Find The Marginal Profit Function If Cost And Revenue Are Given By:$\[ C(x) = 206 + 0.4x \\]$\[ R(x) = 7x - 0.01x^2 \\]$\[ P^{\prime}(x) = \square \\]

Feb 27, 2025 by ADMIN 151 views

**Marginal Profit Function: A Comprehensive Guide**

Introduction

In economics, the marginal profit function is a crucial concept used to determine the additional profit gained from producing one more unit of a product. It is a vital tool for businesses to make informed decisions about production levels and pricing strategies. In this article, we will explore how to find the marginal profit function given the cost and revenue functions.

Cost and Revenue Functions

The cost function, denoted by C(x), represents the total cost of producing x units of a product. On the other hand, the revenue function, denoted by R(x), represents the total revenue generated from selling x units of a product. In this case, we are given the following cost and revenue functions:

${ C(x) = 206 + 0.4x }$

${ R(x) = 7x - 0.01x^2 }$

Marginal Cost and Marginal Revenue

The marginal cost function, denoted by C'(x), represents the rate of change of the cost function with respect to the number of units produced. Similarly, the marginal revenue function, denoted by R'(x), represents the rate of change of the revenue function with respect to the number of units sold.

To find the marginal cost function, we take the derivative of the cost function with respect to x:

${ C'(x) = \frac{d}{dx} (206 + 0.4x) }$

${ C'(x) = 0.4 }$

To find the marginal revenue function, we take the derivative of the revenue function with respect to x:

${ R'(x) = \frac{d}{dx} (7x - 0.01x^2) }$

${ R'(x) = 7 - 0.02x }$

Marginal Profit Function

The marginal profit function, denoted by P'(x), represents the rate of change of the profit function with respect to the number of units produced. The profit function is given by the difference between the revenue function and the cost function:

${ P(x) = R(x) - C(x) }$

${ P(x) = (7x - 0.01x^2) - (206 + 0.4x) }$

${ P(x) = 7x - 0.01x^2 - 206 - 0.4x }$

To find the marginal profit function, we take the derivative of the profit function with respect to x:

${ P'(x) = \frac{d}{dx} (7x - 0.01x^2 - 206 - 0.4x) }$

${ P'(x) = 7 - 0.02x - 0.4 }$

${ P'(x) = 6.6 - 0.02x }$

Conclusion

In conclusion, the marginal profit function is a crucial concept in economics that helps businesses make informed decisions about production levels and pricing strategies. By finding the marginal profit function, businesses can determine the additional profit gained from producing one more unit of a product. In this article, we have explored how to find the marginal profit function given the cost and revenue functions.

Example Problems

Find the marginal profit function if the cost and revenue functions are given by:

${ C(x) = 150 + 0.3x }$

${ R(x) = 8x - 0.02x^2 }$

Find the marginal profit function if the cost and revenue functions are given by:

${ C(x) = 250 + 0.5x }$

${ R(x) = 9x - 0.03x^2 }$

Solutions

To find the marginal profit function, we first need to find the marginal cost and marginal revenue functions.

${ C'(x) = \frac{d}{dx} (150 + 0.3x) }$

${ C'(x) = 0.3 }$

${ R'(x) = \frac{d}{dx} (8x - 0.02x^2) }$

${ R'(x) = 8 - 0.04x }$

The profit function is given by the difference between the revenue function and the cost function:

${ P(x) = R(x) - C(x) }$

${ P(x) = (8x - 0.02x^2) - (150 + 0.3x) }$

${ P(x) = 8x - 0.02x^2 - 150 - 0.3x }$

To find the marginal profit function, we take the derivative of the profit function with respect to x:

${ P'(x) = \frac{d}{dx} (8x - 0.02x^2 - 150 - 0.3x) }$

${ P'(x) = 8 - 0.04x - 0.3 }$

${ P'(x) = 7.7 - 0.04x }$

To find the marginal profit function, we first need to find the marginal cost and marginal revenue functions.

${ C'(x) = \frac{d}{dx} (250 + 0.5x) }$

${ C'(x) = 0.5 }$

${ R'(x) = \frac{d}{dx} (9x - 0.03x^2) }$

${ R'(x) = 9 - 0.06x }$

The profit function is given by the difference between the revenue function and the cost function:

${ P(x) = R(x) - C(x) }$

${ P(x) = (9x - 0.03x^2) - (250 + 0.5x) }$

${ P(x) = 9x - 0.03x^2 - 250 - 0.5x }$

To find the marginal profit function, we take the derivative of the profit function with respect to x:

${ P'(x) = \frac{d}{dx} (9x - 0.03x^2 - 250 - 0.5x) }$

${ P'(x) = 9 - 0.06x - 0.5 }$

${ P'(x) = 8.5 - 0.06x }$

Final Thoughts

Q&A: Marginal Profit Function

Q: What is the marginal profit function?

A: The marginal profit function is a concept in economics that represents the rate of change of the profit function with respect to the number of units produced. It is a crucial tool for businesses to make informed decisions about production levels and pricing strategies.

Q: How is the marginal profit function related to the cost and revenue functions?

A: The marginal profit function is related to the cost and revenue functions through the profit function. The profit function is given by the difference between the revenue function and the cost function:

${ P(x) = R(x) - C(x) }$

To find the marginal profit function, we take the derivative of the profit function with respect to x:

${ P'(x) = \frac{d}{dx} (R(x) - C(x)) }$

Q: What are the key components of the marginal profit function?

A: The key components of the marginal profit function are:

The marginal cost function, which represents the rate of change of the cost function with respect to the number of units produced.
The marginal revenue function, which represents the rate of change of the revenue function with respect to the number of units sold.
The profit function, which represents the difference between the revenue function and the cost function.

Q: How do I find the marginal profit function if the cost and revenue functions are given?

A: To find the marginal profit function, you need to follow these steps:

Find the marginal cost function by taking the derivative of the cost function with respect to x.
Find the marginal revenue function by taking the derivative of the revenue function with respect to x.
Find the profit function by subtracting the cost function from the revenue function.
Take the derivative of the profit function with respect to x to find the marginal profit function.

Q: What are some common applications of the marginal profit function?

A: The marginal profit function has several applications in economics and business, including:

Determining the optimal production level to maximize profit.
Setting prices to maximize revenue.
Analyzing the impact of changes in cost and revenue on profit.

Q: Can you provide some examples of how to find the marginal profit function?

A: Here are some examples:

Find the marginal profit function if the cost and revenue functions are given by:

${ C(x) = 150 + 0.3x }$

${ R(x) = 8x - 0.02x^2 }$

Find the marginal profit function if the cost and revenue functions are given by:

${ C(x) = 250 + 0.5x }$

${ R(x) = 9x - 0.03x^2 }$

Q: What are some common mistakes to avoid when finding the marginal profit function?

A: Some common mistakes to avoid when finding the marginal profit function include:

Failing to take the derivative of the profit function with respect to x.
Failing to account for the marginal cost and marginal revenue functions.
Failing to consider the impact of changes in cost and revenue on profit.

Q: Can you provide some tips for using the marginal profit function in real-world applications?

A: Here are some tips:

Use the marginal profit function to analyze the impact of changes in cost and revenue on profit.
Use the marginal profit function to determine the optimal production level to maximize profit.
Use the marginal profit function to set prices to maximize revenue.

Conclusion

In conclusion, the marginal profit function is a crucial concept in economics that helps businesses make informed decisions about production levels and pricing strategies. By understanding how to find the marginal profit function and using it in real-world applications, businesses can maximize profit and stay competitive in the market.