Question

A rechargeable battery is plugged into a charger. The graph shows \(C\left( t \right)\), 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 \(C'\left( t \right)\)?

(b) Sketch the graph of \(C'\left( t \right)\). What does the graph tell you?

Short answer

(a) The derivative of \(C'\left( t \right)\) represents the instantaneous rate of change of full capacity over an elapsed time in hours.

(b) The graph is shown below:

The graph of \(C'\left( t \right)\) shows that the rate of change of percentage of full capacity is decreasing and reaching 0.

Step-by-step solution

Explain the derivative \(C'\left( t \right)\)

a) The derivative \(C'\left( t \right)\) represents the instantaneous rate of change of full capacity over an elapsed time in hours.

Sketch the graph of \(C'\left( t \right)\)

b) Consider some points \(A,B,C,\) and \(D,\) to plot on the graph of the given function \(C\left( t \right)\). We can estimate the graph of derivatives from these points.

Sketch the given graph of the function with the above points as shown below:

To determine the value of the derivative at any value of \(t\), we can draw the tangent at the point \(\left( {t,C\left( t \right)} \right)\) and estimate the slope.

The slope of the function \(C\left( t \right)\) is \(C'\left( A \right)\) can be determined as the ratio of the rise and run that appears to be equal. The function \(C\left( t \right)\) is increasing at \(x = A\)—the value of \(C'\left( A \right)\) is 40.

The function \(C\left( t \right)\) is increasing at \(x = B\). Hence, the value of \(C'\left( B \right)\) is 20.

It is observed that the tangent at \(t = C,D\) is horizontal. Therefore the slope is zero and it follows that the value of \(C'\left( C \right) = C'\left( D \right) = 0\).

It is observed that the tangent of four points has a positive slope, so \(C'\left( t \right)\) is positive.

Use the above point to estimate the derivative of the function as shown below:

Explain the graph of \(C'\)

The graph of \(C'\) shows that the rate of change of percentage of full capacity in decreasing and reaching to 0.