Problem 5 - Applied Calculus

Question

Suppose that \(q=f(p)\) is the demand curve for a product, where \(p\) is the selling price in dollars and \(q\) is the quantity sold at that price. (a) What does the statement \(f(12)=60\) tell you about demand for this product? (b) Do you expect this function to be increasing or decreasing? Why?

Step-by-Step Solution

Answer

(a) At $12, 60$ units are sold; (b) The function is likely decreasing as price increase leads to lower demand.

Step 1: Understanding the Function Notation

The notation \(f(12) = 60\) means that when the price \(p\) is 12 dollars, the quantity sold \(q\) is 60 units. This tells us that at a unit price of 12 dollars, the product sells 60 units.

Step 2: Analyzing the Demand Curve Behavior

Demand curves typically show the relationship between price and quantity demanded. Generally, as the price \(p\) increases, the quantity demanded \(q\) decreases. This inverse relationship suggests that the demand function \(q = f(p)\) is likely to be decreasing.

Key Concepts

Function Notation

Function notation, such as \( f(p) \), is a way to represent relationships between variables in mathematics. It's a concise method to describe how one variable changes with respect to another. In the context of a demand curve, \( f(p) \) signifies the demand function where \( p \) is the price, and \( q \) or \( f(p) \) is the quantity demanded at that price.
This notation helps us understand the specific relationship without ambiguity.

The given statement \( f(12) = 60 \) specifically tells us that when the price is 12 dollars, the quantity sold is 60 units. This notation elegantly captures complex relationships in a compact format while making analysis straightforward.

• \( f(p) \): This represents the function.\( f \) indicates the function name, and \( p \) is the variable.
• \( f(12) \): This tells you the output value when the input \( p \) equals 12.
• \( =60 \): Shows the outcome of the function when \( p \) is 12 dollars. In this case, it’s the quantity sold.

Understanding this notation is crucial for analyzing how different factors, like changes in price, affect demand.

Inverse Relationship

An inverse relationship is a type of correlation where an increase in one variable leads to a decrease in another. In the case of a demand curve, as the price \( p \) of an item rises, the quantity demanded \( q \) usually decreases. This represents an inverse relationship between price and demand.

People tend to buy less of a product when prices are high and more when prices are low. This principle is central to the law of demand because:

• It reflects consumer behavior, which tends to be sensitive to price changes.
• It explains why demand curves on graphs generally slope downwards from left to right.

Recognizing this pattern helps in understanding market dynamics and price sensitivity. When analyzing or predicting demand, assumptions about this inverse correlation are typical, as deviations might suggest the presence of unusual market conditions or consumer preferences.

Demand Function

The demand function, often presented as \( q = f(p) \), is a mathematical way to represent how much of a product consumers are willing to purchase at various prices. The function encapsulates the relationship between price and quantity demanded, allowing economists and businesses to model consumer behavior.

The main characteristics of demand functions include:

• They can be linear or non-linear, representing different market dynamics.
• They typically showcase the inverse relationship between price and demand.
• Understanding these functions aids in pricing strategies and forecasting sales.

For example, the demand function \( f(p) = 100 - 2p \) signifies that at a price \( p \), the quantity demanded is based on subtracting twice the price from 100.

This systematic approach to understanding demand helps businesses adjust their strategies to meet changing market conditions and optimize pricing to maximize revenue.