Problem 42 - Calculus Early Transcendentals

Question

A paint manufacturing company estimates that it can sell \(g=f(p)\) gallons of paint at a price of \(p\) dollars per gallon. (a) What are the units of \(d g / d p ?\) (b) In practical terms, what does \(d g / d p\) mean in this case? (c) What can you say about the sign of \(d g / d p ?\) (d) Given that \(d g /\left.d p\right|_{p=10}=-100,\) what can you say about the effect of increasing the price from \(\$ 10\) per gallon to \(\$ 11\) per gallon?

Step-by-Step Solution

Answer

(a) Units: gallons per dollar. (b) Change in gallons sold per dollar change. (c) Negative (price up, sales down). (d) 100 fewer gallons sold when price increases by \$1.

Step 1: Identify Units of dg/dp

The derivative \( \frac{dg}{dp} \) represents the rate of change of gallons sold with respect to the price per gallon. Since \( g \) (gallons) is measured in gallons, and \( p \) (price) in dollars, the units will be gallons per dollar.

Step 2: Interpretation of dg/dp

In practical terms, \( \frac{dg}{dp} \) indicates how the quantity of paint sold changes when the price per gallon changes. Specifically, it tells us the amount of gallons that either increase or decrease when the price changes by one dollar.

Step 3: Determine the Sign of dg/dp

The sign of \( \frac{dg}{dp} \) usually depends on the relationship between price and quantity sold. Typically, when price increases the demand decreases, meaning \( \frac{dg}{dp} \) is negative. This reflects the general economic principle of demand.

Step 4: Effect of Price Increase from $10 to $11

Given \( \frac{dg}{dp}\bigg|_{p=10} = -100 \), when the price increases from \\(10 to \\)11, the company will sell 100 fewer gallons of paint. This derivative value indicates a decrease in sales by 100 gallons for every dollar the price is increased when the initial price is \$10.

Key Concepts

Rate of Change

The rate of change is an essential concept in calculus, especially when dealing with real-world problems like those faced in a paint manufacturing company. Here, the rate of change refers to how a certain quantity, such as the number of gallons of paint sold, changes as we vary another quantity, such as the price per gallon. When we express this mathematically, we use the derivative notation \( \frac{dg}{dp} \), which represents the rate at which gallons of paint (\(g\)) change with respect to the price (\(p\)).

• Units of Rate of Change: In this exercise, \( \frac{dg}{dp} \) has the units of gallons per dollar because it measures how many gallons change when there is a change in the price by one dollar.
• Practical Interpretation: Practically, this rate tells a business manager how sensitive their sales are to changes in price. A larger absolute value indicates that sales are very responsive to price changes, while a smaller absolute value suggests less sensitivity.

Understanding this concept helps companies make informed decisions about pricing, as it directly translates to understanding customer behavior in response to price adjustments.

Price Elasticity

Price elasticity is a key concept in economics that relates to the rate of change of demand with respect to price changes. It measures how strongly the quantity demanded of a good responds to changes in its price. Here, we calculate elasticity using the derivative \( \frac{dg}{dp} \). When this derivative is negative, as it often is, it indicates that demand decreases as price increases, consistent with typical consumer behavior.

• Elasticity Interpretation: In this context, when \( \frac{dg}{dp} = -100 \) at \( p = 10 \), it signals that if the paint price rises by one dollar, the company can expect to sell 100 fewer gallons.
• Economic Implications: Understanding elasticity helps businesses set prices strategically. A highly elastic product will have significant changes in demand with small changes in price. Conversely, inelastic products see little change in demand despite price variations.

Price elasticity insights are crucial for optimizing pricing strategies, ensuring competitive advantage, and maximizing revenue.

Demand Function

The demand function, denoted as \( g = f(p) \), is an explicit equation that relates the quantity of a product sold to its price. It captures the relationship between price levels and consumer purchase behaviors. In the context of this exercise, the demand function allows us to compute \( \frac{dg}{dp} \), thus encapsulating the effects of price changes on sales volume.

• Function Characteristics: This function often shows that as prices go up, the quantity demanded decreases, aligning with the law of demand. Businesses frequently analyze this to predict sales at various price points.
• Mathematical Use: By differentiating the demand function, companies can gain insights into how sales will change with price adjustments, which aids in forming pricing strategies.

The demand function is instrumental for companies to project how different pricing strategies could affect overall sales, providing a quantitative basis for decision-making.