Calculus and Its Business Applications
Calculus in business measures change: marginal cost, marginal revenue, elasticity, and the output that maximises profit or minimises average cost. The exam tests limits, standard derivatives, first- and second-order conditions, and basic integration as the reverse of differentiation.
Core rules
- Derivative rules: d/dx(xⁿ) = nx^(n−1); derivative of a constant is 0; (u ± v)′ = u′ ± v′; product and quotient rules as required; chain rule for composite functions.
- Marginal cost MC = C′(q), marginal revenue MR = R′(q), marginal profit = π′(q). Profit is maximised where MR = MC and π″(q) < 0.
- A stationary point has f′(x) = 0. If f″(x) < 0 it is a maximum; if f″(x) > 0 it is a minimum.
- Integration: ∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1. A definite integral gives the area under a curve between limits and can recover total cost from marginal cost.
Worked example
A firm has revenue R(q) = 50q − 0.5q² and cost C(q) = 10q + 200. Profit is π(q) = R(q) − C(q) = −0.5q² + 40q − 200. Differentiate: π′(q) = −q + 40. Setting this to zero gives q = 40. The second derivative is π″(q) = −1 < 0, so q = 40 is a maximum. Profit there is −0.5(40)² + 40(40) − 200 = −800 + 1,600 − 200 = ₹600. The marginal condition also works: MR = 50 − q and MC = 10, so 50 − q = 10 gives q = 40.
Traps
- Differentiating a constant and keeping it instead of writing zero.
- Finding f′(x) = 0 but not checking the second derivative or the endpoints of a restricted domain.
- Omitting the constant of integration in an indefinite integral.
- Treating marginal cost as total cost; MC is the cost of one more unit, not the whole bill.