Mathematically speaking, **Calculus** is the study of change. It comes from the Latin word *calculus*, meaning small pebble. It comprises several related subfields of real analysis:

- Limit analysis — foundation of all of calculus. "The study of infinitesimals"
- Differential calculus — typically the main focus of the course called Calculus I when the courses are numbered. "The algebra of change"
- Integral calculus — typically the main focus of Calculus II, but sometimes begun in Calculus I
- Infinite series — typically covered in Calculus II or Calculus III (the "prerequisite" topic of sequences is sometimes first addressed in Calculus I)
- Multivariable calculus, including vector calculus — typically also in Calculus III

Note that the fundamental concepts of functions, graphs, and limits, which are studied at the beginning of courses in differential calculus, are often first introduced in earlier classes (most notably intermediate algebra and precalculus). Sequences are also typically first studied in earlier classes.

## List of topics

- Review material
- Limits
- Limit (definition)
- Properties of limits
- Continuity

- Derivatives and differentiation
- Derivative (definition)
- Derivative formulas
- Properties of differentiation
- Rules of differentiation
- Higher derivatives
- Implicit differentiation

- Applications of differentiation
- Tangent lines
- Newton's method of finding zeros of functions
- Motion
- Finding velocity from displacement and acceleration from velocity

- Related rates
- Extreme values (maximum and minimum values)
- Mean value theorem and Rolle's theorem
- Monotonicity (increasing/decreaing) and concavity (curvature)
- Curve sketching
- Optimization
- L'Hôpital's rule

- Antiderivatives, integrals and integration
- Antiderivative (definition)
- Antiderivative formulas
- Riemann sums
- Upper sum
- Lower sum

- Definite integral (definition)
- Fundamental theorem of calculus
- Properties of integration

- Techniques of integration
*u*-substitution (or simply "substitution method")- Integration by parts
- Reduction formulas
- Cyclic integrals

- Trigonometric integrals
- Trigonometric substitution ("trig substitution")
- Partial fractions
- Using a table of integrals
- Integration by special substitution
- Continuous Solutions of Implied Integrals

- Applications of integration
- Motion revisited
- Finding displacement from velocity and velocity from acceleration

- Areas
- Volumes

- Motion revisited
- Sequences and infinite series
- Sequence (definition)
- Properties of sequences
- Arithmetic sequences
- Geometric sequences
- Convergence of a sequence
- Series
- Summation notation
- Properties of summation
- Arithmetic series
- Geometric series
- Sum of a finite geometric series

- Infinite series
- Convergence of an infinite series
- Partial sums
- Sum of an infinite geometric series
- Telescoping series

- Basic divergence test
- Convergence tests for infinite series
- Integral test
- Comparison tests
- Direct comparison test
- Limit comparison test

- Alternating series test
- Absolute convergence
- Conditional convergence

- Ratio test
- Root test

- Power series
- Radius of convergence
- Interval of convergence
- Transformation of power series
- Sum of a power series

- Taylor polynomials and Taylor series
- Taylor polynomial (definition)
- Taylor polynomial remainder
- Taylor series (definition)
- Taylor series as power series

- Maclaurin series

- Binomial series

- Convergence of an infinite series

- Other coordinate systems and parametric equations
- Polar coordinate systems
- Tangent line in polar coordinates
- Area in polar coordinates
- Arclength in polar coordinates

- Three-dimensional coordinate systems
- Cartesian coordinate system (3-D)
- Cylindrical coordinate system
- Spherical coordinate system

- Parametric equations
- Tangent line to a parametric curve
- Length of a parametric curve

- Polar coordinate systems
- Elementary vector analysis
- Vector (definition)
- Properties of vectors
- Vector algebra

- Lines and planes in three dimensions
- Equations of lines in space
- Vector equation of a line
- Scalar parametric equations of a line
- Symmetric equation of a line

- Equations of planes in space
- Vector equation of a plane
- Scalar equation of a plane

- Equations of lines in space
- Functions of several variables
- Graphing a function of several variables
- Traces and level curves

- Limit of a function of several variables
- Continuity of a function of several variables

- Graphing a function of several variables
- Partial derivatives
- Partial derivative (definition)
- Gradient
- Directional derivative
- ...

- Multiple integrals
- ...

- Differential equations
- Direction fields
- ...

- History of calculus