Marginal Profit

Business Calculus · Axiom Academy

Combining marginal cost and revenue for optimal decisions Profit is what remains after subtracting costs from revenue: Every business wants to maximize profit. Calculus gives us the tools to find exactly how much to produce. Should we produce one more unit? The answer depends on whether that unit adds more revenue than it costs. This is exactly what marginal profit measures! Marginal Profit is the additional profit from producing and selling one more unit. It equals the derivative of the profit function. Since P(x) = R(x) - C(x) , we can take the derivative: Marginal Profit = Marginal Revenue - Marginal Cost Marginal profit tells you the net benefit of producing one more unit: MP > 0: The extra unit brings in more revenue than it costs. Produce it! MP The extra unit costs more than it brings in. Don't produce it! MP = 0: You've found the sweet spot - profit is maximized. At the profit-maximizing quantity, marginal profit equals zero: The Golden Rule: Profit is maximized when MR = MC If MR > MC, you're leaving money on the table - each additional unit adds to profit. If MR < MC, you're losing money on each additional unit. MR = MC is the balance point! A company produces smartwatches with: Cost: C(x) = 5000 + 30x + 0.02x^2 Find the profit-maximizing production level. The company should produce 600 smartwatches to maximize profit. Cost: 5,000 + 30(600) + 0.02(600)² = 30,200 Visualizing Profit Maximization Let's see how marginal cost and marginal revenue intersect:

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