Math 2.0

statement theorem

Fundamental theorem of calculus

Statement

Let a<ba < b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be continuous. Then the function F(x)=∫axf(t) dtF(x) = \int_a^x f(t) \, dt is differentiable on (a,b)(a, b), with F′(x)=f(x)F'(x) = f(x). If GG is continuous on [a,b][a, b] and G′=fG' = f on (a,b)(a, b), then ∫abf(x) dx=G(b)−G(a).\int_a^b f(x) \, dx = G(b) - G(a).

Remarks

The first part is the first fundamental theorem. The second part is the second fundamental theorem. Together they say that differentiation and integration are inverse operations. The theorem reduces the computation of integrals to the search for antiderivatives.

History

James Gregory published the first statement and proof of a geometric form of the theorem. Isaac Barrow proved a more general version. Isaac Newton and Gottfried Leibniz then built the calculus around it.

Explanations

L1 human

The fundamental theorem connects the two main ideas of calculus. The derivative of an area function is the height of the graph, and an integral is a change in an antiderivative.

What this node needs for W2
  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 36 nodes in this order. The foundations come first, and the list ends with this node.

Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionAntiderivative, definition, no evidence levelAntiderivativedefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementRiemann integral, definition, no evidence levelRiemann integraldefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementIntermediate value theorem, statement, no evidence levelIntermediate value th…statementDefinite integral, definition, no evidence levelDefinite integraldefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementAdditivity of the integral over intervals, statement, no evidence levelAdditivity of the int…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementMean value theorem for integrals, statement, no evidence levelMean value theorem fo…statementRolle's theorem, statement, no evidence levelRolle's theoremstatementFirst fundamental theorem of calculus, statement, no evidence levelFirst fundamental the…statementMean value theorem, statement, no evidence levelMean value theoremstatementSecond fundamental theorem of calculus, statement, no evidence levelSecond fundamental th…statementFundamental theorem of calculus, statement, no evidence levelFundamental theorem o…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Limit point of a set definition
  2. Local maximum definition
  3. Partition of an interval definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Mesh of a partition definition
  8. Riemann sum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Riemann integrable function definition
  14. Bounded sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Continuous function definition
  18. Antiderivative definition
  19. Fermat's theorem on stationary points statement
  20. Riemann integral definition
  21. Convergent sequence definition
  22. Monotone convergence theorem for sequences statement
  23. Every sequence has a monotone subsequence statement
  24. Intermediate value theorem statement
  25. Definite integral definition
  26. Bolzano–Weierstrass theorem statement
  27. Additivity of the integral over intervals statement
  28. Boundedness theorem statement
  29. Monotonicity of the integral statement
  30. Extreme value theorem statement
  31. Mean value theorem for integrals statement
  32. Rolle's theorem statement
  33. First fundamental theorem of calculus statement
  34. Mean value theorem statement
  35. Second fundamental theorem of calculus statement
  36. Fundamental theorem of calculus statement

What depends on this

2 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus, Formal statements

What links here

Used by