Problem 65 - Calculus and its applications

Question

Lessard \& Company finds that the rate at which the quantity of flameless candles that consumers demand changes with respect to price is given by the marginal-demand function $$D^{\prime}(x)=-\frac{4000}{x^{2}}$$ where \(x\) is the price per candle, in dollars. Find the demand function if 1003 candles are demanded by consumers when the price is \(\$ 4\) per candle.

Step-by-Step Solution

Answer

The demand function is \(D(x) = \frac{4000}{x} + 3\).

Step 1: Integration of the Marginal Demand Function

To find the demand function, we must integrate the marginal demand function. Given: \[D'(x) = -\frac{4000}{x^2}\]Integrate with respect to \(x\):\[D(x) = \int -\frac{4000}{x^2} \, dx = \int -4000x^{-2} \, dx\]The integral of \(-4000x^{-2}\) is:\[D(x) = 4000x^{-1} + C = \frac{4000}{x} + C\]

Step 2: Apply the Boundary Condition

We know from the problem statement that 1003 candles are demanded when the price is \$4. Thus:\[D(4) = 1003\]Plug \(x = 4\) into the demand function:\[1003 = \frac{4000}{4} + C\]\[1003 = 1000 + C\]

Step 3: Solve for the Constant of Integration (C)

From the equation:\[1003 = 1000 + C\]Subtract 1000 from both sides to solve for \(C\):\[C = 3\]

Step 4: Write the Explicit Demand Function

Substitute \(C = 3\) back into the demand function:\[D(x) = \frac{4000}{x} + 3\]This is the demand function that gives the number of candles demanded \(D(x)\) as a function of the price \(x\).

Key Concepts

Marginal Demand

Marginal demand is an essential concept in economics, helping businesses understand how the quantity demanded of a product changes with its price. In the context of our exercise, this is represented by the marginal-demand function, \(D'(x)\). Specifically, this tells us the rate at which the demand for flameless candles changes as the price per candle, \(x\), fluctuates.

The given marginal demand function is \(D'(x) = -\frac{4000}{x^{2}}\). The negative sign indicates that as the price increases, the quantity demanded decreases, which is a common real-world scenario. The change is proportional to \(\frac{1}{x^2}\), meaning that as the price increases, the rate of decrease in demand will diminish over time. This reflects how price sensitivity might lessen at higher prices, a concept known as diminishing marginal utility.

• The marginal demand function acts as a snapshot of demand sensitivity.
• Helps companies decide on pricing strategies.
• Facilitates predictions about consumer behavior patterns.

Understanding the marginal demand helps businesses like Lessard & Company set the right price point to maximize profits.

Integration

Integration is a mathematical process used to reverse differentiation and is key to finding the demand function from the marginal demand function. Think of it as piecing together a picture from its outlines. In our case, we integrate \(D'(x) = -\frac{4000}{x^2}\) to find the actual demand function \(D(x)\).

This involves finding a function whose derivative results in the marginal demand function. Integration can be seen as summing up infinitesimal changes in demand across different price points. Here's how it's done:

• Rewrite the integrand: \(-4000x^{-2}\).
• Integrate with respect to \(x\): \(D(x) = \int -4000x^{-2} \, dx = \frac{4000}{x} + C\).

The integration introduces a 'constant of integration' \(C\), which accounts for initial conditions or specific cases, a concept we'll explore next. Through integration, we convert useful instantaneous information into a broad picture that helps predict candle demand based on price.

Constant of Integration

The constant of integration \(C\) is crucial in forming the specific solution to an indefinite integral. While integration gives you the family of functions that could potentially represent demand, the constant \(C\) is what tailors it to the given situation.

For Lessard & Company, when it's known that 1003 candles are demanded at a price of \$4 per candle, you use this known demand to determine \(C\).

• Plug \(x=4\) into the integrated demand function: \(1003 = \frac{4000}{4} + C\).
• Solve for \(C\) to find: \(1003 = 1000 + C\).
• Thus, \(C = 3\).

Incorporating this constant into the demand function refines it to reflect real-world data. Hence, the demand function becomes \(D(x) = \frac{4000}{x} + 3\). Here, \(C\) plays a pivotal role in ensuring that the demand equation precisely meets known conditions, facilitating accurate predictions for future pricing strategies.