Let P(t) represent

Question

Let P(t) represent the number of cars in a school parking lot t hours after 7 a.m. If P(0) = 8 and P′(0) = 12, which of the following statements must be true?

I. The instantaneous rate of change in the number of cars in the parking lot is 12 cars per hour at 7 a.m.

II. The number of cars in the parking lot is increasing at 7 a.m.

III. There are 20 cars in the parking lot at 8 a.m.

none

I only

I and II only

II and III only

Answer 1

Hi Lissie,

This question involves knowing how to interpret the meaning of the derivative. P(0) = 8 means that at time t=0 (7am), the number of cars in the parking lot is 8. P'(0) = 12 means that at time t=0 (7am), the instantaneous rate of change of cars in the parking lot is 12. The rate of change of cars means that that's the rate cars were entering and leaving the parking lot.

Applying the first derivative test (finding t when P'(t) =0) shows us that since P'(0) is positive, P(t) is increasing. We can then say that (I) and (II) are both true. However, there is no way to tell the exact number of cars in the parking lot at 8am. We have no information about P(1). We cannot use what we know about P(0) or P'(0) to make a claim about P(1).

Thus, choice B (I and II only) is correct.