Compare the average rates of change for the graphed square root function p(x)=x 4−−−−√ and the cube root function f(x)=x 4−−−−√3 over the interval [−4,−3] . (1 point) Responses The functions have the same average rate of change of 0. The functions have the same average rate of change of 0. The functions have the same average rate of change of 1. The functions have the same average rate of change of 1. The square root function has the greater average rate of change. The square root function has the greater average rate of change. The cube root function has the greater average rate of change
The average rate of change for a function over an interval is determined by finding the difference in function values divided by the difference in x-values.
For the square root function p(x) = √x, the average rate of change over the interval [−4,−3] is (p(-3) - p(-4))/(-3 - (-4)).
Similarly, for the cube root function f(x) = ∛x, the average rate of change over the same interval is (f(-3) - f(-4))/(-3 - (-4)).
Calculating these values, we find:
p(-3) = √(-3) = √3 ≈ 1.73
p(-4) = √(-4) = √4 = 2
f(-3) = ∛(-3) = -∛3 ≈ -1.44
f(-4) = ∛(-4) = -∛4 ≈ -1.59
Using these values, the average rate of change for the square root function is (1.73 - 2)/(-3 - (-4)) = -0.27.
The average rate of change for the cube root function is (-1.44 - (-1.59))/(-3 - (-4)) = 0.15.
Therefore, the square root function has the greater average rate of change. The correct answer is "The square root function has the greater average rate of change."