Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Find the marginal revenue for producing \(x\) units. (The revenue is measured in dollars.) $$ R=-6 x^{3}+8 x^{2}+200 x $$

Step-by-Step Solution

Answer

The marginal revenue for producing \(x\) units is given by the derivative function \(R'(x) = -18x^{2} + 16x + 200\).

Step 1: Identifying the Total Revenue Function

The total revenue function, \(R(x)\), is given as \(-6 x^{3}+8 x^{2}+200 x\). This equation represents the revenue realized from producing \(x\) units of a product.

Step 2: Applying the Power Rule of Differentiation

The power rule for differentiation states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). We apply this rule separately to every term of the revenue function \(R(x)\).

Step 3: Calculating the Derivative

When we differentiate \(R(x)\) using the power rule, we have: \(R'(x) = -18x^{2} + 16x + 200\). This function represents the marginal revenue for producing \(x\) units.

Key Concepts

Revenue Function

In economics, a revenue function is a mathematical representation showing how revenue is generated through sales at different levels of output. For this exercise, the revenue function is given as \[R(x) = -6x^{3} + 8x^{2} + 200x\]Each term in this equation plays a role in determining how much revenue is earned. - The term \(-6x^{3}\) often suggests that there are diminishing returns at higher levels of production. - The \(8x^{2}\) and \(200x\) terms might indicate initial increases in revenue as production begins.Understanding the structure of a revenue function is crucial. It helps us understand how changes in production levels affect total revenue.

Power Rule

The power rule is a fundamental tool in calculus used to find the derivative of polynomial functions quickly and easily. It's expressed as: If \(f(x) = x^n\) then \(f'(x) = nx^{n-1}\). Here's how to apply the power rule to each term of a polynomial function:- For \(-6x^3\), multiply the exponent (3) by the coefficient (-6) to get \(-18\), and reduce the exponent by 1, resulting in \(-18x^2\).- For \(8x^2\), the derivative is \(16x\) (since 2 times 8 is 16).- Similarly, the derivative of \(200x\) is simply \(200\) (as the exponent of \(x\) here is 1).By applying the power rule to each term, you efficiently find the slope or rate of change, which in this case refers to marginal revenue.

Differentiation

Differentiation is the process of finding the derivative of a function. In economics, it's particularly useful for determining marginal concepts, like marginal revenue, cost, and profit.Here, differentiation helps us understand how changes in production impact revenue. The derivative of the revenue function, \(R(x) = -6x^{3} + 8x^{2} + 200x\), is calculated using the power rule, resulting in: \[R'(x) = -18x^{2} + 16x + 200\]This derivative function \(R'(x)\) represents the marginal revenue. It tells us how much additional revenue will be generated by producing one more unit. Knowing marginal revenue assists businesses in making pricing and production decisions, optimizing their output to maximize profit potential.