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### Given Information:

1. **Cost function**: \( C(x) = -0.2x^2 + 3x - 30 \)
2. **Revenue function**: \( R(x) = 76x \)
3. **Profit function**: \( P(x) = R(x) - C(x) \)

Where \( P(x) \) is the profit for producing and selling \( x \) items.

#### (a) **Find the marginal profit using the formula \( P(x+1) - P(x) \) when \( x = 13 \):**

First, calculate the profit function \( P(x) \):

\[
P(x) = R(x) - C(x) = 76x - (-0.2x^2 + 3x - 30) = 76x + 0.2x^2 - 3x + 30
\]

Simplify:

\[
P(x) = 73x + 0.2x^2 + 30
\]

Now, compute \( P(13) \) and \( P(14) \):

1. **For \( x = 13 \):**
   \[
   P(13) = 73(13) + 0.2(13^2) + 30 = 949 + 0.2(169) + 30 = 949 + 33.8 + 30 = 1012.8
   \]

2. **For \( x = 14 \):**
   \[
   P(14) = 73(14) + 0.2(14^2) + 30 = 1022 + 0.2(196) + 30 = 1022 + 39.2 + 30 = 1091.2
   \]

Now, calculate the marginal profit:

\[
P(14) - P(13) = 1091.2 - 1012.8 = 78.4
\]

Thus, the marginal profit when 13 items are sold is **78.4 dollars**.

#### (b) **Use the derivative to estimate the marginal profit when 13 items are sold:**

The derivative of the profit function \( P(x) \) with respect to \( x \) is the **marginal profit**:

\[
P'(x) = \frac{d}{dx} \left( 73x + 0.2x^2 + 30 \right) = 73 + 0.4x
\]

Now, substitute \( x = 13 \) into the derivative:

\[
P'(13) = 73 + 0.4(13) = 73 + 5.2 = 78.2
\]

Thus, the derivative estimate of the marginal profit at 13 items is **78.2 dollars**.

---

Do you want more details or have any questions? Here are 5 follow-up questions related to this problem:

1. How is the marginal profit related to the derivative of the profit function?
2. Can the marginal profit ever be negative? What does that imply?
3. How would you find the break-even point, where profit is zero?
4. How does the cost function being quadratic affect the profit function?
5. How does the derivative help in optimizing profit?

**Tip**: Using the derivative for marginal profit is more precise as it captures the instantaneous rate of change.

The cost and revenue functions for producing and selling x items are given by C(x) = -0.2x^2 + 3x - 30 and R(x) = 76x. Find the marginal profit when 13 items are sold using the formula P(x + 1) - P(x), and use the derivative to estimate the marginal profit when 13 items are sold.

Learn how to calculate the marginal profit using the formula P(x + 1) - P(x) for cost and revenue functions. Understand the role of derivatives in estimating marginal profit with step-by-step solutions. Ideal for students in Grades 11-12 studying calculus.

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