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 [2,6] left bracket 2 comma 6 right bracket [3,7] left bracket 3 comma 7 right bracket [1,5] left bracket 1 comma 5 right bracket [0,4] left bracket 0 comma 4 right bracket
To compare the estimated average rate of change of the two functions, we need to find the derivative of each function.
The derivative of d(x) = 2x-1-√3 is:
d'(x) = 2
The derivative of g(x) = 2x+1-√ is:
g'(x) = 1/√(2x+1)
To find the interval where the square root function has the lower estimated average rate of change, we need to compare the values of 1/√(2x+1) for each interval.
Let's evaluate 1/√(2x+1) for each interval:
1. For [2,6]:
1/√(2(2)+1) = 1/√(5) ≈ 0.447
1/√(2(6)+1) = 1/√(13) ≈ 0.272
2. For [3,7]:
1/√(2(3)+1) = 1/√(7) ≈ 0.378
1/√(2(7)+1) = 1/√(15) ≈ 0.258
3. For [1,5]:
1/√(2(1)+1) = 1/√(3) ≈ 0.578
1/√(2(5)+1) = 1/√(11) ≈ 0.301
4. For [0,4]:
1/√(2(0)+1) = 1/√(1) = 1
1/√(2(4)+1) = 1/√(9) ≈ 0.333
Thus, the interval where the square root function has the lower estimated average rate of change is [3,7] (left bracket 3, comma 7, right bracket).