Problem 6 - Calculus

Question

An economist is interested in how the price of a certain item affects its sales. At a price of \(\$ p,\) a quantity, \(q,\) of the item is sold. If \(q=f(p),\) explain the meaning of each of the following statements: (a) \(\quad f(150)=2000\) (b) \(\quad f^{\prime}(150)=-25\)

Step-by-Step Solution

Answer

(a) 2000 units are sold at $150; (b) sales decrease by 25 units per $1 increase at $150.

Step 1: Understanding function notation

Function notation is used to describe a relationship between two variables. Here, the function \( f(p) \) represents the quantity \( q \) sold at price \( p \).

Step 2: Analyze statement (a): \( f(150) = 2000 \)

The statement \( f(150) = 2000 \) means that when the price \( p \) is \( \$150 \), the quantity sold \( q \) is \( 2000 \) units. This indicates a specific point on the demand curve for the item.

Step 3: Analyze statement (b): \( f^{\prime}(150) = -25 \)

The derivative \( f^{\prime}(150) = -25 \) represents the rate at which the quantity sold changes with respect to price at \( \\(150 \). Specifically, it means that for each \\)1 increase in price from \$150, the quantity sold decreases by 25 units. This shows the sensitivity of demand to price changes at this price point.

Key Concepts

Function Notation

Function notation is a way to denote functions that depict the relationship between two variables. In this context, the function is noted as \( f(p) \). Here, \( f \) is the function that associates the price \( p \) with the quantity \( q \) sold. The notation is a concise way of expressing how one variable changes as another one varies. This is crucial in economics and many other fields, allowing analysis of dependencies between different quantities.

• \( q = f(p) \): This equation signifies that the quantity \( q \) depends on the price \( p \).
• \( f(150) = 2000 \): This states that at a price of \$150, the quantity sold is 2000 units.

Function notation provides a systematic way to express and evaluate these relationships, making complex interactions more understandable at a glance.

Demand Curve

The demand curve is a graphical representation of the relationship between the price of an item and the quantity demanded by consumers. As prices fluctuate, the quantity demanded will change, typically decreasing as prices rise and increasing as they fall.The demand curve helps to visualize this interaction:

• It generally has a negative slope, indicating an inverse relationship between price and quantity demanded.
• Each point on the curve, such as \( f(150) = 2000 \), represents the quantity demanded at a specific price level.

Understanding the demand curve is essential as it helps businesses determine optimal pricing strategies to maximize revenue or achieve other goals.

Derivative

In calculus, a derivative is a fundamental concept used to determine how a function changes as its input changes. Here, the derivative of the function \( f(p) \), denoted as \( f'(p) \), quantifies the rate of change of the quantity sold with respect to changes in price at any given point.Using derivatives to analyze economic functions:

• \( f'(150) = -25 \): Indicates that at the price of \$150, for each additional dollar the price increases, the quantity sold decreases by 25 units.
• Represents the slope of the demand curve at a given point, providing insights into how sensitive the demand is to price changes.

Derivatives are invaluable tools in economics, giving clear quantitative measures of how variables interact with one another.

Price Sensitivity

Price sensitivity, also known as price elasticity, is a measure of how the quantity demanded of a good responds to changes in its price. It is derived from the concept of the derivative, providing insight into consumer behavior regarding pricing.Key aspects of price sensitivity:

• When \( f'(150) = -25 \), it reflects a high level of sensitivity; a small increase in price results in a significant decrease in quantity demanded.
• Understanding price sensitivity assists businesses in making informed pricing decisions, balancing between consumer demand and sales revenue.

Evaluating price sensitivity allows companies to anticipate market behavior and set prices to align with both consumer expectations and business objectives.