GUIDE of Business Mathematics NEB 12 Derivaties Top Questions

Advanced Business Math Solutions

Example 1: Minimizing Average Cost

Question: If the total cost function is C = 1/3 Q^3 - 3Q^2 + 9Q, find the output level (Q) where average cost is minimum and find that minimum cost.

Step 1: Find Average Cost (AC)
AC = Total Cost / Q = (1/3 Q^3 - 3Q^2 + 9Q) / Q = 1/3 Q^2 - 3Q + 9.

Step 2: Find the Derivative of AC
d(AC)/dQ = 2/3 Q - 3.

Step 3: Solve for Q
Set the derivative to zero: 2/3 Q - 3 = 0. This gives 2Q = 9, so Q = 4.5.

Step 4: Find the Minimum Value
Substitute Q = 4.5 into the AC equation: 1/3(4.5)^2 - 3(4.5) + 9 = 2.25.
Hint: Always divide by Q first before differentiating to find Average Cost Minimums!

Example 2: Profit Maximization

Question: The demand equation is P = 1/3 Q^2 - 10Q + 75. Find the value of P that maximizes total revenue.

Step 1: Find Total Revenue (R)
Revenue = P * Q = (1/3 Q^2 - 10Q + 75) * Q = 1/3 Q^3 - 10Q^2 + 75Q.

Step 2: Find Marginal Revenue (MR)
Differentiate Revenue: MR = Q^2 - 20Q + 75.

Step 3: Solve for Q
Set MR = 0: Q^2 - 20Q + 75 = 0. Factoring gives (Q - 5)(Q - 15) = 0.
Testing shows Q = 5 gives the maximum.

Step 4: Find P
Plug Q = 5 into the demand equation to get P = 33.33.
Tip: Revenue is maximum when its derivative (MR) hits zero.

Example 5: Applied Maxima

Question: Determine two positive numbers whose sum is 30 and whose product is maximum.

Step 1: Set up equations
Let numbers be x and y. x + y = 30, so y = 30 - x.
Product P = x * (30 - x) = 30x - x^2.

Step 2: Differentiate Product
dP/dx = 30 - 2x.

Step 3: Solve
Set to zero: 30 - 2x = 0, so x = 15. If x is 15, then y is also 15.
Hint: When a sum is fixed, the product is always largest when the two numbers are equal (a square)!

Example 3: Monopolist Profit Maximization

Question: For a monopolist, demand is P = 100 - 10Q and Average Cost is AC = (100/Q) + 10Q - 100 + 20Q. Find the max profit.

Steps:
1. Revenue = (100 - 10Q)Q = 100Q - 10Q^2.
2. Total Cost = AC * Q = 100 + 10Q^2 - 100Q + 20Q^2 = 30Q^2 - 100Q + 100.
3. Profit = Revenue - Cost = -40Q^2 + 200Q - 100.
4. Differentiate and set to 0: -80Q + 200 = 0.
5. Result: Q = 2.5.