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
A. The square root function has the greater average rate of change.
B. The cube root function has the greater average rate of change.
C. The functions have the same average rate of change of 1.
D. The functions have the same average rate of change of 0.
To find the average rate of change of a function over an interval, we need to find the difference in the function values at the endpoints of the interval and divide it by the difference in the x-values of the endpoints.
For the square root function, we are given p(x) = √(x+4). Evaluating the function at the endpoints of the interval [−4, -3], we have:
p(-4) = √(-4+4) = √0 = 0
p(-3) = √(-3+4) = √1 = 1
The average rate of change of the square root function over the interval [-4, -3] is (1 - 0)/((-3) - (-4)) = 1/(-1) = -1.
For the cube root function, we are given f(x) = ∛(x+4). Evaluating the function at the endpoints of the interval [−4, -3], we have:
f(-4) = ∛(-4+4) = ∛0 = 0
f(-3) = ∛(-3+4) = ∛1 = 1
The average rate of change of the cube root function over the interval [-4, -3] is (1 - 0)/((-3) - (-4)) = 1/(-1) = -1.
Therefore, the square root function and the cube root function have the same average rate of change of -1 over the interval [−4,−3].
The correct answer is D. The functions have the same average rate of change of 0.