The profit P (in dollars) from selling x units of a product is given by the function below.
P = 36,000 + 2051

x
 − 
1
8x2
,    150 ≤ x ≤ 275
Find the marginal profit for each of the following sales. (Round your answers to two decimal places.)

To calculate the marginal profit, we need to find the derivative of the profit function \( P(x) \), which gives us the rate of change of profit with respect to the number of units sold \( x \). 

Given the profit function: 

\[
P(x) = 36,000 + 2051x - \frac{1}{8}x^2
\]

The marginal profit is represented by the derivative \( P'(x) \). Let's differentiate \( P(x) \) with respect to \( x \):

\[
P'(x) = \frac{d}{dx}(36,000 + 2051x - \frac{1}{8}x^2)
\]

Step-by-step differentiation:

1. The derivative of the constant \( 36,000 \) is 0.
2. The derivative of \( 2051x \) is \( 2051 \).
3. The derivative of \( -\frac{1}{8}x^2 \) is \( -\frac{1}{4}x \) (using the power rule).

Thus, the marginal profit function is:

\[
P'(x) = 2051 - \frac{1}{4}x
\]

Now, we can calculate the marginal profit at specific values of \( x \) by plugging them into \( P'(x) \). Let me know which specific values you would like to calculate the marginal profit for, or if you have any predefined values.

Would you like to see detailed examples for specific \( x \) values? Here are some related questions for practice:

1. What is the marginal profit when \( x = 200 \)?
2. How does the marginal profit change as \( x \) increases?
3. What is the interpretation of negative marginal profit?
4. At which point is the marginal profit zero?
5. How does this quadratic form affect the marginal profit compared to a linear profit model?

**Tip**: The marginal profit function can help determine the optimal production level by identifying where profit maximization occurs.

The profit P (in dollars) from selling x units of a product is given by the function below. P = 36,000 + 2051x − 1/8x^2, 150 ≤ x ≤ 275. Find the marginal profit for each of the following sales. (Round your answers to two decimal places.)

Explore how to calculate the marginal profit from a quadratic profit function P(x) = 36,000 + 2051x - (1/8)x^2. Learn step-by-step how to differentiate the profit function to find the rate of change of profit with respect to sales. This solution applies calculus concepts and is suitable for advanced high school or college students.

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