Optimization is one of the most useful applications of calculus in business because it helps managers choose the best production level, price, or cost strategy. A company often wants to maximize profit, maximize revenue, or minimize cost while facing limits such as demand, labor, materials, and capacity. Calculus turns these decisions into functions that can be analyzed with derivatives.
The key idea is to find where a small change in output no longer improves the business goal.
Key Facts
- Profit is revenue minus cost: P(x) = R(x) - C(x).
- Marginal profit is the derivative of profit: P'(x) = R'(x) - C'(x).
- Profit is maximized at an interior point when marginal revenue equals marginal cost: MR = MC.
- Revenue from price and quantity is R(x) = x p(x), where p(x) is the demand price.
- A critical point occurs where f'(x) = 0 or f'(x) is undefined.
- Use the second derivative test: if f''(a) < 0, f has a local maximum at x = a; if f''(a) > 0, f has a local minimum at x = a.
Vocabulary
- Optimization
- Optimization is the process of finding the input value that makes a quantity as large or as small as possible.
- Revenue
- Revenue is the total money earned from selling goods or services, often modeled as price times quantity.
- Cost function
- A cost function gives the total cost of producing a certain number of units.
- Marginal revenue
- Marginal revenue is the rate at which revenue changes when one more unit is produced or sold.
- Marginal cost
- Marginal cost is the rate at which total cost changes when one more unit is produced.
Common Mistakes to Avoid
- Maximizing revenue instead of profit, which is wrong because high sales can still lose money if costs are too large.
- Setting average cost equal to marginal cost for profit maximization, which is wrong because the profit condition is marginal revenue equals marginal cost.
- Forgetting to check endpoints, which is wrong because a maximum or minimum on a restricted business domain can occur at the boundary.
- Ignoring units, which is wrong because derivatives such as MR and MC are measured in dollars per unit, not just dollars.
Practice Questions
- 1 A company has revenue R(x) = 80x - x^2 and cost C(x) = 200 + 20x. Find the production level x that maximizes profit, and find the maximum profit.
- 2 Demand is p(x) = 120 - 2x, and cost is C(x) = 300 + 10x. Write the revenue and profit functions, then find the value of x that maximizes profit.
- 3 A business finds that MR is greater than MC at its current production level. Explain whether it should increase or decrease production to move toward maximum profit.