To determine
To explain: The meaning of the derivative F′(v).
Answer
The meaning of the derivative F′(v) is the instantaneous rate of change of fuel economy F with respect to speed v.
Explanation
The meaning of derivative of F′(v) is as follows:
This means that the ratio of change of fuel economy in a very infinitesimal change of speed. That is,
F′(v)=limΔv→0ΔFΔv
To determine
To sketch: The graph of F′(v).
Explanation
From the graph, it is clear that the slope of the graph is positive from v=5 to v=50. Thus, the graph of F′(v) is positive from v=5 to v=50.
The graph of F(v) has a horizontal tangent at v=50. Thus, F′(50)=0
The slope of the graph is negative from v=50 towards right. Thus the graph of F′(v) is negative from v=50 towards right.
Graph:
Use the above information and trace the graph of F′(v) as shown below in Figure 1.
Observation:
The graph tells us that the rate of change of full capacity is decreasing and becomes zero and then becomes negative.
To determine
To calculate: The speed at which one should drive to save gas.
Answer
The speed at which one should drive to save gas is 50 miles per hour.
Explanation
Given:
The given graph shows the effects of driving speed on gas mileage.
The unit of the fuel economy is miles per gallon while the unit of speed is miles per hour.
Calculation:
The gas can be saved only when F(v) attain its maximum value.
The function F(v) is maximum when F′(v)=0.
From part (b), it is observed that the derivative F′(v)=0 when v=50 miles per hour.
Thus, the speed at which one should drive to save gas is 50 miles per hour.
