Calculus
Calculus studies change, approximation, accumulation, and the behavior of functions.
It provides the mathematical language needed to describe limits, continuity, derivatives, integrals, sequences, series, and differential equations. These ideas form a central bridge between elementary mathematics and the mathematical structures used throughout physics, engineering, and the sciences.
This section organizes the material by mathematical topic, combining theoretical ideas with detailed worked exercises.
Limits
Limits describe the behavior of functions as a variable approaches a particular value or tends toward infinity.
The resources in this section explore fundamental limits, indeterminate forms, L’Hôpital’s rule, and Taylor expansions through theory and worked examples.
Continuity
Continuity formalizes the idea that a function varies without abrupt breaks near a point.
The exercises examine the definition of continuity, removable and jump discontinuities, infinite and oscillatory discontinuities, piecewise functions, and continuity conditions involving parameters.
Differentiability
Differentiability describes the local rate of change of a function and provides the foundation for differential calculus.
The worked problems explore differentiability at critical points, one-sided derivatives, piecewise functions, absolute values, radicals, and conditions for higher-order smoothness.
Recursively Defined Sequences
Recursive sequences are defined by specifying initial values and a rule that determines each new term from previous ones.
The exercises focus on monotonicity, boundedness, invariant intervals, induction arguments, fixed points, convergence, and divergence.
Series
Infinite series extend the idea of finite addition to sequences of partial sums.
This section studies convergence and divergence, fundamental convergence criteria, and the behavior of important classes of numerical series.
Integration by Parts
Integration by parts transforms integrals of products by reversing the product rule for derivatives.
The worked exercises include polynomial-exponential products, trigonometric functions, logarithmic integrals, and repeated applications of the method.
Integration by Substitution
Integration by substitution simplifies integrals through an appropriate change of variable.
The examples include algebraic, logarithmic, trigonometric, and hyperbolic substitutions, with detailed transformations and final results.
Ordinary Differential Equations
Ordinary differential equations describe relationships between an unknown function and its derivatives.
This section introduces fundamental solution methods for first- and second-order differential equations and explores how differential equations model mathematical and physical processes.
Cauchy Problems
Cauchy problems combine differential equations with initial conditions that select a particular solution.
The worked exercises include first- and second-order equations, separable and linear equations, characteristic equations, and the determination of constants from initial data.
Why Calculus Matters
Calculus provides a framework for reasoning about quantities that change continuously.
Limits make it possible to describe approximation precisely. Derivatives capture local change. Integrals describe accumulation. Sequences and series connect finite procedures with infinite processes, while differential equations express laws of evolution.
Together, these ideas form one of the fundamental mathematical languages of modern science.