theorem 7.38 Taylor's theorem with Lagrange remainder

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:733 · p. 215

Rests on

depends_on proposition 7.30 Leibniz rule ¶
- depends_on definition 7.12 Product of functions ¶
  - depends_on definition 7.9 Real function ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- depends_on equation 7.14 eq:ana-derivh ¶
- depends_on proposition 7.27 Differentiable implies continuous ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - 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 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-11 ¶
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-14 ¶
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 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 7.4 Convergence ¶ ↺
    - 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-22 ¶

Supports

depends_on definition 32.57 The Hénon–Heiles Hamiltonian ¶
depends_on definition 32.12 Linearization ¶
- depends_on proposition 32.17 Classification of planar fixed points ¶
  - depends_on example 32.18 A linear centre that is really a stable focus ¶
  - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on proposition 32.26 The Hopf normal form ¶
- depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
  - depends_on definition 32.22 Bifurcation ¶
    - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
    - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
    - depends_on theorem 32.25 Hopf bifurcation, quoted ¶ ↺
  - depends_on example 32.18 A linear centre that is really a stable focus ¶ ↺
- depends_on theorem 32.13 Linear stability ¶
  - depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
  - depends_on proposition 32.23 Saddle-node bifurcation ¶ ↺
  - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶ ↺
  - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
depends_on lemma A.754 Exact depth expansion ¶
- depends_on proposition A.755 The Boussinesq pair ¶
  - depends_on theorem A.756 The Korteweg–de Vries equation ¶
depends_on lemma A.187 Exponentiation ¶
depends_on lemma A.138 The flat exponential ¶
- depends_on definition A.139 The standard radial mollifier ¶
  - depends_on definition A.141 Mollification ¶
    - depends_on proposition A.143 Reproduction identity ¶
  - depends_on lemma A.140 Properties of the mollifier ¶
    - depends_on proposition A.143 Reproduction identity ¶ ↺
    - depends_on proposition A.142 The mollification is smooth ¶
- depends_on lemma A.140 Properties of the mollifier ¶ ↺
depends_on lemma A.452 Young and Hölder ¶
- depends_on lemma A.489 Courant–Lebesgue ¶
  - depends_on theorem A.490 Equicontinuity of the normalised class ¶
- depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
  - depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶
  - depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
    - depends_on definition A.467 Green function of the shifted problem ¶
      - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶ ↺
  - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶ ↺
- depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
  - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
    - depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
depends_on lemma A.212 Second-order flatness ¶
- depends_on lemma A.213 The three minima ¶
  - depends_on theorem A.214 Wilks, $k$ parameters and $r$ constraints ¶
depends_on lemma A.209 Uniform third-order remainder ¶
- depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
  - depends_on lemma A.213 The three minima ¶ ↺
- depends_on lemma A.211 Root-$n$ localisation ¶
  - depends_on lemma A.213 The three minima ¶ ↺
depends_on phenomenon 28.13 Universality of small oscillations ¶
depends_on proposition 16.50 The second variation ¶
depends_on proposition 17.67 Holomorphy ¶
- depends_on theorem 17.78 Causality implies dispersion relations ¶
  - depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
  - depends_on remark 28.26 Causality is visible in the susceptibility ¶
  - depends_on remark 28.62 Neither velocity is a speed limit ¶
- depends_on theorem 17.93 Fundamental strip and holomorphy ¶
  - depends_on theorem 17.94 Mellin inversion ¶
    - depends_on proposition 17.96 Poles give asymptotics ¶
    - depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
      - depends_on corollary 17.101 Coupled channels ¶
depends_on proposition 9.88 The model problem, and the error of the composite ¶
… 2 more

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Leibniz rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:742
`depends_on`	→	Rolle	declared	parts/02-mathematical-methods/05-real-analysis.tex:742
`depends_on`	←	The Hénon–Heiles Hamiltonian	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1505
`depends_on`	←	Linearization	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:283
`depends_on`	←	Exact depth expansion	declared	appendices/A-long-proofs.tex:36726
`depends_on`	←	Exponentiation	declared	appendices/A-long-proofs.tex:9634
`depends_on`	←	The flat exponential	declared	appendices/A-long-proofs.tex:7665
`depends_on`	←	Young and Hölder	declared	appendices/A-long-proofs.tex:22216
`depends_on`	←	Second-order flatness	declared	appendices/A-long-proofs.tex:10753
`depends_on`	←	Uniform third-order remainder	declared	appendices/A-long-proofs.tex:10567
`depends_on`	←	Universality of small oscillations	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:406
`depends_on`	←	The second variation	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1494
`depends_on`	←	Holomorphy	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2445
`depends_on`	←	The model problem, and the error of the composite	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:3371
`depends_on`	←	Taylor's theorem in several variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3206
`depends_on`	←	Overshoot at a jump	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:835
`proves`	←	ch:05-real-analysis@proof-22	declared	parts/02-mathematical-methods/05-real-analysis.tex:745