Problem 3 - Applied Calculus

Question

Write the Leibniz notation for the derivative of the given function and include units. The number, \(N,\) of gallons of gas left in a gas tank is a function of the distance, \(D,\) in miles, the car has been driven.

Step-by-Step Solution

Answer

The derivative in Leibniz notation is \(\frac{dN}{dD}\) with units of gallons per mile.

Step 1: Understanding the Function

The function given is for the number of gallons, \(N\), of gas left in a gas tank as a function of the distance, \(D\), driven in miles. This means \(N\) depends on \(D\), and we can write this relationship as \(N(D)\).

Step 2: Identifying the Derivative

The task is to find the derivative of \(N\) with respect to \(D\) using Leibniz notation. The derivative represents how the number of gallons of gas left changes as the car is driven additional miles.

Step 3: Writing the Leibniz Notation

In Leibniz notation, the derivative of \(N\) with respect to \(D\) is written as \(\frac{dN}{dD}\). This symbol indicates the rate of change of \(N\) as \(D\) changes.

Step 4: Including Units

Since \(N\) represents gallons of gas and \(D\) represents miles, the derivative \(\frac{dN}{dD}\) has units of gallons per mile. This shows how many gallons of gas are used per mile driven.

Key Concepts

Derivative

In mathematics, a derivative represents how a function changes as its input changes. It's like measuring the sensitivity of one variable, usually dependent, with respect to another. Think of it as a way to determine how fast something is changing at any given point. Derivatives are essential for understanding motion, growth, and any rate of change in various disciplines. For instance, if you have a function that tells you how much of a gas tank is left as you drive, the derivative of that function at any point will tell you how quickly the gas is being used. In this context, finding the derivative involves using specific notations and rules governed by calculus.

Function Notation

Function notation is a concise way to represent the relationship between two variables, often referred to as the input and the output of the function. In this situation, we have a function that shows the number of gallons in a tank depending on how far a car has traveled. If we let the number of gallons be represented by the letter \(N\), and the distance traveled be \(D\), we show this relationship through function notation as \(N(D)\). This notation tells us that \(N\) is a function of \(D\). By using this function, we can calculate the output for any given input, meaning we can find out how much gas remains for each mile driven.

Rate of Change

The rate of change is a critical concept that reflects how one quantity varies in relation to another over time or space. It's like how quickly a car consumes gas as it moves. In calculus, the rate of change is often expressed with derivatives. In our example of miles driven versus gas remaining, the rate of change offers insights into efficiency and consumption patterns. When we find the derivative \(\frac{dN}{dD}\), we are essentially finding the rate at which the gallons of gas decrease with each mile traveled. This provides valuable information for understanding how fuel-efficient a vehicle is over a journey.

Units of Measurement

Units of measurement are crucial for making sense of quantities and rates in the real world. They give us a standardized way to interpret numbers in specific terms. In this exercise, units play a vital role. We have our derivative \(\frac{dN}{dD}\), which expresses how the function output changes. Here, \(N\), measured in gallons, changes as we measure distance in miles with \(D\). Therefore, the units for our derivative are 'gallons per mile'. This unit tells us how many gallons are consumed per mile, giving a clear understanding of fuel usage per distance, which is invaluable when planning long drives or optimizing vehicle performance.