theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:705 · p. 215

Rests on

depends_on definition 7.16 Limit ¶
- depends_on definition 7.2 Absolute value ¶
- depends_on definition 7.9 Real function ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
depends_on theorem 7.34 Rolle ¶
- depends_on lemma 7.33 Fermat: interior extremum ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
    - proves proof ch:05-real-analysis@proof-6 ¶
  - proves proof ch:05-real-analysis@proof-17 ¶
- depends_on theorem 7.24 Extreme value theorem ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on proposition 7.22 Sequential characterization ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
      - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
    - proves proof ch:05-real-analysis@proof-7 ¶
  - depends_on theorem 7.7 Bolzano–Weierstrass ¶
    - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on definition A.36 Cut ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-32 ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-33 ¶
    - proves proof ch:05-real-analysis@proof-4 ¶
  - proves proof ch:05-real-analysis@proof-9 ¶
- proves proof ch:05-real-analysis@proof-18 ¶
proves proof ch:05-real-analysis@proof-21 ¶

Supports

depends_on lemma A.306 The standard smooth bump ¶
- depends_on lemma A.307 Partition of unity on a compact manifold ¶
  - depends_on definition A.308 Integral over the manifold ¶
    - depends_on theorem A.311 General Stokes theorem ¶
      - depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
      - depends_on lemma A.555 The actions are well defined ¶
      - depends_on theorem 24.23 Poincaré–Cartan integral invariant ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Limit	declared	parts/02-mathematical-methods/05-real-analysis.tex:715
`depends_on`	→	Rolle	declared	parts/02-mathematical-methods/05-real-analysis.tex:715
`depends_on`	←	The standard smooth bump	declared	appendices/A-long-proofs.tex:15103
`proves`	←	ch:05-real-analysis@proof-21	declared	parts/02-mathematical-methods/05-real-analysis.tex:718