estratto da matematica verde 4a - mcurie.edu.it
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ESTRATTO DA MATEMATICA VERDE 4A
CAPITOLO 16 - Funzioni e loro proprietà
1 Funzioni reali di variabile reale.........................................................................................776
1.1 Definizione di funzione.....................................................................................776
1.2 Classificazione delle funzioni...........................................................................777
1.3 Dominio, zeri e studio del segno di una funzione............................................777
1.4 Grafici delle funzioni e trasformazioni geometriche.........................................779
2 Proprietà delle funzioni.....................................................................................................780
2.1 Funzioni crescenti e decrescenti......................................................................781
2.2 Funzioni periodiche..........................................................................................782
2.3 Funzioni pari e funzioni dispari........................................................................782
2.4 Proprietà delle principali funzioni trascendenti.................................................783
CAPITOLO 17 - Limiti di funzioni
1 Punti di accumulazione....................................................................................................821
2 Limite finito per x che tende a un numero finito .......................................822
2.1 Definizione e significato..........................................................................................822
2.2 Funzioni continue...................................................................................................825
2.2.1 Funzione costante.....................................................................................................826
2.2.2 Funzione polinomiale................................................................................................826
2.2.3 Funzione radice quadrata.........................................................................................826
2.2.4 Funzioni goniometriche.............................................................................................826
2.2.5 Funzione esponenziale.............................................................................................826
2.2.6 Funzione logaritmica.................................................................................................826
2.3 Limite per eccesso e limite per difetto....................................................................827
2.4 Limite destro e Limite sinistro.................................................................................828
3 Limite infinito per x che tende a un numero finito .................................829
3.1 Definizione e significato..........................................................................................829
3.1.1 Limiti destro e sinistro infiniti.....................................................................................832
3.2 Asintoti verticali......................................................................................................833
4 Limite finito per x che tende all'infinito ......................................................834
4.1 Definizione e significato..........................................................................................834
4.2 Limite per eccesso e limite per difetto....................................................................836
4.3 Asintoti orizzontali..................................................................................................836
5 Limite infinito per x che tende all'infinito ...............................................837
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CAPITOLO 18 - Calcolo dei limiti e continuità delle funzioni
1 Operazioni sui limiti..........................................................................................................868
1.1 Limiti di funzioni elementari....................................................................................868
1.2 Limite della somma................................................................................................869
1.3 Limite del prodotto..................................................................................................870
1.4 Limite del quoziente...............................................................................................871
1.5 Limite della potenza del tipo .....................................................................872
1.6 Limite delle funzioni composte...............................................................................873
2 Forme indeterminate........................................................................................................873
2.1 Forma indeterminata + - .................................................................................874
2.1.1 Limite di una funzione polinomiale............................................................................874
2.1.2 Limite di una funzione irrazionale..............................................................................874
2.2 Forma indeterminata 0 .....................................................................................875
2.3 Forma indeterminata / ....................................................................................875
2.3.1 Limite di una funzione razionale fratta per x .....................................................875
2.4 Forma indeterminata 0 / 0.....................................................................................877
2.5 Forme indeterminate 00, 0, 1..............................................................................878
3 Infinitesimi, infiniti e loro confronto...................................................................................881
3.1 Infinitesimi..............................................................................................................881
3.1.1 Confronto tra infinitesimi...........................................................................................882
3.2 Infiniti......................................................................................................................882
3.2.1 Confronto tra infiniti...................................................................................................883
3.2.2 Gerarchia degli infiniti ...............................................................................................884
4 Funzioni continue.............................................................................................................884
5 Punti di discontinuità di una funzione...............................................................................887
5.1 Punti di discontinuità di prima specie.....................................................................888
5.2 Punti di discontinuità di seconda specie.................................................................888
5.3 Punti di discontinuità di terza specie (o eliminabile)...............................................889
6 Asintoti..............................................................................................................................890
6.1 Asintoti verticali e orizzontali..................................................................................890
6.1.1 Ricerca degli asintoti orizzontali e verticali................................................................890
6.2 Asintoti obliqui........................................................................................................892
6.2.1 Ricerca degli asintoti obliqui......................................................................................892
7 Grafico probabile di una funzione.....................................................................................894
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Matematica www.mimmocorrado.it 1
Grafici delle funzioni elementari La retta
qx
my
x
my
qy
La funzione di proporzionalità inversa xky
0k
0k
q
tgm
xxy 2 2 3 2xy
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Matematica www.mimmocorrado.it 2
La funzione esponenziale xay
1a 1a0
La funzione logaritmica xlogy a
1a 1a0
La funzione irrazionale n xy
2 xy 3 xy
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Matematica www.mimmocorrado.it 4
La funzione potenza nxy
2xy
3xy
4xy
5xy
pari nxy paridis nxy
Al crescere dell’esponente la curva si schiaccia maggiormente nell’origine.
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Semicirconferenza
22 xay raggio = a
Semiellisse
22
22 x
abby
semiasse orizzontale = a semiasse verticale = b
La parabola (asse parallelo all’asse y)
a4Δ;
a2bV
a4Δ1;
a2bF
a41y
direttrice
a2bx
asse
·
d
x·
cbxaxy 2 y a < 0
O
asse
V
F
·
·c·
d
·
cbxaxy 2 y a > 0
O
asse
V
F
·
c
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in generale:
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RIEPILOGO TIPOLOGIE DI LIMITE DI FUNZIONE
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