 Find the marginal cost, marginal revenue, and marginal profit functions.
C(x) = 7x;  R(x) = 9x − 0.001x^2


Find the marginal cost

Marginal Revenue

Marginal Profit

To solve for the marginal cost, marginal revenue, and marginal profit functions, we differentiate the given functions with respect to \(x\), as these represent the marginal rates.

### 1. **Marginal Cost**:
The marginal cost is the derivative of the cost function \(C(x)\).
Given: 
\[
C(x) = 7x
\]
Differentiate with respect to \(x\):
\[
\text{Marginal Cost} = C'(x) = \frac{d}{dx}(7x) = 7
\]

### 2. **Marginal Revenue**:
The marginal revenue is the derivative of the revenue function \(R(x)\).
Given:
\[
R(x) = 9x - 0.001x^2
\]
Differentiate with respect to \(x\):
\[
\text{Marginal Revenue} = R'(x) = \frac{d}{dx}(9x - 0.001x^2) = 9 - 0.002x
\]

### 3. **Marginal Profit**:
The profit function is given by the difference between revenue and cost:
\[
\text{Profit function} = P(x) = R(x) - C(x)
\]
Thus:
\[
P(x) = (9x - 0.001x^2) - 7x = 2x - 0.001x^2
\]
Now, differentiate the profit function \(P(x)\) with respect to \(x\):
\[
\text{Marginal Profit} = P'(x) = \frac{d}{dx}(2x - 0.001x^2) = 2 - 0.002x
\]

### Summary:
- **Marginal Cost**: \(7\)
- **Marginal Revenue**: \(9 - 0.002x\)
- **Marginal Profit**: \(2 - 0.002x\)

Would you like any further details or clarifications?

Here are 5 related questions you could explore:
1. What happens to the marginal profit as \(x\) increases?
2. At what level of production does marginal revenue equal marginal cost?
3. How would the functions change if the cost had a quadratic term?
4. What is the significance of negative marginal profit?
5. How does marginal analysis influence business decisions?

**Tip:** Marginal functions help predict changes for small increases in production or sales, optimizing decision-making.

Find the marginal cost, marginal revenue, and marginal profit functions. C(x) = 7x; R(x) = 9x − 0.001x^2.

This detailed solution covers how to find the marginal cost, marginal revenue, and marginal profit functions using differentiation. The problem involves a linear cost function and a quadratic revenue function, making it suitable for high school or college-level economics and calculus students.

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