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).