Compare the average rates of change for the graphed square root function p\left(x\right)\ =\ \sqrt{x+4} and the cube root function f\left(x\right)\ =\ \sqrt[3]{x+4} over the interval [-4,-3].
• The cube root function has the greater average rate of change.
• The functions have the same average rate of change of 1.
• The square root function has the greater average rate of change.
• The functions have the same average rate of change of 0.
To find the average rate of change of a function over the interval [-4, -3], we need to find the difference in the function values at the endpoints and divide by the difference in the x-values.
For the square root function p(x) = sqrt(x+4):
p(-4) = sqrt(-4+4) = sqrt(0) = 0
p(-3) = sqrt(-3+4) = sqrt(1) = 1
The average rate of change for p(x) over the interval [-4, -3] is:
(1-0)/(-3-(-4)) = 1/1 = 1
For the cube root function f(x) = cbrt(x+4):
f(-4) = cbrt(-4+4) = cbrt(0) = 0
f(-3) = cbrt(-3+4) = cbrt(1) = 1
The average rate of change for f(x) over the interval [-4, -3] is:
(1-0)/(-3-(-4)) = 1/1 = 1
Therefore, the functions p(x) and f(x) have the same average rate of change of 1 over the interval [-4, -3].