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].
• The estimated average rates of change of p (x) and q (x) are both 1/3 over [0.1, 8.9].
• The estimated average rates of change of p (x) and q (x) are both 5/3 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].
The estimated average rate of change of a function over an interval is computed by finding the difference in the function values at the endpoints of the interval and dividing it by the difference in the input values (or "x-values") at the endpoints of the interval.
For function p(x) = √x - 5, the average rate of change over the interval [0.1, 8.9] is:
p(8.9) - p(0.1) / (8.9 - 0.1)
= (√8.9 - 5) - (√0.1 - 5) / 8.8
≈ (2.983 - 5) - (0.316 - 5) / 8.8
≈ -2.017 - 4.684 / 8.8
≈ -6.701 / 8.8
≈ -0.762
For function q(x) = 5√[x-1], the average rate of change over the interval [0.1, 8.9] is:
q(8.9) - q(0.1) / (8.9 - 0.1)
= (5√[8.9-1]) - (5√[0.1-1]) / 8.8
≈ (5√[7.9]) - (5√[-0.9]) / 8.8
≈ (5√7.9) - (5i√0.9) / 8.8
≈ (5√7.9) - (5i√0.9) / 8.8
(Note: The square root of a negative number (√[-0.9]) is typically represented by the imaginary unit "i".)
We cannot simplify the expression further because the square root of 7.9 is not a rational number, and the square root of -0.9 involves an imaginary component. However, we can conclude that the estimated average rate of change of q(x) is a complex number (i.e., a number with a real and imaginary part).
Therefore, the correct answer is: 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].