Consider the cubic function d\left(x\right)\ =\ \sqrt[3]{2x-1} and the square root function g\left(x\right)\ =\ \sqrt{2x+1}. Select the interval where the square root function has the lower estimated average rate of change.
• [1,5]
• [0,4]
• [2,6]
• [3,7]
To find the average rate of change of a function on an interval, we can use the formula:
Average rate of change = (f(b) - f(a)) / (b - a)
where f(b) is the value of the function at the right endpoint of the interval, f(a) is the value of the function at the left endpoint of the interval, and (b - a) is the length of the interval.
We need to compare the average rate of change of the square root function on each of the given intervals.
For the interval [1,5]:
f(b) = g(5) = sqrt(2(5) + 1) = sqrt(11)
(1) = sqrt(2(1) + 1) = sqrt(3)
(b - a) = 5 - 1 = 4
Average rate of change = (sqrt(11) - sqrt(3)) / 4
For the interval [0,4]:
f(b) = g(4) = sqrt(2(4) + 1) = sqrt(9) = 3
(0) = sqrt(2(0) + 1) = sqrt(1) = 1
(b - a) = 4 - 0 = 4
Average rate of change = (3 - 1) / 4 = 2 / 4 = 1/2
For the interval [2,6]:
f(b) = g(6) = sqrt(2(6) + 1) = sqrt(13)
(2) = sqrt(2(2) + 1) = sqrt(5)
(b - a) = 6 - 2 = 4
Average rate of change = (sqrt(13) - sqrt(5)) / 4
For the interval [3,7]:
f(b) = g(7) = sqrt(2(7) + 1) = sqrt(15)
(3) = sqrt(2(3) + 1) = sqrt(7)
(b - a) = 7 - 3 = 4
Average rate of change = (sqrt(15) - sqrt(7)) / 4
To determine which interval has the lower estimated average rate of change, we need to compare the numerical values of the fractions:
(sqrt(11) - sqrt(3)) / 4 ≈ 0.871 (rounded to 3 decimal places)
(3 - 1) / 4 = 1/2 = 0.5
(sqrt(13) - sqrt(5)) / 4 ≈ 0.7 (rounded to 1 decimal place)
(sqrt(15) - sqrt(7)) / 4 ≈ 0.641 (rounded to 3 decimal places)
Therefore, the interval [1,5] has the lowest estimated average rate of change for the square root function.