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Calculus I / Applications of Differentiation

Overview

Calibrating implied volatility from a market option price is a root-finding problem; selecting the portfolio weight that maximises risk-adjusted return is an optimisation problem; proving that (1+r/n)n→er(1 + r/n)^n \to e^r as n→∞n \to \infty requires evaluating an indeterminate limit. This topic is where the derivative becomes a decision tool. Having built the full differentiation toolkit in Topic 3, we now deploy it to locate extreme values, test whether they are maxima or minima, read the shape of a function from its first and second derivatives, handle limits that resist direct substitution, solve applied optimisation problems systematically, find roots numerically, and reverse the differentiation process to recover functions from their rates of change.
By the end of this topic you should be able to:
  • state the Extreme Value Theorem, identify critical points, and find absolute extrema on a closed interval via the closed-interval method;
  • apply Rolle's Theorem and the Mean Value Theorem, and use the sign of f′f' to determine where a function is increasing or decreasing;
  • classify critical points using the First and Second Derivative Tests; identify concavity and inflection points from the sign of f′′f'';
  • execute the full curve-sketching algorithm on a polynomial, rational, or exponential function;
  • evaluate indeterminate limits of the form 0/00/0, ∞/∞\infty/\infty, 0⋅∞0 \cdot \infty, ∞−∞\infty - \infty, 000^0, ∞0\infty^0, and 1∞1^\infty using L'Hôpital's Rule;
  • set up and solve applied optimisation problems using the systematic four-step approach;
  • implement the Newton–Raphson iteration and identify when it converges or fails;
  • find the general antiderivative of a function and solve initial-value problems with a given condition.

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