Problem 2 - Applied Calculus

Question

Write the Leibniz notation for the derivative of the given function and include units. The cost, \(C,\) of a steak, in dollars, is a function of the weight, \(W,\) of the steak, in pounds.

Step-by-Step Solution

Answer

The derivative in Leibniz notation is \(\frac{dC}{dW}\), with units dollars per pound.

Step 1: Understand the Function Relationship

The function given is the cost, \(C\), of a steak which depends on its weight, \(W\). We can express this relationship with the function \(C(W)\). It signifies that the cost \(C\) is a function of weight \(W\).

Step 2: Identify the Derivative

The derivative of the function \(C(W)\) with respect to \(W\) is written as \(\frac{dC}{dW}\). This expression indicates how the cost \(C\) changes with respect to changes in weight \(W\).

Step 3: Include Units

Since \(C\) is measured in dollars and \(W\) is measured in pounds, the units for the derivative \(\frac{dC}{dW}\) would be dollars per pound. This means for every pound change in weight, the change in cost is measured in dollars.

Key Concepts

Understanding Derivatives in Leibniz Notation

The derivative of a function provides critical information about how a function behaves. In Leibniz notation, it is expressed as \(\frac{dy}{dx}\), where \(y\) is a function of \(x\). In our example, the cost \(C\) is a function of weight \(W\). So, the derivative is written as \(\frac{dC}{dW}\). This notation indicates how much the cost of the steak changes for each unit change in weight. Derivatives show us the rate of change, which is crucial in many real-world contexts. Whether you are examining how fast a car accelerates or how a disease spreads, derivatives are the mathematical way to describe these changes.

Function Relationships: Inputs and Outputs

A function relationship is an association between two quantities. One is an input, and the other is an output. For the steak example, weight \(W\) is the input, and cost \(C\) is the output. We write this relationship as \(C(W)\), which means the cost depends on the weight. Understanding this dependency is crucial because it allows us to predict the output for a given input.

• The function \(C(W)\) signifies the relationship between cost and weight.
• The input variable \(W\) is used to compute the output \(C\).
• This relationship follows the generic format \(y=f(x)\), where \(y\) is the output and \(x\) is the input.

Grasping function relationships helps in forming mathematical models to represent real-world scenarios.

Units in Calculus: Expressing Change

Units play a vital role in interpreting derivatives. In the context of the problem, the derivative \(\frac{dC}{dW}\) carries units of dollars per pound. This unit reveals that for every pound of steak weight added, the cost changes by a certain number of dollars.

• Units indicate the nature of change: \(\frac{dC}{dW}\) represents dollars per pound.
• It shows the cost change per unit weight, making the derivative practical and interpretable.
• Units are an essential part of any physical quantities, providing context and meaning.

By including units, the derivative becomes a powerful tool for understanding and applying mathematical findings to real-world situations.