Problem 18 - Calculus

Question

Let \(p\) be the price of an item and \(q\) be the number of items sold at that price, where \(q=f(p) .\) What do the following quantities mean in terms of prices and quantities sold? (a) \(f(25)\) (b) \(f^{-1}(30)\)

Step-by-Step Solution

Answer

(a) Items sold at a price of 25; (b) Price for selling 30 items.

Step 1: Understand the Function

The function \( f(p) \) is a relationship where the number of items sold, \( q \), depends on the price \( p \). This function provides \( q \) when given a specific price \( p \). Thus, \( f(p) = q \) for different price values \( p \).

Step 2: Evaluate f(25)

To evaluate \( f(25) \), we consider \( f(p) \) with \( p = 25 \). So \( f(25) \) represents the number of items sold when the price per item is \( 25 \) units (dollars, euros, etc.).

Step 3: Understand the Inverse Function

The inverse function \( f^{-1}(q) \) indicates which price \( p \) results in \( q \) items being sold. It reverses the roles of prices and quantities from the original function.

Step 4: Evaluate f^{-1}(30)

To evaluate \( f^{-1}(30) \), consider the price \( p \) at which exactly 30 items are sold. Therefore, \( f^{-1}(30) \) represents the price at which the quantity sold is 30 items.

Key Concepts

Function Evaluation

Function evaluation is the process of determining the output of a function for a specific input. In the context of our exercise, the function \( f(p) \) represents a relationship between the price \( p \) of an item and the number of items \( q \) sold at that price. Evaluating \( f(p) \) means finding out how many items are sold when the item is priced at a certain level.

For instance, when we evaluate \( f(25) \), we are essentially asking, "How many items will be sold if each is priced at 25 units?" Here, it tells us the precise quantity of items sold at this specific price. This simple concept helps businesses make strategic decisions, by evaluating how pricing can influence sales volume.

In terms of real-world applications, function evaluation helps in:

• Predicting sales based on different price points
• Adjusting pricing strategies to increase sales
• Understanding market demand at various price levels

Function evaluations are powerful because they provide actionable insights that directly affect sales forecasting and inventory management.

Price-Quantity Relationship

The price-quantity relationship is a fundamental concept in economics and marketing. It describes how the amount of a product that consumers buy changes in response to changes in its price.

In mathematical terms, this relationship can be expressed as a function \( q = f(p) \), where \( q \) is the quantity sold and \( p \) is the price level. This function shows whether the item is a luxury, necessity, or inferior good, based on how sensitive the quantity is to changes in price.

Understanding the price-quantity relationship involves:

• Assessing demand elasticity, which measures how much the quantity demanded will change with a change in price
• Setting optimal prices that balance profitability and competitiveness
• Analyzing consumer behavior and market trends

Through this relationship, businesses can determine how various prices might lead to different sales volumes. It helps in designing pricing strategies that maximize revenue while ensuring customers remain satisfied.

Mathematical Modeling

Mathematical modeling is the process of creating a mathematical representation of a real-world situation to predict and analyze behaviors and outcomes.

In the exercise, \( f(p) \) and its inverse \( f^{-1}(q) \) are examples of mathematical models used to capture the relationship between price and quantity. Each model offers a different perspective: while \( f(p) \) tells us how quantity varies with price, the inverse function \( f^{-1}(q) \) reveals what price corresponds to a specific sales volume.

Mathematical modeling allows us to:

• Simulate different pricing scenarios and their potential effects on sales
• Understand the complex dynamics between consumer choices and pricing
• Optimize pricing strategies to maximize profitability

By employing mathematical models, businesses gain insights that drive data-driven decision-making, ensuring they adapt effectively to market changes and consumer needs.