Problem 39 - Calculus

Question

If \(g(v)\) is the fuel efficiency, in miles per gallon, of a car going at \(v\) miles per hour, what are the units of \(g^{\prime}(90) ?\) What is the practical meaning of the statement \(g^{\prime}(55)=-0.54 ?\)

Step-by-Step Solution

Answer

The units of \(g'(90)\) are miles per gallon per mile per hour. \(g'(55)=-0.54\) means fuel efficiency decreases by 0.54 mpg for each 1 mph increase at 55 mph.

Step 1: Understand the Function

The given function is denoted by \(g(v)\) where \(v\) represents the velocity of a car in miles per hour, and \(g(v)\) represents the fuel efficiency in miles per gallon. Thus, \(g\) is a function that translates velocity into fuel efficiency.

Step 2: Define the Derivative and Units of g'(90)

The derivative \(g'(v)\) represents how the fuel efficiency changes with respect to changes in speed. This means that \(g'(90)\) describes the rate of change of fuel efficiency when the car is traveling at 90 miles per hour. Hence, the units for \(g'(90)\) are \(\frac{\text{miles per gallon}}{\text{miles per hour}}\) or "miles per gallon per mile per hour."

Step 3: Interpret Practical Meaning of g'(55)=-0.54

The statement \(g'(55) = -0.54\) implies that at a speed of 55 miles per hour, the fuel efficiency of the car is decreasing by 0.54 miles per gallon for every additional mile per hour increase in speed. This negative sign indicates the efficiency is dropping as speed increases at that point.

Key Concepts

Fuel Efficiency

Fuel efficiency is a measure of how far a vehicle can travel on a specific amount of fuel. For instance, when we say a car has a fuel efficiency of 35 miles per gallon (mpg), it means the car can travel 35 miles using one gallon of fuel.
In the context of the problem, fuel efficiency is given by the function \(g(v)\), which signifies how efficient the car is at utilizing fuel at different speeds \(v\).

• Higher fuel efficiency means longer distances per gallon.
• Different speeds impact how efficiently fuel is used.
• Economical driving can reduce fuel consumption, saving costs and resources.

Understanding fuel efficiency helps drivers optimize their speed for the best mileage.

Rate of Change

The rate of change is a mathematical way to describe how one quantity changes in relation to another. In terms of derivatives, it states how the function's output changes when there is a small change in the input.
The derivative \(g'(v)\) in this context represents the rate of change of fuel efficiency with speed. For example, the practical meaning of \(g'(55) = -0.54\) is:

• It tells us that at 55 mph, if we increase the speed slightly, the fuel efficiency decreases by 0.54 mpg for each additional mph.
• A negative rate of change indicates a decrease in fuel efficiency as speed increases at this point.

This helps drivers understand the impact of driving faster or slower on fuel usage.

Units of Measurement

Units of measurement are crucial in quantifying and understanding the relationships between different variables. In our exercise, \(g'(v)\) is expressed in the units \(\frac{\text{miles per gallon}}{\text{miles per hour}}\).
These units provide a clear picture of how efficiency changes with speed:

• Miles per gallon (mpg) measures how far a car can travel with a gallon of fuel.
• Miles per hour (mph) measures how fast the car is moving.
• The derivative \(g'(v)\) combines these to show efficiency change per speed change.

Understanding these units is essential in interpreting how adjustments in speed affect fuel economy.

Velocity

Velocity is a vector quantity that represents how fast something is moving and in what direction. For the problem at hand, velocity \(v\) is considered as the speed of the car in miles per hour.
The role of velocity in this exercise is multi-faceted:

• It serves as the input for the function \(g(v)\), determining fuel efficiency at that speed.
• The rate of change or derivative \(g'(v)\) gives insights into how efficiency varies with small changes in velocity.
• Understanding velocity helps in analyzing dynamic conditions which affect fuel consumption.

By interpreting velocity effectively, drivers and engineers can make informed decisions regarding driving speeds for optimal fuel efficiency.