Problem 71 - College Algebra and Calculus: An Applied Approach

Question

Find the profit function for the given marginal profit and initial condition. $$ \frac{d P}{d x}=-24 x+805 \quad P(12)=\$ 8000 $$

Step-by-Step Solution

Answer

The profit function \(P(x)\) is \(P(x) = -12x^2 + 805x - 13224\)

Step 1: Perform the Integral

To get from a derivative back to the original function, an integral is performed. Hence, the integral of \( \frac{d P}{d x} = -24x + 805 \) with respect to \(x\) will give the profit function \(P(x)\) up to some constant \(C\) (because the constant would disappear when taking the derivative). The integral is computed using the antiderivative rule. \(\int -24x + 805 \, dx = -12x^2 + 805x + C\)

Step 2: Apply the Initial Condition

The constant of integration \(C\) can be determined using the given initial condition \(P(12) = 8000\). Substitute \(x = 12\) into the result from Step 1: \( -12(12)^2 + 805(12) + C = 8000 \) Then solve for \(C\). The result is \(C = -13224\)

Step 3: Write the Final Profit Function

Substitute the value for \(C\) into the equation from Step 1. The final profit function is then \(P(x) = -12x^2 + 805x - 13224\)

Key Concepts

Marginal Profit

Marginal profit is an important concept in economics and calculus that is essential for understanding how profits change with respect to changes in production levels. It is the derivative of the profit function with respect to the number of units produced, essentially showing how much profit is expected to change if one more unit is produced. This means

• If you're producing more units, your marginal profit helps you see exactly how much more profit or loss each additional unit might bring.
• When the marginal profit is zero, it typically means you've reached the level of production where you're maximizing profit – producing more could start reducing your overall profit.

In our exercise, the marginal profit function given is \( \frac{dP}{dx} = -24x + 805 \), meaning each additional unit changes profit depending on both a linear term \(-24x\) and a constant term \(805\). Understanding this initially allows us to estimate and manage the increases or decreases in profit efficiently.

Integral Calculus

Integral calculus is a branch of calculus that deals with the concept of "integration." It is essentially about finding the overall quantity from the rate of change, much like moving from speed to distance, or in our case, from marginal profit back to the profit function. This means

• We use integration to find unknown functions when we know their rate of change, which in financial calculus, helps us determine the original profit function from its derivative.
• The integration process involves finding anti-derivatives, which reverse the effect of differentiation.

In this exercise, we performed the integration on \( \frac{dP}{dx} = -24x + 805 \) with respect to \(x\). As a result, we obtained the function \(-12x^2 + 805x + C\). Here, \(C\) represents the constant of integration that can be determined with more information, specifically the initial conditions.

Initial Conditions

Initial conditions are a critical aspect of solving problems in calculus that involve differential equations. They provide specific values at a given point, which allows us to solve for any constants introduced during integration. Essentially, without them, we wouldn’t be able to find the precise equation, just a family of potential equations. This means:

• Initial conditions turn a general solution into a specific one by allowing us to determine unknown constants.
• For business calculations, knowing specific values like the exact profit at a given production level makes our predictions and models much more reliable.

In our scenario, we used the initial condition \(P(12) = 8000\) to solve for the constant \(C\) in the integrated equation. By substituting these into \(-12x^2 + 805x + C\), we found \(C\) to be \(-13224\). This finalizes the specific profit function as \(P(x) = -12x^2 + 805x - 13224\), accurately matching the conditions unique to this problem's scenario.