Math 2.0

statement theorem

Cauchy mean value theorem

Statement

Let a<ba < b and let f,g ⁣:[a,b]→Rf, g \colon [a, b] \to \mathbb{R} be continuous on [a,b][a, b] and differentiable on (a,b)(a, b). Then there is c∈(a,b)c \in (a, b) with (f(b)−f(a)) g′(c)=(g(b)−g(a)) f′(c).(f(b) - f(a)) \, g'(c) = (g(b) - g(a)) \, f'(c).

Remarks

Apply Rolle's theorem to h(x)=(f(b)−f(a))g(x)−(g(b)−g(a))f(x)h(x) = (f(b) - f(a)) g(x) - (g(b) - g(a)) f(x), which has h(a)=h(b)h(a) = h(b). With g(x)=xg(x) = x it is the mean value theorem. It is the main step in the proof of L'Hôpital's rule. Mathlib states it as exists_ratio_deriv_eq_ratio_slope.

Explanations

L1 human

Think of (g(t),f(t))(g(t), f(t)) as a point moving along a curve in the plane. At some time, the direction of motion is parallel to the chord joining the start and the end.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded 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 functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent 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 …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementCauchy mean value theorem, statement, no evidence levelCauchy mean value the…statement
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  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Bounded sequence definition
  11. Limit of a sequence definition
  12. Monotone sequence definition
  13. Continuous function definition
  14. Fermat's theorem on stationary points statement
  15. Convergent sequence definition
  16. Monotone convergence theorem for sequences statement
  17. Every sequence has a monotone subsequence statement
  18. Bolzano–Weierstrass theorem statement
  19. Boundedness theorem statement
  20. Extreme value theorem statement
  21. Rolle's theorem statement
  22. Mean value theorem statement
  23. Cauchy mean value theorem statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Mean_value_theorem, Cauchy's mean value theorem

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