Antiderivatives and Initial-Value Problems

Derivatives move forward in time: given position, find velocity; given velocity, find acceleration. Many finance problems run in the opposite direction. Given that the marginal cost function $C'(x)$ is known, recover total cost $C(x)$. Given that the continuous dividend rate $q(t)$ is observed, recover cumulative dividends. Given that a stock's instantaneous return $\mu$ and volatility $\sigma$ are specified, the path of the stock price is the solution to a differential equation. All of these are antidifferentiation problems — the machine at the heart of integral calculus.

This note introduces antiderivatives, the table of fundamental antidifferentiation formulas, the notation of the indefinite integral, and the technique of solving first- and second-order initial-value problems (IVPs). The IVP is the bridge between the abstract antiderivative and the specific function that matches real-world conditions (a starting price, a known cost at zero output).

Learning objectives. By the end of this note you should be able to: