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
- Applications of differentiation
- Tangent lines
- Newton's method of finding zeros of functions
- Motion
- 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
- 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
- Sequences and infinite series
- Sequence (definition)
- Properties of sequences
- Arithmetic sequences
- Geometric sequences
- Convergence of a sequence
- Series
- Infinite series
- Other coordinate systems and parametric equations
- Elementary vector analysis
- Vector (definition)
- Properties of vectors
- Vector algebra
- Lines and planes in three dimensions
- Functions of several variables
- Partial derivatives
- Partial derivative (definition)
- Gradient
- Directional derivative
- ...
- Multiple integrals
- ...
- Differential equations
- Direction fields
- ...
- History of calculus