Problem 51 - Calculus Volume 1

Question

Suppose that \(N(x)\) computes the number of gallons of gas used by a vehicle traveling \(x\) miles. Suppose the vehicle gets 30 \(\mathrm{mpg.}\) a. Find a mathematical expression for \(N(x)\) b. What is \(N(100) ?\) Explain the physical meaning. c. What is \(N^{\prime}(100) ?\) Explain the physical meaning.

Step-by-Step Solution

Answer

a. \(N(x) = \frac{x}{30}\). b. \(N(100) = 3.33\), meaning about 3.33 gallons used for 100 miles. c. \(N'(100) = \frac{1}{30}\), showing gas mileage is \(\frac{1}{30}\) gallons/mile.

Step 1: Establishing the Mathematical Expression for N(x)

The problem states that the vehicle gets 30 miles per gallon (mpg). This means for every gallon of gas, the vehicle can travel 30 miles. Therefore, to compute the number of gallons of gas used, we need to divide the number of miles traveled by the vehicle by the miles per gallon. Thus, the mathematical expression for \( N(x) \), representing the gallons of gas used for traveling \( x \) miles, is:\[N(x) = \frac{x}{30}.\]

Step 2: Calculating N(100)

To find \( N(100) \), substitute \( x = 100 \) miles into the expression obtained for \( N(x) \):\[N(100) = \frac{100}{30} = \frac{10}{3} \approx 3.33.\]This calculation means that the vehicle uses approximately 3.33 gallons of gas to travel 100 miles.

Step 3: Understanding N'(x) and Calculating N'(100)

To understand \( N^{\prime}(x) \), observe that \( N(x) \) is a linear function. Differentiating both sides of the function \( N(x) = \frac{x}{30} \) with respect to \( x \), we get:\[N^{\prime}(x) = \frac{d}{dx}\left(\frac{x}{30}\right) = \frac{1}{30}.\]Thus, \( N^{\prime}(100) = \frac{1}{30} \), which indicates the rate of change of gas consumption per mile is \( \frac{1}{30} \) gallons per mile.

Step 4: Interpreting the Physical Meaning of N'(100)

The derivative \( N^{\prime}(100) = \frac{1}{30} \) represents the rate at which gas is consumed as the vehicle travels. Specifically, it tells us that on average, the vehicle consumes \( \frac{1}{30} \) of a gallon for every additional mile traveled, which corresponds to the efficiency of the vehicle in terms of fuel consumption.

Key Concepts

Derivatives

In calculus, a derivative represents the rate at which a function is changing at any given point. It is a fundamental tool for understanding how variables change in relation to each other. In our exercise, when we take the derivative of the function \(N(x) = \frac{x}{30}\), which describes the number of gallons used for traveling \(x\) miles, we get \(N'(x) = \frac{1}{30}\). This derivative is constant, indicating that the rate at which the vehicle consumes fuel remains the same per additional mile.

For most linear functions, as in this example, the derivative will be constant. This simplification helps us understand the efficiency of processes or actions that exhibit uniform change. By knowing the rate, or derivative, we can make predictions about consumption or cost over any journey distance.

Linear Functions

Linear functions are a type of mathematical expression where the relationship between the two variables involved is proportional. These functions create straight lines when graphed. In our exercise, \(N(x) = \frac{x}{30}\) is a linear function, meaning that the gallons of gas used increases at a constant rate as the miles traveled increase.

One of the easiest ways to recognize a linear function is by its format, \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. In the exercise, \(m = \frac{1}{30}\) represents miles per gallon efficiency. Linear functions are immensely helpful in various real-world applications, from predicting costs to engineering efficiency.

Fuel Efficiency

Fuel efficiency is a measure of how effectively a vehicle converts fuel into motion. In our problem, the vehicle's fuel efficiency is 30 miles per gallon (mpg). This means for every 30 miles driven, exactly one gallon of fuel is used.

Understanding fuel efficiency not only helps in controlling fuel costs but also reduces environmental impact. Higher fuel efficiency translates to traveling more miles with less fuel consumption, which generally results in fewer emissions. As shown in the exercise, knowing the miles per gallon allows us to create mathematical models, like \(N(x) = \frac{x}{30}\), which can predict fuel requirements and help in planning journeys efficiently.

Mathematical Modeling

Mathematical modeling involves using mathematical equations to represent real-world scenarios. By translating physical situations into mathematical symbols, we gain a more precise understanding of their dynamics.

In the exercise, we modeled the vehicle's fuel usage with \(N(x) = \frac{x}{30}\). This model allows us to quickly compute the gallons of gas used for any distance \(x\). Furthermore, by differentiating this model, we got \(N'(x)\), which provides insights into the rate of change or efficiency of fuel usage.
Through models, complex systems can be analyzed, predicted, and optimized using calculus concepts like derivatives and linear functions, making them indispensable in fields like engineering, economics, and science.