\contentsline {section}{\numberline {1}Operazioni}{1}{section.1}
\contentsline {subsection}{\numberline {1.1}Valore assoluto}{1}{subsection.1.1}
\contentsline {subsubsection}{\numberline {1.1.1}Proprieta'}{1}{subsubsection.1.1.1}
\contentsline {subsection}{\numberline {1.2}Logaritmi}{2}{subsection.1.2}
\contentsline {subsubsection}{\numberline {1.2.1}Proprieta'}{2}{subsubsection.1.2.1}
\contentsline {section}{\numberline {2}Limiti}{3}{section.2}
\contentsline {subsection}{\numberline {2.1}Proprieta' d'Archimede}{3}{subsection.2.1}
\contentsline {subsection}{\numberline {2.2}Densita'}{3}{subsection.2.2}
\contentsline {subsubsection}{\numberline {2.2.1}Proprieta'}{4}{subsubsection.2.2.1}
\contentsline {subsection}{\numberline {2.3}Partizione di $\mathbb {R}$}{5}{subsection.2.3}
\contentsline {subsubsection}{\numberline {2.3.1}Proprieta'}{5}{subsubsection.2.3.1}
\contentsline {subsection}{\numberline {2.4}X limitato superiormente (maggiorante)}{5}{subsection.2.4}
\contentsline {subsection}{\numberline {2.5}X limitato inferiormente (minorante)}{5}{subsection.2.5}
\contentsline {subsection}{\numberline {2.6}X limitato}{5}{subsection.2.6}
\contentsline {subsection}{\numberline {2.7}Massimo di X}{6}{subsection.2.7}
\contentsline {subsubsection}{\numberline {2.7.1}Proprieta'}{6}{subsubsection.2.7.1}
\contentsline {subsection}{\numberline {2.8}Minimo di X}{6}{subsection.2.8}
\contentsline {subsubsection}{\numberline {2.8.1}Proprieta'}{6}{subsubsection.2.8.1}
\contentsline {subsection}{\numberline {2.9}Estremo superiore}{6}{subsection.2.9}
\contentsline {subsubsection}{\numberline {2.9.1}Proprieta'}{6}{subsubsection.2.9.1}
\contentsline {subsection}{\numberline {2.10}Estremo inferiore}{8}{subsection.2.10}
\contentsline {subsubsection}{\numberline {2.10.1}Proprieta'}{8}{subsubsection.2.10.1}
\contentsline {subsection}{\numberline {2.11}X illimitato superiormente}{10}{subsection.2.11}
\contentsline {subsection}{\numberline {2.12}X illimitato inferiormente}{10}{subsection.2.12}
\contentsline {subsection}{\numberline {2.13}Insiemi numerici separati}{10}{subsection.2.13}
\contentsline {subsubsection}{\numberline {2.13.1}Proprieta'}{10}{subsubsection.2.13.1}
\contentsline {subsection}{\numberline {2.14}Insiemi contigui}{10}{subsection.2.14}
\contentsline {subsubsection}{\numberline {2.14.1}Proprieta'}{11}{subsubsection.2.14.1}
\contentsline {section}{\numberline {3}I numeri Complessi}{11}{section.3}
\contentsline {subsection}{\numberline {3.1}$\mathbb {C}$}{11}{subsection.3.1}
\contentsline {subsubsection}{\numberline {3.1.1}Proprieta'}{12}{subsubsection.3.1.1}
\contentsline {subsection}{\numberline {3.2}Addizione}{12}{subsection.3.2}
\contentsline {subsubsection}{\numberline {3.2.1}Proprieta'}{12}{subsubsection.3.2.1}
\contentsline {subsection}{\numberline {3.3}Moltiplicazione}{12}{subsection.3.3}
\contentsline {subsubsection}{\numberline {3.3.1}Proprieta'}{12}{subsubsection.3.3.1}
\contentsline {subsection}{\numberline {3.4}Forma algebrica}{13}{subsection.3.4}
\contentsline {subsection}{\numberline {3.5}Coniugato di $z$}{13}{subsection.3.5}
\contentsline {subsubsection}{\numberline {3.5.1}Proprieta'}{13}{subsubsection.3.5.1}
\contentsline {subsection}{\numberline {3.6}Divisione}{13}{subsection.3.6}
\contentsline {subsection}{\numberline {3.7}Rappresentazione cartesiana}{13}{subsection.3.7}
\contentsline {subsection}{\numberline {3.8}Modulo}{13}{subsection.3.8}
\contentsline {subsubsection}{\numberline {3.8.1}Proprieta'}{13}{subsubsection.3.8.1}
\contentsline {subsection}{\numberline {3.9}Argomento}{14}{subsection.3.9}
\contentsline {subsection}{\numberline {3.10}Forma trigonometrica}{14}{subsection.3.10}
\contentsline {subsubsection}{\numberline {3.10.1}Proprieta}{14}{subsubsection.3.10.1}
\contentsline {subsubsection}{\numberline {3.10.2}Moltiplicazione}{14}{subsubsection.3.10.2}
\contentsline {subsubsection}{\numberline {3.10.3}Divisione}{14}{subsubsection.3.10.3}
\contentsline {subsubsection}{\numberline {3.10.4}Potenza}{14}{subsubsection.3.10.4}
\contentsline {subsubsection}{\numberline {3.10.5}Radice n-esima}{14}{subsubsection.3.10.5}
\contentsline {section}{\numberline {4}Funzioni}{15}{section.4}
\contentsline {subsection}{\numberline {4.1}Funzione limitata}{15}{subsection.4.1}
\contentsline {subsection}{\numberline {4.2}Massimo e punto di massimo}{15}{subsection.4.2}
\contentsline {subsection}{\numberline {4.3}Minimo e punto di minimo}{15}{subsection.4.3}
\contentsline {section}{\numberline {5}Successioni}{15}{section.5}
\contentsline {subsection}{\numberline {5.1}Limite di successione}{16}{subsection.5.1}
\contentsline {subsection}{\numberline {5.2}Unicita' del limite}{16}{subsection.5.2}
\contentsline {subsection}{\numberline {5.3}Permanenza del segno}{16}{subsection.5.3}
\contentsline {subsection}{\numberline {5.4}Successione convergente e limitata}{17}{subsection.5.4}
\contentsline {subsection}{\numberline {5.5}Limite di successioni prodotto}{17}{subsection.5.5}
\contentsline {subsubsection}{\numberline {5.5.1}Successione infinitesima per una successione limitata}{17}{subsubsection.5.5.1}
\contentsline {subsection}{\numberline {5.6}Teorema del limite assoluto}{18}{subsection.5.6}
\contentsline {subsection}{\numberline {5.7}Successione infinitamente grande}{18}{subsection.5.7}
\contentsline {subsection}{\numberline {5.8}Operazioni con i limiti}{18}{subsection.5.8}
\contentsline {subsection}{\numberline {5.9}Successioni divergenti}{19}{subsection.5.9}
\contentsline {subsubsection}{\numberline {5.9.1}Successione somma}{19}{subsubsection.5.9.1}
\contentsline {subsubsection}{\numberline {5.9.2}Successione prodotto}{19}{subsubsection.5.9.2}
\contentsline {subsubsection}{\numberline {5.9.3}Successione reciproca}{20}{subsubsection.5.9.3}
\contentsline {subsubsection}{\numberline {5.9.4}Successione quoziente}{20}{subsubsection.5.9.4}
\contentsline {subsubsection}{\numberline {5.9.5}Successione esponenziale}{20}{subsubsection.5.9.5}
\contentsline {subsubsection}{\numberline {5.9.6}Forme indeterminate}{21}{subsubsection.5.9.6}
\contentsline {subsection}{\numberline {5.10}Successioni monotone}{21}{subsection.5.10}
\contentsline {subsubsection}{\numberline {5.10.1}Proprieta'}{21}{subsubsection.5.10.1}
\contentsline {subsection}{\numberline {5.11}Teoremi di confronto}{21}{subsection.5.11}
\contentsline {subsubsection}{\numberline {5.11.1}Confronto di successioni convergenti}{21}{subsubsection.5.11.1}
\contentsline {subsubsection}{\numberline {5.11.2}Confronto di successioni divergenti}{21}{subsubsection.5.11.2}
\contentsline {subsection}{\numberline {5.12}Successione estratta}{22}{subsection.5.12}
\contentsline {subsubsection}{\numberline {5.12.1}Esempio}{22}{subsubsection.5.12.1}
\contentsline {subsubsection}{\numberline {5.12.2}Teorema sulle succ. estratte}{22}{subsubsection.5.12.2}
\contentsline {subsection}{\numberline {5.13}Teorema di Bolzano-Weierstrass}{22}{subsection.5.13}
\contentsline {subsection}{\numberline {5.14}Criterio di convergenza di Cauchy}{22}{subsection.5.14}
\contentsline {section}{\numberline {6}Limiti di funzioni}{24}{section.6}
\contentsline {subsection}{\numberline {6.1}Intorno}{24}{subsection.6.1}
\contentsline {subsection}{\numberline {6.2}Punto di accumulazione}{24}{subsection.6.2}
\contentsline {subsubsection}{\numberline {6.2.1}Esempi}{24}{subsubsection.6.2.1}
\contentsline {subsection}{\numberline {6.3}Limiti convergenti}{24}{subsection.6.3}
\contentsline {subsection}{\numberline {6.4}Limiti divergenti positivamente}{24}{subsection.6.4}
\contentsline {subsection}{\numberline {6.5}Limiti divergenti negativamente}{25}{subsection.6.5}
\contentsline {subsection}{\numberline {6.6}Confronto di limiti}{25}{subsection.6.6}
\contentsline {subsection}{\numberline {6.7}Intorno destro (sinistro)}{25}{subsection.6.7}
\contentsline {subsection}{\numberline {6.8}Teorema dell'intorno destro e sinistro}{25}{subsection.6.8}
\contentsline {subsection}{\numberline {6.9}Limiti di funzioni composte}{26}{subsection.6.9}
\contentsline {subsection}{\numberline {6.10}Limiti di funzioni e successioni}{26}{subsection.6.10}
\contentsline {subsection}{\numberline {6.11}Limiti notevoli}{28}{subsection.6.11}
\contentsline {section}{\numberline {7}Funzioni continue}{29}{section.7}
\contentsline {subsubsection}{\numberline {7.0.1}Esempio}{29}{subsubsection.7.0.1}
\contentsline {subsection}{\numberline {7.1}Teorema dell'esistenza degli zeri}{29}{subsection.7.1}
\contentsline {subsubsection}{\numberline {7.1.1}Corollario}{30}{subsubsection.7.1.1}
\contentsline {subsection}{\numberline {7.2}Teorema dei valori intermedi}{31}{subsection.7.2}
\contentsline {subsection}{\numberline {7.3}Limiti di funzioni continue e successioni}{31}{subsection.7.3}
\contentsline {subsection}{\numberline {7.4}Teorema di continuita' delle funzioni composte}{31}{subsection.7.4}
\contentsline {subsection}{\numberline {7.5}Teorema di Weierstrass}{31}{subsection.7.5}
\contentsline {subsection}{\numberline {7.6}Punti di discontinuita'}{32}{subsection.7.6}
\contentsline {subsubsection}{\numberline {7.6.1}Discontinuita' di prima specie}{33}{subsubsection.7.6.1}
\contentsline {subsubsection}{\numberline {7.6.2}Discontinuita' di seconda specie}{33}{subsubsection.7.6.2}
\contentsline {subsubsection}{\numberline {7.6.3}Discontinuita' di terza specie}{33}{subsubsection.7.6.3}
\contentsline {section}{\numberline {8}Funzioni monotone}{33}{section.8}
\contentsline {section}{\numberline {9}Funzioni uniformemente continue}{34}{section.9}
\contentsline {section}{\numberline {10}Derivata}{35}{section.10}
\contentsline {subsection}{\numberline {10.1}Derivate notevoli (regole di derivazione)}{36}{subsection.10.1}
\contentsline {subsubsection}{\numberline {10.1.1}Alcune dimostrazioni}{37}{subsubsection.10.1.1}
\contentsline {subsection}{\numberline {10.2}Derivate di funzioni composte}{39}{subsection.10.2}
\contentsline {section}{\numberline {11}Funzioni invertibili}{40}{section.11}
\contentsline {subsubsection}{\numberline {11.0.1}Proprieta'}{40}{subsubsection.11.0.1}
\contentsline {subsection}{\numberline {11.1}Monotonia e invertibilita'}{40}{subsection.11.1}
\contentsline {subsection}{\numberline {11.2}Continuita' e invertibilita'}{41}{subsection.11.2}
\contentsline {subsection}{\numberline {11.3}Teorema di derivazione delle funzioni inverse}{41}{subsection.11.3}
\contentsline {subsection}{\numberline {11.4}Teorema di Rolle}{42}{subsection.11.4}
\contentsline {subsection}{\numberline {11.5}Teorema di Cauchy}{42}{subsection.11.5}
\contentsline {subsection}{\numberline {11.6}Teorema di Lagrange}{43}{subsection.11.6}
\contentsline {subsubsection}{\numberline {11.6.1}Corollario I}{43}{subsubsection.11.6.1}
\contentsline {subsubsection}{\numberline {11.6.2}Corollario II}{43}{subsubsection.11.6.2}
\contentsline {subsubsection}{\numberline {11.6.3}Corollario III}{44}{subsubsection.11.6.3}
\contentsline {subsection}{\numberline {11.7}Funzioni crescenti e derivate}{44}{subsection.11.7}
\contentsline {subsection}{\numberline {11.8}Teorema di De l'Hopital}{45}{subsection.11.8}
\contentsline {section}{\numberline {12}Massimi e minimi relativi}{45}{section.12}
\contentsline {subsection}{\numberline {12.1}Teorema di Fermat}{46}{subsection.12.1}
\contentsline {section}{\numberline {13}Concavita', convessita' e flessi}{47}{section.13}
\contentsline {section}{\numberline {14}Asintoti}{48}{section.14}
\contentsline {section}{\numberline {15}Studio del grafico di una funzione}{49}{section.15}