Question

F ′ (x) is the average rate of change of f over the interval [a,b] for a small enough ℎ.

A) True
B) False
C) Cannot be determined
D) Not mentioned

Answer 1

The statement that f'(x) is the average rate of change of f over an interval [a,b] is false because f'(x) represents the instantaneous rate of change, not the average rate over an interval.

Step-by-step explanation:

The statement that f ′ (x) is the average rate of change of f over the interval [a,b] for a small enough h is False. The average rate of change of a function over the interval [a,b] is the change in the function value divided by the change in the variable, which is given by the formula (f(b) - f(a)) / (b - a). On the other hand, f ′ (x) represents the instantaneous rate of change or the derivative of the function at a particular point x.

The instantaneous rate of change can be approximated using the difference quotient as the interval becomes infinitesimally small, essentially as h approaches zero, but it is not equivalent to the average rate of change over an interval unless the function is linear.