2. A firm has an average cost function, $$ A C=Q^{2}-9 Q+\frac{150}{Q}+75 $$ (a) Find an expression for the TC function and calculate TC when $$ Q=15 $$. 4 marks (b) Write down the equations for FC and TVC. 2 marks (c) Find an expression for the MC function. 3 marks 3. Find the maximum and/or minimum values (if any) for each of the functions below. (a) $$ P=-2 Q^{2}+8 Q $$ 5 marks (b) $$ y=x^{3}-3 x^{2}-9 x $$ 5 marks 1 Question Two 1. Find $$ \frac{d y}{d x} $$ from the first principles of differentiation. (a) $$ \mathrm{y}=2 \mathrm{x}^{2}+3 x $$ 2 marks (b) $$ \mathrm{y}=\sqrt{x} $$ 3 marks 2. Determine the intervals along which each of the following curves is increasing or decreasing (consider the positive half of the plane, $$ x0 $$ ) (a) $$ A C=Q^{2}-20 Q+120 $$ 5 marks (b) $$ T R=50 Q-Q^{2} $$ 5 marks 3. Sketch the following functions over the stated interval. (a) $$ Q=100-P^{2}, \quad-1

Question

2. A firm has an average cost function, $$ A C=Q^{2}-9 Q+\frac{150}{Q}+75 $$ (a) Find an expression for the TC function and calculate TC when $$ Q=15 $$. 4 marks (b) Write down the equations for FC and TVC. 2 marks (c) Find an expression for the MC function. 3 marks 3. Find the maximum and/or minimum values (if any) for each of the functions below. (a) $$ P=-2 Q^{2}+8 Q $$ 5 marks (b) $$ y=x^{3}-3 x^{2}-9 x $$ 5 marks 1 Question Two 1. Find $$ \frac{d y}{d x} $$ from the first principles of differentiation. (a) $$ \mathrm{y}=2 \mathrm{x}^{2}+3 x $$ 2 marks (b) $$ \mathrm{y}=\sqrt{x} $$ 3 marks 2. Determine the intervals along which each of the following curves is increasing or decreasing (consider the positive half of the plane, $$ x>0 $$ ) (a) $$ A C=Q^{2}-20 Q+120 $$ 5 marks (b) $$ T R=50 Q-Q^{2} $$ 5 marks 3. Sketch the following functions over the stated interval. (a) $$ Q=100-P^{2}, \quad-1

UpStudy AI · Accepted Answer

Sure, let's tackle **Question 2** step by step.

---

## **Question 2**

**A firm has an average cost function:**

$$
AC = Q^{2} - 9Q + \frac{150}{Q} + 75
$$

### **(a) Find an expression for the TC function and calculate TC when $$ Q = 15 $$.** *(4 marks)*

**Solution:**

1. **Understanding the Relationship:**
   
   The **Average Cost (AC)** is related to **Total Cost (TC)** by the formula:
   
   $$
   AC = \frac{TC}{Q}
   $$
   
   Therefore, to find **TC**, multiply **AC** by **Q**:
   
   $$
   TC = AC \times Q
   $$

2. **Substituting the Given AC Function:**
   
   $$
   TC = \left(Q^{2} - 9Q + \frac{150}{Q} + 75\right) \times Q
   $$
   
3. **Simplifying the Expression:**
   
   $$
   TC = Q^3 - 9Q^2 + 150 + 75Q
   $$
   
   So, the **Total Cost (TC)** function is:
   
   $$
   TC = Q^{3} - 9Q^{2} + 75Q + 150
   $$

4. **Calculating TC When $$ Q = 15 $$:**
   
   Substitute $$ Q = 15 $$ into the TC function:
   
   $$
   TC = (15)^3 - 9(15)^2 + 75(15) + 150
   $$
   
   $$
   TC = 3375 - 2025 + 1125 + 150
   $$
   
   $$
   TC = 3375 - 2025 = 1350
   $$
   
   $$
   1350 + 1125 = 2475
   $$
   
   $$
   2475 + 150 = 2625
   $$
   
   **Therefore, when $$ Q = 15 $$, $$ TC = 2625 $$.**

---

### **(b) Write down the equations for FC and TVC.** *(2 marks)*

**Solution:**

1. **Total Cost (TC) Breakdown:**
   
   $$
   TC = FC + TVC
   $$
   
2. **From the TC Function:**
   
   $$
   TC = Q^{3} - 9Q^{2} + 75Q + 150
   $$
   
   - **Fixed Cost (FC):** This is the part of TC that does not depend on $$ Q $$.
     
     $$
     FC = 150
     $$
   
   - **Total Variable Cost (TVC):** This comprises all terms in TC that depend on $$ Q $$.
     
     $$
     TVC = Q^{3} - 9Q^{2} + 75Q
     $$
   
**Therefore:**
   
$$
FC = 150
$$
$$
TVC = Q^{3} - 9Q^{2} + 75Q
$$

---

### **(c) Find an expression for the MC function.** *(3 marks)*

**Solution:**

1. **Marginal Cost (MC):** It's the derivative of Total Cost (TC) with respect to $$ Q $$.
   
   $$
   MC = \frac{dTC}{dQ}
   $$

2. **Differentiate the TC Function:**
   
   $$
   TC = Q^{3} - 9Q^{2} + 75Q + 150
   $$
   
   $$
   \frac{dTC}{dQ} = 3Q^{2} - 18Q + 75
   $$
   
   So,
   
   $$
   MC(Q) = 3Q^{2} - 18Q + 75
   $$

**Therefore, the Marginal Cost (MC) function is:**

$$
MC(Q) = 3Q^{2} - 18Q + 75
$$

---

If you need assistance with the other questions or further clarifications, feel free to ask!
**Question 2:** **(a) Total Cost (TC) when $$ Q = 15 $$ is 2625.** **(b) Fixed Cost (FC) is 150, and Total Variable Cost (TVC) is $$ Q^{3} - 9Q^{2} + 75Q $$.** **(c) Marginal Cost (MC) is $$ 3Q^{2} - 18Q + 75 $$.**

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