Compare the estimated average rate of change for the graphed cubic function b(x)=\root(3)(3x+9) to the estimated average rate of change of the square root function d(x)=\sqrt(-3x+9) over the interval [−12,−3] . Which comparison is true? The estimated average rate of change of b times x is greater than the estimated average rate of change of d times x, but both rates are positive. The estimated average rate of change of d times x is greater than the estimated average rate of change of b times x because d times x is positive over the interval but b times x is negative. The estimated average rate of change of b times x is greater than the estimated average rate of change of d times x because b times x is increasing over the interval but d times x is decreasing. The estimated average rate of change of d(x) is greater than the estimated average rate of change of b(x) , but both rates are negative.
To find the average rate of change of a function over an interval, we find the difference in the function values at the endpoints of the interval and divide by the difference in the x-values of the endpoints.
For b(x) = \root(3)(3x+9), the average rate of change over the interval [-12, -3] is given by:
average rate of change of b(x) = (b(-3) - b(-12)) / (-3 - (-12))
= (\root(3)(0) - \root(3)(-27)) / (-3 + 12)
= 0 - (-3) / 9
= 3/9
= 1/3
For d(x) = \sqrt(-3x+9), the average rate of change over the interval [-12, -3] is given by:
average rate of change of d(x) = (d(-3) - d(-12)) / (-3 - (-12))
= (\sqrt(-3(-3)+9) - \sqrt(-3(-12)+9)) / (-3 + 12)
= (\sqrt(0+9) - \sqrt(0+9)) / 9
= 0/9
= 0
From these calculations, we see that the average rate of change of b(x) is 1/3, while the average rate of change of d(x) is 0.
Therefore, the comparison "The estimated average rate of change of b times x is greater than the estimated average rate of change of d times x because b times x is increasing over the interval but d times x is decreasing." is true.