Question

The graph (from the US Department of Energy) shows how driving speed affects gas mileage. Fuel economy F is measured in miles per gallon and speed \(v\) is measured in miles per hour.

a) What is the meaning of the derivative \(F'\left( v \right)\)?

b) Sketch the graph of \(F'\left( v \right)\).

c) At what speed should you drive if you want to save on gas?

Short answer

(a) The derivative \(F'\left( v \right)\) represents the instantaneous rate of change of fuel in respective of speed.

(b) There will be variation in graph depending on estimates of \(F'\), however, at \(v = 50\), it will turn from positive to negative.

The graph is shown below:

(c) You can save gas when you drive at a speed of 50 \({\mathop{\rm mi}\nolimits} /h\) where \(F\) is maximum and \(F'\) is 0.

Step-by-step solution

Explain the derivative \(F'\left( v \right)\)

a) The derivative of \(F'\left( v \right)\) represents the instantaneous rate of change of fuel in respective of speed.

Sketch the graph of \(F'\left( v \right)\)

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

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

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

It is observed from the graph that the tangent at \(v = A\) is 10. Then the value of \(F'\left( A \right) = 1\).

\(F'\left( v \right)\) can be determined as the ratio of the rise and run that appears to be equal. The function \(F\left( v \right)\) is increasing at \(v = B\)—the value of \(F'\left( B \right)\) is 2.

It is observed that the tangent at \(v = C\) is horizontal, therefore the slope is zero and it follows that the value of \(F'\left( C \right) = 0\).

The slope of the function \(F\left( v \right)\) is \(F'\left( v \right)\) can be determined as the ratio of the fall and run. The function \(F\left( v \right)\) is decreasing at \(v = D\)—the value of \(F'\left( D \right)\) is 0.

The function \(F\left( v \right)\) is decreasing at \(v = E\). The value of \(F'\left( E \right) = - 1\).

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:

There will be variation in graph depending on estimates of \(F'\), however, at \(v = 50\), it will turn from positive to negative.

Determine at what should speed to save on gas

c) You can save gas when you drive at a speed of 50 \({\mathop{\rm mi}\nolimits} /h\) where \(F\) is maximum and \(F'\) is 0.