $y=\sqrt{x}$, the estimated average rate over the interval [2,7] is equal to the slope of the secant line passing through the points (2, $\sqrt{2}$) and (7, $\sqrt{7}$).
The slope of a line passing through two points $(x_1, y_1)$ and $(x_2 , y_2)$ is given by:
$slope = \frac{y_2 - y_1}{x_2 - x_1}$
So, for the square root function, the slope is:
$slope_{square root} = \frac{\sqrt{7} - \sqrt{2}}{7 - 2} = \frac{\sqrt{7} - \sqrt{2}}{5}$
Now, let's find the estimated average rate of the cube root function over the same interval. The cube root function is $y = 3\sqrt{x} + 2$.
The estimated average rate for this function is equal to the slope of the secant line passing through the points (2, $3\sqrt{2} + 2$) and (7, $3\sqrt{7} + 2$).
Thus, the slope for the cube root function is:
$slope_{cube root} = \frac{3\sqrt{7} + 2 - (3\sqrt{2} + 2)}{7 - 2} = \frac{3\sqrt{7} - 3\sqrt{2}}{5}$
The difference between the estimated average rates of the two functions is:
$Difference = slope_{square root} - slope_{cube root} = \frac{\sqrt{7} - \sqrt{2}}{5} - \frac{3\sqrt{7} - 3\sqrt{2}}{5}$
Simplifying this expression further:
$Difference = \frac{\sqrt{7} - \sqrt{2} - 3\sqrt{7} + 3\sqrt{2}}{5} = \frac{-2\sqrt{7} + 2\sqrt{2}}{5}$
To the nearest tenth, the difference between the estimated average rates of the graphed square root function and the estimated average rates of the cube root function is $\boxed{0.6}$.