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? (1 point) Responses
A. The estimated average rate of change of b(x) is greater than the estimated average rate of change of d(x) , but both rates are positive.
B. The estimated average rate of change of d(x) is greater than the estimated average rate of change of b(x) because d(x) is positive over the interval but b(x) is negative.
C. The estimated average rate of change of b(x) is greater than the estimated average rate of change of d(x) because b(x) is increasing over the interval but d(x) is decreasing.
D. 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 estimated average rate of change of a function over an interval, we need to find the difference in the function values at the endpoints of the interval and divide by the difference in the input values.
For the function b(x) = 3x + 9√3, the difference in function values at the endpoints of the interval [-12, -3] is:
b(-3) - b(-12) = (3(-3) + 9√3) - (3(-12) + 9√3)
= (-9 + 9√3) - (-36 + 9√3)
= -9 + 9√3 + 36 - 9√3
= 27
The difference in input values is -3 - (-12) = 9
Therefore, the estimated average rate of change of b(x) over the interval is 27/9 = 3.
For the function d(x) = -3x + 9√3, the difference in function values at the endpoints of the interval is:
d(-3) - d(-12) = (-3(-3) + 9√3) - (-3(-12) + 9√3)
= (9 + 9√3) - (36 + 9√3)
= 9 + 9√3 - 36 - 9√3
= -27
The difference in input values is -3 - (-12) = 9
Therefore, the estimated average rate of change of d(x) over the interval is -27/9 = -3.
Comparing the two estimated average rates of change, we can see that the estimated average rate of change of b(x) is greater than the estimated average rate of change of d(x) because 3 > -3. However, both rates are positive since the values of the functions at the endpoints of the interval are positive.
Therefore, the correct comparison is:
A. The estimated average rate of change of b(x) is greater than the estimated average rate of change of d(x), but both rates are positive.