#### To determine

**To explain:** The meaning of the derivative C′(t).

#### Answer

The meaning of the derivative C′(t) is the instantaneous rate of change of percentage of full capacity with respect to time elapsed in hours.

#### Explanation

The meaning of derivative of C′(t) is as follows:

This means that the ratio of change of percentage of full capacity in a very infinitesimal change of time. That is,

C′(t)=limΔt→0ΔCΔt

#### To determine

**To sketch:** The graph of C′(t) and illustrate the graph.

#### Explanation

**Given:**

A rechargeable battery is connected to a charger.

The graph represents the function C(t) which is the percentage of full capacity that the battery reaches as a function of time *t* elapsed in hours.

**Calculation:**

From the figure, it is very clear that the slope of the graph is always positive. This implies that the derivative of the graph must have positive functional value.

Draw the slope at t=0 of the graph C(t) as shown below in Figure 1.

From the Figure 1, compute the slope of the function C(t) at t=0,

m=40−01−0=40

Thus, the slope m=40 at t=0.

Recall the fact that, the slope of the function at the point x=a is equal to the derivative of the function at that point.

Therefore, C′(t)=40.

Initially, the value of C(t) increases up to t=4 while the slope of C(t) decreases.

After t=4, the slope decreases and the slope becomes zero after a very long time.

**Graph:**

Use the above information and trace the graph of C′(t) as shown below in Figure 2.

**Illustration:**

The graph tells us that the rate of change of full capacity is decreasing and becomes zero after very longtime.