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 = 3square root of 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 the graphed square root function over the interval [2, 7], we need to find the slope of the secant line connecting the points (2, √2) and (7, √7). The slope of this line is given by:
(√7 - √2) / (7 - 2) = (√7 - √2) / 5
To find the estimated average rate of the cube root function over the same interval, we need to find the slope of the secant line connecting the points (2, 3√2 + 2) and (7, 3√7 + 2). The slope of this line is given by:
(3√7 + 2 - 3√2 - 2) / (7 - 2) = (3√7 - 3√2) / 5
Now we need to find the difference between these two slopes:
(√7 - √2) / 5 - (3√7 - 3√2) / 5
= (√7 - 3√7 + 3√2 - √2) / 5
= (-2√7 + 2√2) / 5
To the nearest tenth, the difference between the estimated average rates of the graphed square root function and the cube root function over the interval [2, 7] is -0.7.