Alphabeta MathDarboux, L'Hôpital, and Taylor's Theorem
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✓ 17 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- Topology of ℝ
2 · Logical flowchart
3 · Items
Definition
Higher derivatives and the classes Cᵏ and C^∞
Theorem
The general Leibniz rule for the n-th derivative of a product
Lemma
Higher-order Rolle theorem
Theorem
Darboux's theorem: every derivative has the intermediate-value property
Corollary
A derivative has neither a removable discontinuity nor a jump discontinuity
Lemma
An injective Darboux function on an interval is strictly monotone
Corollary
An injective or monotone derivative on an interval is continuous
Lemma
Cauchy's mean-value theorem in quotient form when the denominator derivative is nonzero
Theorem
L'Hôpital's rule for the 0/0 form at finite or infinite, one-sided endpoints
Theorem
L'Hôpital's rule for the ∞/∞ form at finite or infinite, one-sided endpoints
Definition
Taylor polynomials and their remainders
Lemma
Taylor polynomials match the prescribed derivatives at the centre
Theorem
Taylor's Schlömilch–Roche remainder formula
Corollary
The Lagrange and Cauchy forms of Taylor's remainder
Theorem
Peano's form: the normalized Taylor remainder tends to zero
Corollary
A uniform derivative bound gives a uniform Taylor remainder bound
Theorem
The second-derivative test for strict local extrema
Theorem
The first nonzero higher derivative classifies a stationary point
Remark
Scope, endpoint, factorial, and deferred-remainder conventions
False statement
If f'(a)=f''(a)=0, then the second derivative test still decides whether a is an extremum
Sources
- J. Lebl, Basic Analysis I, Taylor's theorem and related calculus
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes
- UTSA Mathematics, Differentiation rules
- University of Chicago MATH 395 notes
- University of Florida note, Generalized Rolle's theorem
- J. Lebl, Basic Analysis I, Mean value theorem
- Colgate University MATH 323, Chapter 5 notes
- University of Pennsylvania, derivatives and discontinuities
- Peer-reviewed article on injective Darboux functions (DOI Serbia)
- J. Lebl, Basic Analysis I, Monotone functions
- UC Davis, L'Hopital's rule
- MathWorld, Schlömilch's remainder
- Taylor's theorem (Wikipedia): statement and Peano remainder
- Taylor's theorem (Wikipedia): Lagrange remainder and error estimate
- University of Minnesota MATH 5615, higher derivative test