Mathematical Analysis
[O] Olmsted, J. M.H.: The Real Number System , 1962, 216 pages [G2] Giusti, E.: Analisi Matematica 1, Seconda Edizione Bollati Boringhieri, 1991 (edizione fuori commercio; Libreria Efesto ) [G3] Giusti, E.: Analisi Matematica 1, Terza Edizione Bollati Boringhieri, 2002 [R] Rudin, W.: Principi di analisi matematica, Milano 1991 (edizione fuori commercio) [B] Bertsch, Dal Passo, Giacomelli - Analisi Matematica - McGraw-Hill (2011) - piattaforma Connect (esercizi a scelta multipla)
1 . Number Systems
2. Infinite Sequences
- Sequences
- Convegence of Sequences; Definition of limit
- Asymptotic Estimates
- Fundamental theorems on sequences
- Subsequences: The Bolzano–Weierstrass theorem
- Cauchy Sequence
- The Stolz–Cesàro theorem
3. Series
4. Real functions
- Limits and Continuity of Functions
- Asymptotes
- The Extreme Value Theorem (Weirstrass' theorem)
- The Intermediate Value Theorem
- Exponential and logarithmic functions
- Funzioni monòtone
- Trigonometric functions; even, odd, periodic
- Hyperbolic functions
- Inverse functions
- Inverse trigonometric functions
- Remarkable Limits: Trigonometric Limits
- Confronti e stime asintotiche: simbolo ~
5. Derivatives
- Derivative of a function
- Differentiability and Continuity: Non-differentiable maps
- The algebra of derivatives
- Fermat's Theorem (Critical Point Theorem)
- Rolle's theorem
- Lagrange's Mean Value Theorem
- Cauchy’s Form of the Mean Value Theorem
- L’Hopital’s Rule
- Finding the maximum/minimum values of a function
- Second derivative
7. Integrals
- The Riemann integral
- Properties of the integral
- The fundamental theorem of calculus
- Integration by parts
- Integration by substitution