Problem 98 - Calculus Volume 1

Question

Suppose the total profit of a company is \(y=P(x)\) thousand dollars when \(x\) units of an item are sold. a. What does \(\frac{P(b)-P(a)}{b-a}\) for \(0 < a < b\) measure, and what are the units? b. What does \(P^{\prime}(x)\) measure, and what are the units? c. Suppose that \(P^{\prime}(30)=5, \) what is the approximate change in profit if the number of items sold increases from 30 to 31\(?\)

Step-by-Step Solution

Answer

a: Average profit change, thousand dollars per item. b: Instant profit change, thousand dollars per item. c: 5 thousand dollars.

Step 1: Understanding Commission Calculation

The expression \( \frac{P(b)-P(a)}{b-a} \) is the formula for the average rate of change of the profit function, \( P(x) \), over the interval \([a, b]\). This measures the average profit change per unit sold between \( x = a \) and \( x = b \). Since profit \( y \) is in thousand dollars, the units are thousand dollars per item.

Step 2: Meaning of the Derivative

The derivative \( P'(x) \) represents the instantaneous rate of change of profit with respect to the number of items sold at a particular sales level. The units are also thousand dollars per item, as it signifies how much the profit changes for each additional unit sold.

Step 3: Calculating the Approximate Profit Change

Given the information that \( P'(30) = 5 \), we interpret this as the profit increases by approximately 5 thousand dollars when one additional unit is sold, starting from 30 units. The change in profit from selling the 31st item is, therefore, approximately 5 thousand dollars.

Key Concepts

Average Rate of Change

When we talk about the average rate of change, we are trying to understand how a function behaves over a certain interval. If we have a function like the company's profit function, \( P(x) \), the average rate of change between two points \( a \) and \( b \) is given by the formula:\[\frac{P(b) - P(a)}{b-a}\]This tells us how much the profit changes, on average, for each unit increase in sales between \( x=a \) and \( x=b \).
Think of it as finding the slope of a straight line connecting two points on the graph of the profit function. It's like determining how steep the hill is when you hike from point \( a \) to point \( b \), where steeper hills mean more profit per unit.

• The units here are thousand dollars per item because both the numerator and the denominator are divided appropriately to reflect this change over units.
• This method is very useful in various fields because it provides a way to quantify changes over time or over intervals, giving insights into the performance of the profit strategy.

Derivative Interpretation

A derivative in calculus is a powerful tool that tells us the instantaneous rate of change of a function. Specifically, when we calculate \( P'(x) \), we know how the company's profit is changing at exactly any sales level \( x \).
While the average rate of change gives us information over an interval, the derivative gives us insight into what's happening precisely at a single point.

• The units for \( P'(x) \) are the same as for the average rate of change: thousand dollars per item.
• It helps businesses make decisions based on real-time data regarding profitability as they tweak their sales strategy, production rate, or pricing.

The derivative can be thought of as zooming in on the graph to see exactly how steep it is right at one particular point. This helps companies decide if selling one more unit will be profitable and by how much.

Profit Function Analysis

Analyzing a profit function using calculus allows for more accurate predictions and strategies. Given that \( P'(30) = 5 \), we understand that at the moment when 30 items are sold, each additional item sold will increase profit by approximately 5 thousand dollars.

• This interpretation can guide decisions about pricing, inventory, and marketing strategies, helping maximize profits.
• It provides a framework for understanding how sensitive profits are to changes in sales, which is crucial for financial planning and forecasting.

In essence, knowing how profits will behave as we sell more products helps a company aim for optimal performance.
This analysis serves as a continual feedback loop, enabling businesses to adapt quickly and efficiently to market demand changes, potentially leading to more informed decision-making and better financial outcomes.