Understanding the Demand Function

Oct 13, 2025 by miodeya.com 65 views

Understanding the Demand Function

The demand function, denoted as D(x), represents the relationship between the quantity of a product demanded by consumers and the price of the product. In this case, we are given the demand function D(x) = \frac{167}{5x + 7}. This function indicates that the quantity demanded is inversely proportional to the price, which is a common phenomenon in economics.

Marginal Revenue

Marginal revenue is the change in total revenue that results from selling one additional unit of a product. It is a measure of the additional revenue generated by selling one more unit of a product. Mathematically, marginal revenue (MR) is represented as the derivative of the total revenue (TR) function with respect to the quantity sold (x).

Calculating Marginal Revenue

To calculate the marginal revenue, we need to find the derivative of the total revenue function. The total revenue function is given by the product of the price and the quantity sold. Since the price is given by the demand function D(x), we can write the total revenue function as:

TR(x) = P(x) * x = D(x) * x = \frac{167}{5x + 7} * x

To find the marginal revenue, we need to differentiate the total revenue function with respect to x. Using the product rule of differentiation, we get:

MR(x) = \frac{d}{dx} (TR(x)) = \frac{d}{dx} (\frac{167x}{5x + 7}) = \frac{(5x + 7)(167) - 167x(5)}{(5x + 7)^2} = \frac{167(5x + 7 - 5x)}{(5x + 7)^2} = \frac{167(7)}{(5x + 7)^2} = \frac{1169}{(5x + 7)^2}

Evaluating Marginal Revenue at x = 5

Now that we have the marginal revenue function, we can evaluate it at x = 5 to find the marginal revenue when x = 5.

MR(5) = \frac{1169}{(5(5) + 7)^2} = \frac{1169}{(25 + 7)^2} = \frac{1169}{(32)^2} = \frac{1169}{1024} ≈ 1.14

Rounding the Answer to the Nearest Cent

Since we are asked to round the answer to the nearest cent, we can round 1.14 to 1.14.

Conclusion

In this article, we have discussed the concept of marginal revenue and how to calculate it using the demand function. We have also evaluated the marginal revenue function at x = 5 and rounded the answer to the nearest cent. The marginal revenue when x = 5 is approximately 1.14.

References

[1] Demand Function. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Demand_function
[2] Marginal Revenue. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Marginal_revenue

Additional Resources

[1] Economics 101. (n.d.). Retrieved from https://www.khanacademy.org/economics-finance-topic
[2] Microeconomics. (n.d.). Retrieved from https://www.britannica.com/topic/microeconomics
Marginal Revenue Q&A =========================

Frequently Asked Questions

Q: What is marginal revenue?

A: Marginal revenue is the change in total revenue that results from selling one additional unit of a product. It is a measure of the additional revenue generated by selling one more unit of a product.

Q: How do I calculate marginal revenue?

A: To calculate marginal revenue, you need to find the derivative of the total revenue function with respect to the quantity sold (x). The total revenue function is given by the product of the price and the quantity sold.

Q: What is the formula for marginal revenue?

A: The formula for marginal revenue is:

MR(x) = \frac{d}{dx} (TR(x))

where TR(x) is the total revenue function.

Q: How do I evaluate marginal revenue at a specific value of x?

A: To evaluate marginal revenue at a specific value of x, you need to substitute the value of x into the marginal revenue function and simplify.

Q: What is the relationship between marginal revenue and demand function?

A: The marginal revenue function is related to the demand function through the total revenue function. The demand function represents the relationship between the quantity demanded and the price, while the marginal revenue function represents the change in total revenue resulting from a change in quantity sold.

Q: Can you provide an example of how to calculate marginal revenue?

A: Let's consider the demand function D(x) = \frac{167}{5x + 7}. To calculate the marginal revenue, we need to find the derivative of the total revenue function TR(x) = P(x) * x, where P(x) is the price function given by the demand function.

MR(x) = \frac{d}{dx} (TR(x)) = \frac{d}{dx} (\frac{167x}{5x + 7}) = \frac{(5x + 7)(167) - 167x(5)}{(5x + 7)^2} = \frac{1169}{(5x + 7)^2}

Q: How do I round the answer to the nearest cent?

A: To round the answer to the nearest cent, you need to round the decimal value of the marginal revenue to the nearest hundredth.

Q: What is the significance of marginal revenue in economics?

A: Marginal revenue is an important concept in economics as it helps businesses and policymakers understand the relationship between the quantity sold and the revenue generated. It is a key factor in decision-making, particularly in pricing and production decisions.

Q: Can you provide additional resources for learning more about marginal revenue?

A: Yes, there are many resources available for learning more about marginal revenue, including textbooks, online courses, and articles. Some recommended resources include:

Economics 101 by Khan Academy
Microeconomics by Britannica
Marginal Revenue by Investopedia

Conclusion

In this article, we have provided answers to frequently asked questions about marginal revenue, including its definition, calculation, and significance in economics. We have also provided examples and additional resources for learning more about marginal revenue.