Question

The demand function for a product is modeled by \(p=75-0.25 x\) (a) If \(x\) changes from 7 to 8 , what is the corresponding change in \(p\) ? Compare the values of \(\Delta p\) and \(d p\). (b) Repeat part (a) when \(x\) changes from 70 to 71 units.

Short answer

For \(x\) changing from 7 to 8, \(\Delta p = -0.25\) and \(dp = -0.25\). For \(x\) changing from 70 to 71, \(\Delta p = -0.25\) and \(dp = -0.25\).

Step-by-step solution

Calculate \(\Delta p\) for \(x\) from 7 to 8

First, find the values of \(p\) when \(x=7\) and \(x=8\) by replacing \(x\) in the function \(p=75-0.25x\) and subtract the two to find \(\Delta p\).

Calculate \(dp\) for \(x\) from 7 to 8

Derive the demand function to get \(dp/dx = -0.25\). This derivative is constant so the value of \(dp\) for \(x\) is the same for any \(x\).

Calculate \(\Delta p\) for \(x\) from 70 to 71

Repeat the process in step one with new \(x\) values for 70 and 71.

Calculate \(dp\) for \(x\) from 70 to 71

Given that the derivative is constant, the value of \(dp\) will be the same as that obtained in step two.

Key concepts

Derivative

In mathematics, derivatives are a fundamental tool for examining how functions change. They provide a precise way of finding the rate at which one quantity changes with respect to another. For a demand function like the one given, which is modeled by \(p = 75 - 0.25x\), the derivative \(\frac{dp}{dx}\) tells us how the price \(p\) changes as the quantity (number of units, \(x\)) changes.

• To find \(\frac{dp}{dx}\), we derive the function with respect to \(x\) and obtain \(\frac{dp}{dx} = -0.25\).
• This derivative value of \(-0.25\) means that for each additional unit produced or sold, the price drops by \(0.25\) units.

Derivatives are particularly useful for predicting how small changes in one variable will affect another, particularly when these variables are tightly linked, like price and quantity in demand functions.

Price Elasticity

Price elasticity of demand is a measure that assesses the responsiveness of the quantity demanded to a change in price. It is quantified as the percentage change in quantity demanded divided by the percentage change in price. Understanding whether the demand is elastic or inelastic helps businesses set prices that maximize revenue.

• If demand is elastic, consumers are more responsive to price changes. A small drop in price could result in a large increase in quantity demanded.
• If demand is inelastic, consumers are less responsive to price changes, meaning price changes don't significantly affect the quantity demanded.

In the context of the demand function \(p = 75 - 0.25x\), the derivative \(\frac{dp}{dx} = -0.25\) represents a constant rate of change in price with respect to quantity. While this specific equation doesn't directly calculate elasticity, analyzing the interplay of \(x\) and price changes can shed light on elasticity behaviors within a given interval.

Marginal Analysis

Marginal analysis is an examination of the additional benefits of an activity compared to the additional costs incurred by that same activity. It plays a crucial role in economics, especially in decision-making processes. This analysis identifies optimal levels of production by assessing the marginal cost and marginal benefit.

In demand functions, marginal analysis helps determine how changes in production levels will impact prices and revenues.

• For the function \(p = 75 - 0.25x\), marginal analysis involves using the derivative \(\frac{dp}{dx}\) to assess the impact of unit increases in quantity \(x\).
• A constant marginal cost, like \(0.25\) per unit in this case, signals a predictable decrease in price per additional unit produced.
• This allows businesses to better manage production strategies by knowing exactly how price might respond to changes in output levels.

Ultimately, through marginal analysis, companies can fine-tune their operations to align with demand and optimize profits in competitive markets.