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.
[1,5]
[0,4]
[2,6]
[3,7]
To find the estimated average rate of change of a function on an interval, we can find the difference in the function values at the endpoints of the interval and divide it by the difference in the input values of the endpoints.
For the square root function g(x) = 2√(x+1), let's find the estimated average rate of change for each interval:
Interval [1,5]:
g(5) - g(1) = 2√(5+1) - 2√(1+1) = 2√6 - 2√2
Change in input values = 5 - 1 = 4
Interval [0,4]:
g(4) - g(0) = 2√(4+1) - 2√(0+1) = 2√5 - 2
Change in input values = 4 - 0 = 4
Interval [2,6]:
g(6) - g(2) = 2√(6+1) - 2√(2+1) = 2√7 - 2√3
Change in input values = 6 - 2 = 4
Interval [3,7]:
g(7) - g(3) = 2√(7+1) - 2√(3+1) = 2√8 - 2√4
Change in input values = 7 - 3 = 4
Now let's compare these values to determine the interval with the lowest estimated average rate of change. We need to find the minimum value among these:
2√6 - 2√2
2√5 - 2
2√7 - 2√3
2√8 - 2√4
Since we are looking for the interval with the lowest estimated average rate of change, we want to minimize these values. By inspection, we can see that the minimum value occurs when x = sqrt(2), because both terms in 2√6 - 2√2 will have the same value, and all other terms will have larger values.
Therefore, the interval [1,5] will have the lowest estimated average rate of change for the square root function g(x) = 2√(x+1).