Consider the cubic function d(x)=2x−1−−−−−√3 and the square root function g(x)=2x 1−−−−−√ . Select the interval where the square root function has the lower estimated average rate of change.(1 point) Responses [1,5] left bracket 1 comma 5 right bracket [3,7] left bracket 3 comma 7 right bracket [2,6] left bracket 2 comma 6 right bracket [0,4] left bracket 0 comma 4 right bracket
To find the interval where the square root function has the lower estimated average rate of change, we need to compare the average rate of change of both functions on each interval.
For the cubic function d(x) = 2x - 1 - √3, the average rate of change on any interval [a, b] can be found by finding the difference in function values at the endpoints and dividing by the difference in x-values:
average rate of change = [d(b) - d(a)] / [b - a]
For the square root function g(x) = 2√(x - 1), the average rate of change on any interval [a, b] is simply the derivative of the function, which is given by:
g'(x) = 2 / (2√(x - 1))
To compare the average rate of change on each interval, we'll evaluate the average rate of change for both functions on each interval and compare the values.
Let's evaluate the average rate of change for both functions on each interval:
Interval [1, 5]:
For d(x) = 2x - 1 - √3:
average rate of change = [d(5) - d(1)] / [5 - 1] = [9·5 - 9 - √3 - (1·2 - 9 - √3)] / 4 = [45 - 9 - √3 - 18 + 9 + √3] / 4 = 27/4
For g(x) = 2√(x - 1):
average rate of change is the derivative, which is a constant 2/2 = 1
Since 27/4 is greater than 1, the square root function has a lower estimated average rate of change on the interval [1, 5].
Interval [3, 7]:
For d(x) = 2x - 1 - √3:
average rate of change = [d(7) - d(3)] / [7 - 3] = [13·7 - 9 - √3 - (7·2 - 9 - √3)] / 4 = [91 - 9 - √3 - 14 + 9 + √3] / 4 = 68/4 = 17/1
For g(x) = 2√(x - 1):
average rate of change is the derivative, which is a constant 2/2 = 1
Since 17/1 is greater than 1, the square root function has a lower estimated average rate of change on the interval [3, 7].
Interval [2, 6]:
For d(x) = 2x - 1 - √3:
average rate of change = [d(6) - d(2)] / [6 - 2] = [13·6 - 9 - √3 - (5·2 - 9 - √3)] / 4 = [78 - 9 - √3 - 10 + 9 + √3] / 4 = 68/4 = 17/1
For g(x) = 2√(x - 1):
average rate of change is the derivative, which is a constant 2/2 = 1
Since 17/1 is greater than 1, the square root function has a lower estimated average rate of change on the interval [2, 6].
Interval [0, 4]:
For d(x) = 2x - 1 - √3:
average rate of change = [d(4) - d(0)] / [4 - 0] = [9·4 - 9 - √3 - (0·2 - 9 - √3)] / 4 = [36 - 9 - √3 - 0 + 9 + √3] / 4 = 36/4 = 9/1
For g(x) = 2√(x - 1):
average rate of change is the derivative, which is a constant 2/2 = 1
Since 9/1 is equal to 1, the square root function has the same estimated average rate of change as the cubic function on the interval [0, 4].
Therefore, the interval where the square root function has the lower estimated average rate of change is [1, 5].