Elementary Calculus and Its Applications
Elementary calculus supplies the marginal analysis used in business decisions. The exam tests limits, continuity, standard derivatives, product/quotient/chain rules, higher-order derivatives, maxima and minima, marginal cost and revenue, elasticity of demand, and simple integration as the reverse of differentiation.
Core rules
- d/dx (xⁿ) = nxⁿ⁻¹; d/dx (eˣ) = eˣ; d/dx (log x) = 1/x; derivative of a constant is zero.
- Product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².
- Marginal cost = dC/dx, marginal revenue = dR/dx; average revenue = R/x.
- Maximum or minimum where dy/dx = 0: d²y/dx² < 0 gives a maximum, > 0 gives a minimum.
- Elasticity of demand = (dx/dp) × (p/x).
- ∫xⁿ dx = xⁿ⁺¹/(n + 1) + c for n ≠ −1.
Worked example
Revenue is R(x) = 100x − x²/2. Differentiate: MR = 100 − x. Setting MR = 0 gives x = 100 units. The second derivative is −1, which is negative, so revenue is maximised there. Maximum revenue = 100(100) − 100²/2 = 10,000 − 5,000 = ₹5,000.
Traps
Setting MR equal to price instead of zero when maximising revenue; forgetting the constant of integration; dropping the chain rule on composite functions; and testing only the first derivative, which locates a stationary point but does not classify it.