Problem 2 - Calculus Concepts

Question

Write an equation or differential equation for the given information. The marginal cost of producing window panes (that is, the rate of change of \(\operatorname{cost} c\) with respect to the number of units produced) is inversely proportional to the number of panes \(p\) produced.

Step-by-Step Solution

Answer

The differential equation is \( \frac{dc}{dp} = \frac{k}{p} \).

Step 1: Understanding the Problem

We're tasked with finding an equation describing the marginal cost of producing window panes. Marginal cost refers to the derivative of the cost function with respect to the number of units, i.e. \( \frac{dc}{dp} \). The problem states this marginal cost is inversely proportional to the number of panes produced, \( p \).

Step 2: Setting Up the Proportionality Equation

The phrase 'inversely proportional' implies a relationship of the form \( \frac{dc}{dp} = \frac{k}{p} \), where \( k \) is a proportionality constant. Here, \( \frac{dc}{dp} \) is the marginal cost and \( \frac{k}{p} \) quantitatively expresses 'inversely proportional to the number of panes \( p \)'.

Step 3: Forming the Differential Equation

From the proportionality relationship established, we form the differential equation: \( \frac{dc}{dp} = \frac{k}{p} \). This equation represents the marginal cost \( \frac{dc}{dp} \) being inversely proportional to \( p \), proportional constant being \( k \).

Key Concepts

Marginal Cost

Marginal cost is a fundamental concept in economics, especially when discussing production costs. It refers to the additional cost incurred by producing one more unit of a good or service. Put simply, it's the cost of increasing production by one unit. This concept is pivotal in decision-making because:

• It helps businesses determine the level of production that maximizes profits.
• It predicts how costs will change with changes in production levels.
• Understanding marginal costs can guide pricing strategies.

To calculate it mathematically, you observe how the total cost changes when you slightly increase the number of units produced. If you denote the cost function as \( c(p) \), the marginal cost is the derivative \( \frac{dc}{dp} \), showing how cost changes with respect to the number of units, \( p \). In many economic models, analyzing this rate of change helps determine optimal production levels.

Inverse Proportionality

Inverse proportionality is a relationship between two variables where the product of the variables is a constant. In other words, as one variable increases, the other decreases at a rate that maintains a constant product. This relationship is expressed as \( y = \frac{k}{x} \), where \( k \) is the constant of proportionality.In the context of the given problem, the marginal cost \( \frac{dc}{dp} \) is inversely proportional to the number of panes \( p \). This means as more panes are produced, the marginal cost per pane decreases, implying a certain efficiency in production as volume increases.Many real-world relationships exhibit inverse proportionality, including:

• The relationship between speed and time taken for a journey.
• Supply and price in market conditions with constant demand (as supply increases, price typically drops).

Understanding inverse proportionality is crucial for comprehending how variables interact dynamically.

Differentiation

Differentiation is a key calculus tool allowing us to understand rates of change. When we differentiate a function, we compute its derivative, which represents the rate at which the function's value changes as its input changes.For instance, the derivative \( \frac{dc}{dp} \) in our problem is a differentiation example. It shows how the cost changes with every additional window pane produced. Differentiation thus enables:

• Finding and analyzing marginal costs, as it directly examines how costs evolve with production volumes.
• Optimizing functions to find minimum and maximum points, pertinent in cost minimization and revenue maximization.
• Tracking the behavior of functions, vital in forecasting and strategic planning.

This concept is more than just a mathematical procedure; it's a window into understanding change, which is essential in economics and numerous other fields.

Economics in Calculus

Economics uses calculus to model and solve a range of economic problems. Marginal cost as a concept directly arises from calculus, demonstrating how small changes in production influence overall costs.Incorporating calculus in economics allows for:

• More accurate modeling of complex economic systems, like maximization of profits or minimization of costs.
• Understanding dynamic relationships between economic variables over time.
• Increasing precision in economic analysis and predictions.

For example, using the differential equation \( \frac{dc}{dp} = \frac{k}{p} \), we can better understand the cost dynamics in production processes. Calculus thus transforms abstract economic theories into quantifiable, actionable insights, aiding economists in crafting strategies for efficient resource allocation and market behavior analysis.