Check

FEEN Regular Weekly Test

Business Mathematics: Comprehensive Case Study

Topic: Complete Application of Derivatives

Case Study • 20 Marks • 45 Minutes

CASE STUDY: The "FEEN SmartWatch" Launch

FEEN Electronics Ltd. has started the production of a new gadget called the "FEEN SmartWatch". The company's financial analysts have derived the following mathematical models for their production and sales:

• 📉 Demand Function: P = 100 - 2Q
  (Where P is price per unit in Rs. '000 and Q is output in thousands)
• 🏭 Total Cost Function: C = (1/3)Q³ - 5Q² + 28Q + 10
  (Where C is Total Cost in Rs. '000)

As the Chief Financial Officer (CFO), you are required to analyze the business using calculus. Answer the following questions based on the functions above.

1. Market Analysis (Elasticity)

Calculate the Price Elasticity of Demand (Ed) when the quantity sold is Q = 10. Interpret whether the demand is elastic, inelastic, or unitary at this point.

💡 Tip: First find P when Q=10. Then find derivative dP/dQ. Use formula: Ed = -(P/Q) × (1 ÷ (dP/dQ)).

2. Revenue Formulation

Construct the Total Revenue Function (R) and the Marginal Revenue Function (MR) in terms of Q.

💡 Tip: Revenue is simply Price × Quantity (R = P × Q). Marginal Revenue is the first derivative of R (dR/dQ).

3. Cost Structure Analysis

Analyze the Marginal Cost (MC) function:

• Find the Marginal Cost (MC) function.
• Find the Stationary Point of the Marginal Cost function.
• Find the Point of Inflection of the Total Cost function. What does this point represent in business terms?

💡 Tip: MC is the derivative of Total Cost. To find the stationary point of MC, take the derivative of MC and set to 0. The inflection point of TC is where the 2nd derivative is zero (d²C/dQ² = 0).

4. Profit Optimization

Determine the level of output (Q) that maximizes the Total Profit (π). Also, calculate the maximum profit amount.

💡 Tip: Profit (π) = Revenue (R) - Cost (C).

1. Form the equation π = R - C.
2. Find dπ/dQ and set it to 0 to find critical Q.
3. Crucial Step: Find d²π/dQ² and check if it is NEGATIVE to confirm Maximum Profit.

🔥 Quick Check: Answers

Q1: Elasticity
|Ed| = 4 (Elastic)

Q3: Point of Inflection
At Q = 5

Q4: Max Profit Output
Q = 9 (Check validity!)

© Focus Edge Education Network (FEEN) | Regular Weekly Test Series