Total Cost from Marginal Cost

Business Calculus · Axiom Academy

LESSON Total Cost from Marginal Cost Using integration to recover the total cost function Differentiation and integration are inverse operations: Where FC is the fixed cost (the constant of integration) Marginal cost tells us how much each additional unit costs. By "adding up" all these marginal costs (integrating), we get the total variable cost. Add the fixed costs, and we have the complete cost function. 2 Finding the Constant (Fixed Costs) When integrating, we get + C . In business contexts, this constant represents fixed costs . "Fixed costs are 5,000" means C = 5000 "It costs 5,000 to produce 0 units" → C(0) = 5000 "Producing 100 units costs 12,000" → C(100) = 12000 Fixed costs (FC) are costs that don't change with production level: rent, insurance, salaries, equipment. Variable costs come from the integral of marginal cost. 3 Example: Manufacturing Plant A factory's marginal cost function is: Fixed costs (rent, insurance, etc.) total 8,000 per month. Find the total cost function. Verify: Check with Marginal Cost Differentiate the total cost to confirm we recover the marginal cost: C'(x) = 2(0.03)x + 20 = 0.06x + 20 = MC(x) ✓ 4 Example: Finding FC from a Known Point A company's marginal cost is MC(x) = + 5 dollars per unit. If producing 100 units costs 1,500, find C(x). 5 Common MC Forms and Their Integrals

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