Compare the estimated average rate of change for the graphed cubic function b(x)=3x+9−−−−−√3 to the estimated average rate of change of the square root function d(x)=−3x+9−−−−−−−√ over the interval [−12,−3]. Which comparison is true?
To find the average rate of change of a function over an interval, we need to calculate the difference in the function values at the endpoints of the interval and divide by the difference in the input values.
For the cubic function b(x) = 3√(x+9), the average rate of change over the interval [−12,−3] can be found as follows:
b(-3) = 3√(-3+9) = 3√6
b(-12) = 3√(-12+9) = 3√(-3)
Average rate of change for b(x) = (b(-3) - b(-12))/(-3 -(-12)) = (3√6 - 3√(-3))/9 = (3/9)√6 - (3/9)√(-3) = (1/3)√6 - (1/3)√(-3).
Next, for the square root function d(x) = -3√(x+9), the average rate of change over the interval [−12,−3] is calculated as follows:
d(-3) = -3√(-3+9) = -3√6
d(-12) = -3√(-12+9) = -3√(-3)
Average rate of change for d(x) = (d(-3) - d(-12))/(-3 -(-12)) = (-3√6 - (-3√(-3)))/9 = (-3/9)√6 - (-3/9)√(-3) = (-1/3)√6 + (1/3)√(-3).
Comparing the two average rates of change, we have:
(1/3)√6 - (1/3)√(-3) vs (-1/3)√6 + (1/3)√(-3)
From here, we can observe that the two expressions are equal in magnitude but have opposite signs. Therefore, the comparison that is true is: The estimated average rate of change of the cubic function b(x) = 3√(x+9) over the interval [−12,−3] is equal in magnitude but opposite in sign to the estimated average rate of change of the square root function d(x) = -3√(x+9) over the same interval.