Question

discuss the validity of the following statement. if the statement is always true, explain why. if not, give a counterexample. the average rate of change of a function f from x = a to x = a + h is less than the instantaneous rate of change at x = a + \\(\frac{h}{2}\\). choose the correct answer below. 〇 a. the statement is false. for example, a car could drive at an average speed of 40 mph from x = a to x = a + h, but be driving 45 mph for an instant at x = a + \\(\frac{h}{2}\\). 〇 b. the statement is true. the average rate of change over smaller and smaller time intervals approaches the instantaneous rate of change, which is always greater than the average rate of change. 〇 c. the statement is true. the instantaneous rate of change is always less than the average rate of change. 〇 d. the statement is false. for example, if the average rate of change is 20, then x = a + 20. thus, the instantaneous rate of change at x = a + \\(\frac{h}{2}\\) is x = a + 10.

Explanation:

To determine the validity, we analyze each option:

• Option B claims the average rate of change is always less than the instantaneous rate, but this isn't true (e.g., a decreasing function could have a negative average rate and a more negative instantaneous rate, making the average greater).
• Option C is incorrect as the instantaneous rate isn't always less than the average.
• Option D's reasoning is flawed (average rate of change isn't related to \(x = a + 20\) in that way).
• Option A provides a valid counterexample: a car with an average speed (average rate of change) of 40 mph and an instantaneous speed (instantaneous rate) of 45 mph at \(x=a+\frac{h}{2}\) shows the average rate isn't always less than the instantaneous rate at that point.

Answer:

A. The statement is false. For example, a car could drive at an average speed of 40 mph from \(x = a\) to \(x = a + h\), but be driving 45 mph for an instant at \(x=a+\frac{h}{2}\).