Compare the estimated average rates of change for the functions p\left(x\right)\ =\sqrt{x}-5 and q\left(x\right)\ =5\sqrt[3]{x-1}over the interval [0.1, 8.9].

Question

• The estimated average rates of change of p (x) and q (x) are both 1 over [0.1, 8.9].
• The estimated average rates of change of p (x) and q (x) are both - over [0.1, 8.9].
• The estimated average rate of change of q (x) is greater than the estimated average rate of change of p (x) over [0.1, 8.9].
• The estimated average rate of change of q (x) is less than the estimated average rate of change of p (x) over [0.1, 8.9].

Answer 1

The estimated average rate of change of a function over an interval is given by the slope of the secant line between the two endpoints of the interval.

For p(x) = sqrt(x) - 5, the slope of the secant line between x = 0.1 and x = 8.9 is equal to 1.

For q(x) = 5 * (x - 1)^(1/3), the slope of the secant line between x = 0.1 and x = 8.9 is also equal to 1.

Therefore, the statement "The estimated average rates of change of p (x) and q (x) are both 1 over [0.1, 8.9]" is true.