Question

a rechargeable battery is plugged into a charger. the graph shows c(t), the percentage of full capacity that the battery reaches as a function of time t elapsed (in hours). (a) what is the meaning of the derivative c(t)? c(t) is elapsed time in hours required for a given percentage of full capacity to be reached. c(t) is the average rate of change of percentage of full capacity with respect to elapsed time in hours. c(t) is the total percentage of full capacity with respect to elapsed time in hours. c(t) is the instantaneous rate of change of percentage of full capacity with respect to elapsed time in hours. (b) sketch the graph of c(t).

Explanation:

Step1: Recall derivative definition

The derivative of a function $y = f(x)$ represents the instantaneous rate of change of $y$ with respect to $x$. Here, $C(t)$ is the percentage of full - capacity as a function of time $t$. So, $C^{\prime}(t)$ is the instantaneous rate of change of the percentage of full capacity with respect to the elapsed time in hours.

Step2: Analyze the shape of $C(t)$ graph

The graph of $C(t)$ starts at $C(0)=0$ and increases towards $C = 100$ as $t$ increases. At the beginning ($t = 0$), the battery is charging quickly, so $C^{\prime}(t)$ is large. As the battery gets closer to being fully charged (as $t$ approaches a large value), the rate of charging slows down, so $C^{\prime}(t)$ approaches 0.

Answer:

(a) $C^{\prime}(t)$ is the instantaneous rate of change of percentage of full capacity with respect to elapsed time in hours.
(b) The graph of $C^{\prime}(t)$ starts at a relatively large positive value (since the battery charges quickly at first), and then decreases monotonically towards 0 as $t$ increases. It is a decreasing curve that starts above the $t$-axis and approaches the $t$-axis as $t$ goes to a large value.