Question

What is the relationship between instantaneous rate of change and the derivative of f at x?

a) The instantaneous rate of change is the same as the derivative.
b) The derivative measures the average rate of change.
c) The derivative measures the instantaneous rate of change.
d) The derivative and instantaneous rate of change are unrelated.

Answer 1

The derivative of a function at a point represents c) the instantaneous rate of change at that point, which is exemplified by both physics (instantaneous acceleration) and chemistry (instantaneous rate of reaction).

Step-by-step explanation:

The relationship between the instantaneous rate of change and the derivative of a function f at x is that the derivative represents the instantaneous rate of change.

This is analogous to how in physics the instantaneous acceleration is the derivative of the velocity function with respect to time, showing the acceleration at a specific instant.

Similarly, in chemistry, the instantaneous rate of reaction is the derivative of concentration with respect to time, indicating the rate of reaction at a particular instant.

Thus, the correct answer to the student's question is: (c) The derivative measures the instantaneous rate of change.