Compare the average rates of change for the graphed square root function p(x)=x+4−−−−−√ and the cube root functionf(x)=x+4−−−−−√3 over the interval [−
To compare the average rates of change for the two functions, we need to find the slopes of the secant lines over the given intervals.
For the square root function p(x) = √(x+4), let's evaluate the function at two points: x₁ = -3 and x₂ = 1.
p(-3) = √(1) = 2
p(1) = √(1+4) = √(5) ≈ 2.236
The average rate of change for p(x) over the interval [-3, 1] is:
(√(5) - 2)/(1 - (-3)) = (√(5) - 2)/4 ≈ 0.059
For the cube root function f(x) = ∛(x+4), again let's evaluate the function at two points: x₁ = -3 and x₂ = 1.
f(-3) = ∛(1) = 1
f(1) = ∛(1+4) = ∛(5) ≈ 1.709
The average rate of change for f(x) over the interval [-3, 1] is:
(∛(5) - 1)/(1 - (-3)) = (∛(5) - 1)/4 ≈ 0.177
Therefore, the average rate of change for the square root function p(x) is approximately 0.059, while the average rate of change for the cube root function f(x) is approximately 0.177.