Theorems · Theorem · global analysis
taylor_mean_remainder_lagrange_iteratedDeriv
∀ {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ},
x₀ ≠ x →
ContDiffOn ℝ (↑n + 1) f (Set.uIcc x₀ x) →
∃ x' ∈ Set.uIoo x₀ x,
f x - taylorWithinEval f n (Set.uIcc x₀ x) x₀ x =
iteratedDeriv (n + 1) f x' * (x - x₀) ^ (n + 1) / ↑(n + 1).factorialA corollary of Taylor's theorem with the Lagrange form of the remainder.
- Defined in
- Mathlib.Analysis.Calculus.Taylor
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- ENatstatement · cited by 4,985
- WithTopstatement · cited by 3,754
- Nat.cast_oneproof · cited by 2,501
- le_of_ltproof · cited by 1,175
- ContinuousMultilinearMapproof · cited by 1,016
- Nat.factorialstatement and proof · cited by 616
- DifferentiableOnproof · cited by 419
- Set.uIccstatement and proof · cited by 393
- ContDiffOnstatement and proof · cited by 294
Cited by1
Results whose statement or proof uses this declaration.
- ProbabilityTheory.exists_cgf_eq_iteratedDeriv_two_cgf_mulproof · cited by 1