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Let's solve the problem by finding the marginal and average functions for each of the provided total functions. We will also explain how to graph them.

### (a) Total Cost Function: \( C(Q) = 3Q^2 + 7Q + 12 \)

#### 1. **Marginal Cost (MC)**:
The marginal cost is the derivative of the total cost function \( C(Q) \) with respect to \( Q \).

\[
MC(Q) = \frac{dC(Q)}{dQ} = \frac{d}{dQ} \left( 3Q^2 + 7Q + 12 \right) = 6Q + 7
\]

#### 2. **Average Cost (AC)**:
The average cost is the total cost divided by \( Q \):

\[
AC(Q) = \frac{C(Q)}{Q} = \frac{3Q^2 + 7Q + 12}{Q} = 3Q + 7 + \frac{12}{Q}
\]

---

### (b) Total Revenue Function: \( R(Q) = 10Q - Q^2 \)

#### 1. **Marginal Revenue (MR)**:
The marginal revenue is the derivative of the total revenue function \( R(Q) \) with respect to \( Q \).

\[
MR(Q) = \frac{dR(Q)}{dQ} = \frac{d}{dQ} \left( 10Q - Q^2 \right) = 10 - 2Q
\]

#### 2. **Average Revenue (AR)**:
The average revenue is the total revenue divided by \( Q \):

\[
AR(Q) = \frac{R(Q)}{Q} = \frac{10Q - Q^2}{Q} = 10 - Q
\]

---

### (c) Total Product Function: \( Q(L) = aL + bL^2 - cL^3 \)

#### 1. **Marginal Product (MP)**:
The marginal product is the derivative of the total product function \( Q(L) \) with respect to labor \( L \).

\[
MP(L) = \frac{dQ(L)}{dL} = \frac{d}{dL} \left( aL + bL^2 - cL^3 \right) = a + 2bL - 3cL^2
\]

#### 2. **Average Product (AP)**:
The average product is the total product divided by labor \( L \):

\[
AP(L) = \frac{Q(L)}{L} = \frac{aL + bL^2 - cL^3}{L} = a + bL - cL^2
\]

---

Now that we have calculated the marginal and average functions, you can graph these functions for visualization by plotting:

1. **Marginal Cost (MC) and Average Cost (AC) vs. Quantity \( Q \)**.
2. **Marginal Revenue (MR) and Average Revenue (AR) vs. Quantity \( Q \)**.
3. **Marginal Product (MP) and Average Product (AP) vs. Labor \( L \)**.

Would you like help generating the graphs or do you have any questions about the solutions?

### Related Questions:
1. How do marginal and average cost functions help in optimizing production decisions?
2. Why does the marginal revenue curve have a downward slope in this example?
3. How does the marginal product influence labor demand in a firm?
4. What is the economic interpretation of the average cost having a \( \frac{12}{Q} \) term?
5. How do marginal and average functions differ conceptually and mathematically in terms of economic analysis?

### Tip:
Remember, when marginal functions (cost, revenue, product) intersect their respective average functions, they indicate key points such as maximum efficiency (for marginal product) or profit maximization (for marginal revenue and cost).

Find the marginal and average functions for the following total functions and graph the results. Total-cost function: (a) C = 3Q^2 + 7Q + 12. Total-revenue function: (b) R = 10Q - Q^2. Total-product function: (c) Q = aL + bL^2 - cL^3, where a, b, c > 0.

Learn how to find and graph the marginal and average functions for cost, revenue, and product functions. This problem includes detailed solutions using calculus and economic analysis for functions like C(Q) = 3Q^2 + 7Q + 12, R(Q) = 10Q - Q^2, and Q(L) = aL + bL^2 - cL^3. Suitable for undergraduate economics students.

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