Problem 46 - College Algebra and Calculus: An Applied Approach

Question

The number \(N\) of gallons of regular unleaded gasoline sold by a gasoline station at a price of \(p\) dollars per gallon is given by \(N=f(p)\). (a) Describe the meaning of \(f^{\prime}(2.959)\) (b) Is \(f^{\prime}(2.959)\) usually positive or negative? Explain.

Step-by-Step Solution

Answer

The value \(f^{\prime}(2.959)\) corresponds to the rate of change in the number of gallons of gas sold in response to a small change in price, when the price is $2.959 per gallon. It's typically expected to be negative, because increasing prices are generally associated with decreasing demand.

Step 1: Understand the meaning of the derivative

The derivative of a function at a certain point gives us the rate of change of that function at that point. In this context, \(f^{\prime}(2.959)\) represents the rate of change of the number of gallons of regular gasoline sold by the station with respect to small changes in the price per gallon when the price is $2.959.

Step 2: Interpret the derivative as rate of change

If \(f^{\prime}(2.959) > 0\), that means the higher the price, the more gallons are sold, and if \(f^{\prime}(2.959) < 0\), that means the higher the price, the fewer gallons are sold. If \(f^{\prime}(2.959) = 0\), that means the number of gallons sold does not change with small changes in price when the price is $2.959.

Step 3: Predict the sign of the derivative (optional)

Typically, as price increases, demand decreases. Therefore, it's likely that \(f^{\prime}(2.959)\) is negative, since it is expected that as the price of gas increases, fewer gallons will be sold.

Key Concepts

Derivative Interpretation

When we talk about derivatives in the context of calculus, we are essentially discussing the rate at which one quantity changes with respect to another. In the given exercise, the derivative \( f^{\prime}(2.959) \) represents the rate of change of the number of gallons of gasoline sold as the price per gallon changes, specifically around the price of $2.959.

To interpret this derivative, consider it as the 'speed' at which sales are increasing or decreasing as the price shifts slightly. If \( f^{\prime}(2.959) \) is positive, it suggests an unusual market phenomenon where higher prices might oddly be driving more sales. Conversely, if it's negative, it aligns with more common economic principles, indicating fewer sales as prices rise. If it equals zero, sales remain stable despite minor price adjustments.

Rate of Change

The concept of rate of change is crucial for understanding how derivatives function. In everyday terms, it's like measuring how quickly or slowly a variable adjusts when another variable alters. In our exercise, this means analyzing how sales volume reacts to price changes at precisely $2.959 per gallon.

Here's how it breaks down:

• If \( f^{\prime}(2.959) > 0 \), then every small increase in price leads to an increase in gasoline sales, a situation that could occur if lower prices are perceived as a loss of quality.
• If \( f^{\prime}(2.959) < 0 \), an increase in price results in decreased sales, reflecting typical consumer behavior where demand falls as prices rise.
• If \( f^{\prime}(2.959) = 0 \), sales numbers remain unaffected by slight price changes, suggesting a perfectly balanced supply and demand at this price point.

Understanding the rate of change helps businesses adjust their pricing strategies to optimize sales and meet market expectations.

Pricing Model Analysis

Pricing model analysis involves examining how different pricing schemes impact sales performance. This analysis is grounded in derivative interpretation and the rate of change, employing them to forecast customer reactions to price adjustments.

For a gas station operating at a price point of $2.959 per gallon, analyzing \( f^{\prime}(2.959) \) becomes vital. By understanding the sign and magnitude of the derivative, the station can predict customer behavior:

• Negative derivative suggests a sensitivity where reducing prices could enhance total sales volume.
• Positive derivative, although rare, might indicate a niche market where higher prices are correlated with increased perceived value.
• A zero derivative indicates a potential price threshold where neither increasing nor decreasing price will significantly impact sales.

This analysis helps in developing effective pricing strategies to maximize profits without alienating customers.