Problem 66 - Calculus and its applications

Question

Keans Corporation finds that the rate at which a seller's quantity supplied changes with respect to price is given by the marginalsupply function $$S^{\prime}(x)=0.24 x^{2}+4 x+10$$ where \(x\) is the price per unit, in dollars. Find the supply function if it is known that the seller will sell 121 units of the product when the price is \(\$ 5\) per unit.

Step-by-Step Solution

Answer

The supply function is \( S(x) = 0.08x^3 + 2x^2 + 10x + 11 \).

Step 1: Understand the Given Information

We are provided with the marginal supply function, which is the derivative of the supply function, as \( S'(x) = 0.24x^2 + 4x + 10 \). Additionally, we know that when the price \( x = 5 \), the supply is 121 units. We need to find the original supply function \( S(x) \).

Step 2: Integrate the Marginal Supply Function

To find the supply function \( S(x) \), we need to integrate the marginal supply function \( S'(x) \). Thus, we compute:\[ S(x) = \int (0.24x^2 + 4x + 10) \, dx \]

Step 3: Solve the Integration

Carrying out the integration, we find:\[ S(x) = 0.08x^3 + 2x^2 + 10x + C \]where \( C \) is the constant of integration.

Step 4: Use Given Condition to Find the Constant

We know that \( S(5) = 121 \). Plug this into our equation to solve for \( C \):\[ 0.08(5)^3 + 2(5)^2 + 10(5) + C = 121 \]Calculate each term:\[ 0.08(125) + 50 + 50 + C = 121 \]\[ 10 + 50 + 50 + C = 121 \]\[ 110 + C = 121 \]\( C = 11 \).

Step 5: Write the Final Supply Function

Substitute \( C = 11 \) back into the supply function:\[ S(x) = 0.08x^3 + 2x^2 + 10x + 11 \]

Key Concepts

Marginal Supply Function

In economics, the concept of a marginal supply function is crucial for understanding how changes in price affect the quantity of goods that suppliers are willing to offer. The marginal supply function, often denoted as \(S'(x)\), represents the rate at which the quantity supplied changes with respect to price. Think of it as a speedometer for supply: it tells you how fast the supply is increasing or decreasing as the price changes.

In simple terms, if we have a marginal supply function given by \(S'(x) = 0.24x^2 + 4x + 10\), this equation gives us the instantaneous change in the quantity supplied per unit change in price. Here, \(x\) represents the price per unit in dollars.

Understanding the marginal supply function helps businesses adjust their production levels in response to price fluctuations. It can guide pricing strategies by indicating when a price increase or decrease could maximize profit. Remember, a higher marginal supply indicates a higher rate of change in supply as price changes, which could be due to various factors like production capacity or cost efficiencies.

Integration in Calculus

Integration is a fundamental concept in calculus, used to find the original function from its derivative. In our exercise, we have the marginal supply function, which is the derivative of the supply function, and our task is to find the supply function itself.

Integrating involves finding a function whose derivative matches the given function. For example, to find the supply function \( S(x) \), we set up the integral of the marginal supply function:
\[ S(x) = \int (0.24x^2 + 4x + 10) \, dx \]

When you perform this integration, it essentially reverses differentiation by finding the antiderivative. You solve for each term:

• The integral of \(0.24x^2\) is \(0.08x^3\)
• The integral of \(4x\) is \(2x^2\)
• The integral of 10 is \(10x\)

After integration, we add a constant \( C \), known as the constant of integration. This constant represents the unknown initial value of the function. Finding this constant requires additional information about the function at a particular point, as we will see next.

Constant of Integration

The constant of integration, represented by \( C \), is a key concept when working with indefinite integrals. After integration, the constant of integration needs to be determined using any additional conditions provided.

In our exercise, once we integrate the marginal supply function, we end up with the following equation:
\[ S(x) = 0.08x^3 + 2x^2 + 10x + C \]
We must determine \( C \) to find the complete supply function. This is done by applying a boundary condition, which is an extra piece of information about the function at a particular value of \( x \). Here, we know \( S(5) = 121 \).

So we substitute \( x = 5 \) in the equation:
\[ 0.08(5)^3 + 2(5)^2 + 10(5) + C = 121 \]
And solve for \( C \):

• Calculate each term: \( 0.08(125) + 50 + 50 + C = 121 \)
• This simplifies to \( 10 + 50 + 50 + C = 121 \)
• Further simplifies to \( 110 + C = 121 \)

Thus, \( C = 11 \). This value plugs back into our integrated function to give us the complete supply function:
\[ S(x) = 0.08x^3 + 2x^2 + 10x + 11 \]
This comprehensive function now accurately models the supplier's behavior in relation to price, thanks to solving for the constant of integration.