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(x) is greater than the estimated average rate of change of d(x) , but both rates are positive. 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(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. 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(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. 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.

Question

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(x) is greater than the estimated average rate of change of d(x) , but both rates are positive. 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(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. 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(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. 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.

Answer 1

To find the estimated average rate of change for a function over an interval, we can use the formula:

Average rate of change = (f(b) - f(a)) / (b - a),

where f(b) represents the value of the function at the endpoint b of the interval, f(a) represents the value of the function at the endpoint a of the interval, and (b - a) represents the length of the interval.

For the cubic function b(x) = \root(3)(3x + 9), let's find the average rate of change over the interval [-12, -3]:

b(-12) = \root(3)(3(-12) + 9) = \root(3)(-21) ≈ -2.758,
b(-3) = \root(3)(3(-3) + 9) = \root(3)(0) = 0,

Average rate of change = (0 - (-2.758)) / (-3 - (-12)) = 2.758 / 9 ≈ 0.306.

For the square root function d(x) = \sqrt(-3x + 9), let's find the average rate of change over the same interval:

d(-12) = \sqrt(-3(-12) + 9) = \sqrt(57),
d(-3) = \sqrt(-3(-3) + 9) = \sqrt(18),

Average rate of change = (\sqrt(18) - \sqrt(57)) / (-3 - (-12)).

To determine which comparison is true, we need to compare these two average rates of change.

Let's calculate the value of each average rate of change:

Average rate of change for b(x) = 0.306,
Average rate of change for d(x) = (\sqrt(18) - \sqrt(57)) / (-3 - (-12)).

Comparing the two average rates of change, we can see that the estimated average rate of change for b(x) is greater than the estimated average rate of change for d(x).

Therefore, the comparison "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" is the correct choice.