Consider the cubic function d (x) = ^3square root of 2x - 1 and the square root function g (x) = square root of 2x + 1. Select the interval where the square root function has the lower estimated average rate of change.
A. [1, 5]
B. [0, 4]
C. [3, 7]
D. [2, 6]
To compare the average rate of change of two functions on a given interval, we can find the derivative of each function and evaluate it at the endpoints of the interval.
First, let's find the derivative of the cubic function d(x):
d'(x) = (1/3)(2x - 1)^(-2/3) * 2
Next, let's find the derivative of the square root function g(x):
g'(x) = (1/2)(2x + 1)^(-1/2) * 2
Now, let's evaluate the derivatives at the endpoints of each interval:
For interval A [1, 5]:
d'(1) = (1/3)(2(1) - 1)^(-2/3) * 2 = (1/3)(1)^(-2/3) * 2 = 2/3
g'(5) = (1/2)(2(5) + 1)^(-1/2) * 2 = (1/2)(11)^(-1/2) * 2 ≈ 0.31
For interval B [0, 4]:
d'(0) = (1/3)(2(0) - 1)^(-2/3) * 2 = (1/3)(-1)^(-2/3) * 2 = -2/3
g'(4) = (1/2)(2(4) + 1)^(-1/2) * 2 = (1/2)(9)^(-1/2) * 2 ≈ 0.22
For interval C [3, 7]:
d'(3) = (1/3)(2(3) - 1)^(-2/3) * 2 = (1/3)(5)^(-2/3) * 2 ≈ 0.26
g'(7) = (1/2)(2(7) + 1)^(-1/2) * 2 = (1/2)(15)^(-1/2) * 2 ≈ 0.15
For interval D [2, 6]:
d'(2) = (1/3)(2(2) - 1)^(-2/3) * 2 = (1/3)(3)^(-2/3) * 2 ≈ 0.39
g'(6) = (1/2)(2(6) + 1)^(-1/2) * 2 = (1/2)(13)^(-1/2) * 2 ≈ 0.18
Comparing the estimated average rate of change at the endpoints of each interval, we can see that the square root function has the lower rate of change on interval C [3,7] which had the values 0.26 and 0.15.
Therefore, the correct answer is C. [3,7].