Section 4.14

Business Applications

Marginal Cost. Marginal Revenue. Elasticity. Calculus is the engine of modern economics, allowing businesses to find the perfect price point to maximize profit.

1

The "Marginal" Concept

Cost & Revenue

  • Cost C(x): Total cost to produce items.
  • Revenue R(x): Total money made from items ().
  • Profit P(x): .

Marginal Analysis

  • Marginal Cost C'(x): Cost to make one more item.
  • Marginal Revenue R'(x): Revenue from selling one more item.
Max Profit occurs when Marginal Revenue = Marginal Cost
2

Worked Example

Maximizing Factory Profit

Given and .

The Profit Gap

Profit: $2000.00

Green (Revenue) vs Red (Cost). Max Profit occurs where the gap is widest.

Notice the profit peaks exactly where the tangent slopes (marginal rates) are parallel.

1. Form Revenue
2. Find Marginals
3. Equate & Solve

Produce 750 units for max profit.

3

Level Up Examples

Example A: Average Cost

Minimize Average Cost for .

1. Formula:
.
2. Derivative:
.
3. Solve:
.

Example B: Elasticity of Demand

If demand is , find Elasticity at .

1. Formula:
. (Measure of sensitivity).
2. Derivative:
.
3. Compute:
.
.

Unit Elasticity (Ideal Revenue spot!).

5

Practice Quiz

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