Calculus

About these notes

Calculus studies change and accumulation. These notes begin with limits, use them to define derivatives, and connect derivatives to integrals through the Fundamental Theorem of Calculus. Read with pencil and paper: checking a hypothesis is as important as manipulating a formula.

Prerequisites

You should be comfortable with algebraic fractions, inequalities, functions and their domains, trigonometry, exponentials and logarithms. All variables are real unless stated otherwise. An interval such as \((a,b)\) excludes its endpoints; \([a,b]\) includes them.

Written chapters

  1. Limits and continuity: precise definitions, limit laws and the Intermediate Value Theorem.
  2. Derivatives and differentiation: local linear approximation, differentiation rules and the Mean Value Theorem.
  3. Integration and the Fundamental Theorem: Riemann sums, signed area, accumulation and substitution.

Each chapter has worked examples and exercises with answers. These are introductory notes, not a complete calculus course.

Planned chapters

The following material is not yet written:

  • Integration techniques: integration by parts, partial fractions and trigonometric substitution; improper integrals and convergence.
  • Sequences and infinite series: convergence tests, power series, Taylor polynomials and remainder bounds.
  • Multivariable differentiation: partial derivatives, differentiability, the chain rule and gradients for optimization.
  • Multiple integrals, Jacobians and change of variables; constrained and unconstrained optimization.

Line and surface integrals, divergence, curl and the major vector calculus theorems belong to Fourier Analysis & PDE (planned). For the algebra of vectors and linear maps, see Linear Algebra.

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