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Calculus

From limits and continuity through single-variable and multivariable differentiation and integration.

For: Learners at different starting points, including first-time and returning students.

After this page you can: Explain how limits, derivatives and integrals connect, solve introductory problems, and choose a next learning path.

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Part of: Mathematics

About Calculus

Calculus studies how quantities change and accumulate. It centers on limits, derivatives and integrals: limits describe nearby behavior, derivatives capture instantaneous rates of change, and integrals represent accumulation through limits of sums. These ideas connect in the Fundamental Theorem of Calculus.

Learners study calculus by moving between formulas, graphs, geometric pictures and worked problems. A single-variable route develops limits and continuity, differentiation and integration; multivariable calculus extends these ideas to functions of several variables. Use the sections as a map, choose an entry point that matches your goal, and try the practice problem before moving on.

Start here

route

I’m new to calculus

Start with the open textbook’s first volume: it introduces functions and limits before moving into derivatives and integration.

Start with: Calculus Volume 1

route

I’m returning to single-variable calculus

Use MIT’s self-study course to revisit differentiation, applications, integration and the infinite; its sequence includes problem sets and exams.

Start with: Single Variable Calculus

route

I want to study several variables

Begin MIT’s multivariable course after single-variable calculus; it covers partial derivatives, multiple integrals and vector calculus.

Start with: Multivariable Calculus

Learn the foundations

1.Limits and continuityLimits describe function behavior near an input, and continuity at a point requires that the function value, limit and their equality all hold.

Limits describe function behavior near an input, and continuity at a point requires that the function value, limit and their equality all hold.

  • 2.4 Continuity

    This section defines continuity at a point and gives a procedure for checking it.

Limit

The value a function approaches as its input approaches a specified point, when that value exists.

Try it

Evaluate lim(x→2) x.

The limit is 2.

2.Derivatives and rates of changeA derivative is defined through a limit of difference quotients and represents an instantaneous rate of change.

A derivative is defined through a limit of difference quotients and represents an instantaneous rate of change.

Difference quotient

A ratio of a change in function values to a change in input values, used to define the derivative by taking a limit.

Try it

For f(x)=x², use the derivative definition to find f′(3).

f′(3)=6.

3.Integrals and accumulationA definite integral can be defined as a limit of sums, and the Fundamental Theorem connects integration with differentiation.

A definite integral can be defined as a limit of sums, and the Fundamental Theorem connects integration with differentiation.

Definite integral

A number defined by the limit of Riemann sums over an interval, when that limit exists.

Fundamental Theorem of Calculus

A theorem relating derivatives and integrals, including a method for evaluating definite integrals using antiderivatives.

Try it

Evaluate ∫₀² 2x dx using an antiderivative.

An antiderivative is x², so the value is 2²−0²=4.

4.Calculus with several variablesMultivariable calculus extends differentiation and integration to functions of two or more independent variables, including partial derivatives and multiple in…

Multivariable calculus extends differentiation and integration to functions of two or more independent variables, including partial derivatives and multiple integrals.

  • Calculus Volume 3

    The volume includes functions of several variables, multiple integration and vector calculus.

Partial derivative

A derivative of a function of several variables that measures change with respect to one variable.

Multiple integral

An integral used for a function of several variables, including double and triple integrals.

Try it

For f(x,y)=x²+y, find the partial derivative with respect to x.

Holding y constant, the partial derivative is 2x.

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Books

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Every free Calculus course and book on RedLectern

Free Calculus online courses

  1. Advanced Calculus for EngineersMIT 18.075 · Browse materials
  2. Calculus of Several VariablesMIT 18.022 · Browse materials
  3. Calculus Revisited: Complex Variables, Differential Equations, and Linear AlgebraMIT RES.18-008 · Browse materials · Video
  4. Calculus Revisited: Multivariable CalculusMIT RES.18-007 · Browse materials · Video
  5. Calculus Revisited: Single Variable CalculusMIT RES.18-006 · Browse materials · Video
  6. Calculus with TheoryMIT 18.014 · Browse materials
  7. Highlights of CalculusMIT RES.18-005 · Browse materials · Video
  8. Matrix Calculus for Machine Learning and BeyondMIT 18.S096 · Browse materials · Video
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  1. Measure and IntegrationMIT 18.125 · Browse materials
  2. Multivariable CalculusMIT 18.02 · Browse materials · Video
  3. Multivariable CalculusMIT 18.02 · Browse materials
  4. Multivariable CalculusMIT 18.02SC · Browse materials · Video
  5. Real AnalysisMIT 18.100A · Browse materials · Video
  6. Real AnalysisMIT 18.100B · Browse materials · Video
  7. Real AnalysisMIT 18.100C · Browse materials
  8. Single Variable CalculusMIT 18.01 · Browse materials · Video
  9. Single Variable CalculusMIT 18.01SC · Browse materials · Video
  10. Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of VariationsMIT 2.035 · Browse materials
  11. System Functions and the Laplace TransformMIT 18.031 · Browse materials · Video
  12. Calculus for Beginners and ArtistsMIT RES.18-003 · Open on OCW
  13. Calculus I: Single Variable CalculusMIT 18.01 · Open on OCW · Video
  14. Calculus Open TextbookMIT RES.18-001 · Open on OCW
  15. Calculus with ApplicationsMIT 18.013A · Open on OCW
  16. Multivariable Calculus Recitation NotesMIT RES.18-016 · Open on OCW
  17. Multivariable Calculus with TheoryMIT 18.024 · Open on OCW
  18. Single Variable CalculusMIT 18.01 · Open on OCW
How these courses are chosen

Listed as Calculus courses:

  • those whose title contains “Calculus”, or
  • those tagged “Calculus” as one of at most 2 topic tags, or as the first of 3.

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