Introduction to Calculus

Calculus is the mathematical study of continuous change. It is divided into two major branches: differential calculus and integral calculus, which are related by the Fundamental Theorem of Calculus.

Core Concepts

1. Limits: The foundation of calculus. A limit tells us the value that a function approaches as the input approaches some value.
2. Derivatives: The rate of change of a quantity. Geometrically, the derivative is the slope of the tangent line to the graph of the function at a point. Used in finance for sensitivities (the "Greeks" in options pricing) and optimization.
3. Integrals: The accumulation of quantities. The definite integral of a function can be interpreted as the signed area of the region bounded by its graph.
4. Taylor Series: A representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. Vital for approximating complex functions in risk management.
5. Multivariable Calculus: Extending calculus to functions of multiple variables, using partial derivatives and multiple integrals. Essential for portfolios with many assets.