Problem 85 - Calculus and its applications

Question

At a price of \(x\) dollars, the demand, in thousands of units, for a certain turntable is given by the demand function $$ q=240 e^{-0.003 x} $$ a) How many turntables will be bought at a price of \(\$ 250 ?\) Round to the nearest thousand. b) Graph the demand function for \(0 \leq x \leq 400\). c) Find the marginal demand, \(q^{\prime}(x)\). d) Interpret the meaning of the derivative.

Step-by-Step Solution

Answer

a) 166,000 units. b) Graph shows exponential decay. c) Marginal demand is \(-0.72 e^{-0.003x}\). d) Derivative shows rate of decrease in demand with price increase.

Step 1: Demand Calculation

To find the number of turntables sold at a price of \(\$250\), substitute \(x = 250\) into the demand function \(q = 240 e^{-0.003x}\). Calculate \(q = 240 e^{-0.003 \times 250}\), which evaluates to approximately 165.665. Rounding to the nearest thousand, the demand is 166,000 units.

Step 2: Graphing the Demand Function

The demand function \(q = 240 e^{-0.003x}\) is an exponential decay function. To graph this, calculate values of \(q\) for a range of \(x\) values from 0 to 400. Plot these points on a graph to visualize how demand decreases with increasing price.

Step 3: Finding the Marginal Demand

To find the marginal demand \(q^{\prime}(x)\), differentiate the demand function \(q = 240 e^{-0.003x}\) with respect to \(x\). Use the chain rule: \(q^{\prime}(x) = \frac{d}{dx}[240 e^{-0.003x}] = 240 \times -0.003 e^{-0.003x}\), which simplifies to \(-0.72 e^{-0.003x}\).

Step 4: Interpret the Derivative

The derivative \(q^{\prime}(x) = -0.72 e^{-0.003x}\) represents the marginal demand, which indicates the rate of change of demand concerning price. This derivative tells us how the demand for turntables decreases as the price increases by one dollar.

Key Concepts

Demand Function

In economics, the demand function is a crucial tool that reflects the relationship between the price of a good and the quantity demanded by consumers. For the given exercise, the demand function is expressed as:

• \( q = 240 e^{-0.003x} \)

This function specifically represents an exponential decay model. Here, \(q\) is the number of turntables (in thousands) demanded at a price level \(x\) (in dollars). The exponential component, \(e^{-0.003x}\), implies that demand decreases as price increases, a common scenario in economics.
Imagine you are interested in calculating how many turntables will be sold at a particular price, say \(\$250\). By substituting \(x = 250\) into the function, you can find the corresponding demand, which highlights consumer behavior dynamically as price changes. This function is immensely valuable for businesses to predict sales and plan production according to different pricing strategies.

Marginal Demand

Marginal demand is an essential concept in understanding how demand changes with respect to price. It is represented by the derivative of the demand function with respect to price, \(x\).

• The marginal demand function in our exercise is \(q^{\prime}(x) = -0.72 e^{-0.003x}\).

This expression tells us how fast the demand changes as the price goes up by one dollar. Specifically, the negative sign indicates that as price increases, demand decreases, which is consistent with typical demand behavior.
In simple terms, marginal demand provides insight into consumer sensitivity to price changes. For instance, a high absolute value of the derivative signifies that small changes in price lead to significant changes in demand. Companies use this information to optimize pricing strategies, aiming to maximize revenue or manage inventory effectively.

Exponential Decay Function

An exponential decay function is a mathematical model used to describe a process that decreases at a consistent rate over time. Within the demand function context, this model effectively captures halved quantities or rapid reductions in demand as prices increase.

• The core feature of the exponential decay function is the exponent's negative sign, narrowly linked to the concept \(q = 240 e^{-0.003x}\).

This means that as \(x\) increases, the term \(e^{-0.003x}\) rapidly approaches zero, illustrating the steep decline in demand.
For businesses and economists, understanding exponential decay is pivotal. It provides quantitative backing for the expected decline in sales with increasing prices, allowing for informed decision-making. Whether used to set competitive prices or assess market risks, the exponential decay model is an effective representation of real-world economic phenomena.