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differenziale,gradiente,matrice Jacobiana
titolo
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x
y
xo
f(xo)
y = f(x) non lineare
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h h
k
f( Xo)(h)
L(h)
)(f)(f)()(f ooo XXX hh
funzione differenza di f in Xo
df( Xo)
differenziale di f in Xo
Xo Xo+h
f(Xo+h)
f(Xo)
k = L(h)
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L : Rn Rm
)(L)(L))((L XXX hh
lineare
)(L)(L)(L XX h)(L h
pj : Rn R
jn1j x:)x,...,x(p
jjj h)(p))((p hhXjx
jj h)(dx h
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nn11 ha...ha))((df hX
jj h)(dx h
f : Rn R
)(dxa...)(dxa))((df nn11 hhh X
))(dxa...dxa( nn11 h
nn11 dxa...dxa)(df X dxa)x(df oo
f : R R ( n = 1 )
dx)x('f o
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f : R R dx)x('f)x(df oo
dx
)x(df)x('f o
o
nn11o dxa...dxa)(df Xf : Rn R
j
oj x
)(fa
X derivata parziale
rispetto ad xj
notazione di Leibnitz
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nn11o dxa...dxa)(df Xf : Rn R
j
oj x
)x(fa
derivata parziale
rispetto ad xj
n
o
1
oo x
)(f,...,
x
)(f:)(f
XXX
n1
o1
1
oo dx
x
)(f...dx
x
)(f)(df
XXX
GRADIENTE di f in Xo
XXX d)(f)(df oo
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df(Xo)
f(Xo)
f(X ) := potenziale elettrico in X
Xo
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AB a1
a2
f(a1 , a2)
P
Q
RR22 RRff campo scalarecampo scalare
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00
punto di minimo
punto di massimo
00
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PUNTO DI SELLAPUNTO DI SELLA
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x
y
OO
0
)(f oX
punto stazionario
sella
minimo
massimo
di puntoX o
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RRn :f
RRn :)(df oX
)(df XM
trasformazione lineare
campo scalare
n
o
2
o
1
o
x
)(f
x
)(f
x
)(f XXX
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RRnn RRmmff )f,...,f,f( m21f
Rn Rmdf(Xo)
M( df(Xo) ) =
campo vettoriale
f
f
f
o
o
m o
1
2
( )
( )
. . . . . . . . .
( )
X
X
X
trasformazione lineare
Jf(Xo) =
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
f
x
o o o
n
o o o
n
m o m o m o
n
1
1
1
2
1
2
1
2
2
2
1 2
( ) ( ). . . . .
( )
( ) ( ). . . . .
( )
. . . . . . . . . . . . . . . . . . . .( ) ( )
. . . . .( )
X X X
X X X
X X X
matrice Jacobiana
di f in Xo
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RR RRmmff )f,...,f,f( m21f
R Rmdf(to)
M( df(to) ) =
campo vettoriale
trasformazione lineare
n = 1
)t('f
)t('f
)t('f
om
o2
o1
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3RR :f
f(R)
f (to)
to
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3RR :f
f (to+ h)
to
f(R)
to+ h
)h(f
f (to)
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3RR :f
f (to+ h)
to
f(R)
to+ h
)h(f
f (to)
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f (to+ h) )h(f
f(R)
f (to)
3RR :f to+ h
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f (to+ h) )h(f
f(R)
f (to)
3RR :f to+ h
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f (to+ h)h
)h(f
f(R)
f (to)
3RR :f to+ h
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f (to+ h)h
)h(f
f(R)
f (to)
3RR :f to+ h
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f (to+ h)
h
)h(f
f(R)
f (to)
3RR :f to+ h
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f(R)
h
)h(flim
0h
f (to)
3RR :f to+ h
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f(R)
f (to)
3RR :f to+ h
)t(z,)t(y,)t(x)t(f
)t('z,)t('y,)t('x)t('f
)('f otvelocità
istantanea
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XX)(f X
)t('
Rn RR f
)t(')(f)t)(f(D X
)t(
dt)t(d
x)(f
dt)t(d
x)(f
dt)t)(f(d n
n
1
1
XX
regola della catenaregola della catena
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integrali edequazioni differenziali
titolo
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x
y
x1xo x2
x3
f(xo)
)x(df)x(f)(f oo X )x(df 1 )x(df 2 )x(df 3
n
0iio )x(df)x(f)(f X )x(df
oxX dx)x('f
integrale definito
di tra xo ed X
)x('f
Integrale definito
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dx)x('f
)x(f)(fdx)x('f oxo
XX
c
n
0iio )x(df)x(f)(f X )x(df
oxX dx)x('f
insieme delle primitive di 'f
integrale indefinito diintegrale indefinito di 'f
Rc
Integrale indefinito
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Tabella degli integrali
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Un’applicazione:
velocità media tra gli istanti to e to+h :
spazio percorso dopo un tempo t :
oggetto in moto rettilineo
h
)t(s)ht(s oo
velocità istantanea nell’istante to :
h
)t(s)ht(slim)t(v oo
0ho
)t(s
)t('s o
Moto rettilineo
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Un’applicazione:
accelerazione di gravità costante: gvelocità raggiunta dopo un tempo t : v(t) = g tspazio percorso dopo un tempospazio percorso dopo un tempo t : s(t)s(t)
xg)x('s
t
0
2
2
xg
t
0dx)x('s)t(s
t
0dxxg
t
0dxxg
2tg2
1)t(s 2tg
2
1)t(s
oggetto in caduta libera con velocità iniziale nulla
??
2
tg
2
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Esercizio
Calcolare la derivata della funzione:
|x|log:)x(f
|x|D|x|
1)x('f
Per la regola della catena :
x
|x|
|x|
1
x
1
x
1|x|logD }0{x R
Esercizio
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dxx
1
x
1|x|logD }0{x R
c|x|log
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CRESCITA CRESCITA DI UNA POPOLAZIONE ISOLATA DI UNA POPOLAZIONE ISOLATA
IN UN AMBIENTE CON RISORSE ILLIMITATEIN UN AMBIENTE CON RISORSE ILLIMITATE( ad esempio: batteri in coltura ) ( ad esempio: batteri in coltura )
x(t) = numero di batteri vivi nell’istante t
variazionevariazione x x nell’intervallonell’intervallo t t ::
x r x t
tasso di crescita
tasso di natalità tasso di mortalità
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0tlim
xr
x(t)
variazionevariazione x x nell’intervallonell’intervallo t t ::
t
CRESCITA CRESCITA DI UNA POPOLAZIONE ISOLATA DI UNA POPOLAZIONE ISOLATA
IN UN AMBIENTE CON RISORSE ILLIMITATEIN UN AMBIENTE CON RISORSE ILLIMITATE( ad esempio: batteri in coltura ) ( ad esempio: batteri in coltura )
x
= numero di batteri vivi nell’istante t
'xEQUAZIONE DIFFERENZIALE
Equazione differenziale
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dtrx
dx
log| |x r t c | |x e r t c x e ec r t x c e r t 1
xrxdt
dx
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x c e r t 1tr
1ec)t(G
1
0r
1 cec)0(F F(t F( e r t) ) 0
condizione iniziale :
INTEGRALE PARTICOLAREINTEGRALE PARTICOLARE
INTEGRALE GENERALEINTEGRALE GENERALE
Condizione iniziale
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NkNt
DECADIMENTO RADIOATTIVODECADIMENTO RADIOATTIVO
N(t)= nuclei radioattivi nell’istante t
variazionevariazione N N nell’intervallonell’intervallo t t ::
0tlim
N Nk t ( k > 0 )
'Ntke)0(N)t(N
Decadimento radioattivo
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dx)x(gdx)x(fdx)x(g)x(f
dx)x(fcdx)x(fc
dfgdgf)gf(d
dfgdgf)gf(d
dfggfdgfintegrazione per parti
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Risolvere gli esercizi da pagina 387 a pagina 390 sul testo consigliato
(le pagine non possono essere presentate sul web, perché appartengono all’Editore)
Esercizi sugli integrali indefiniti
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a b
y = f(x)
1n
0ii )t(Fd)a(F)b(F
rettangoloide
di f su [a, b]
x
y
Area di un rettangoloide
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y = f(x)
xa b
y
1n
0ii )t(Fd)a(F)b(F
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y = f(x)
xa b
y
1n
0ii )t(Fd)a(F)b(F dt)t(f
b
a
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xa b
y
c
b
c
c
a
b
adt)t(fdt)t(fdt)t(f
additività
a
b
b
adt)t(fdt)t(f
0dt)t(fa
a
additività
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Teorema della media
Teorema della media
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Soluzioni degli esercizi proposti a pagina 404
Esercizi sugli integrali definiti
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Soluzioni degli esercizi proposti a pagina 472
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Polinomi di Taylor
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no )xx(
infinitesimo di ordine n per x che tende ad xo
)xx)(x('f)x(f)x(f ooo
)xx)(x('f)x(f)x(f ooo
Se f è una funzione differenziabile in xo , allora :
2o2 )xx(a n
on )xx(a
infinitesimo di ordine 2
infinitesimo di ordine 1
infinitesimo
di ordine n
!k
)x(fa o
)k(
k derivata di ordine k
no
o)n(
2o
oooo )xx(
!n
)x(f)xx(
2
)x("f)xx)(x('f)x(f
polinomio di Taylor di f di ordine n con punto iniziale xo
)x(fTn
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Esempio
f(x) = sin x
xo = 0
xcos)x('f
xsin)x("f
xcos)x('''f
xsin)x(f IV
0
1
0
1
0
0
x
0
!3
x3
0
xcos)x(f V 1!5
x5
ko
o)k(
)xx(!k
)x(f
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Esempio
f(x) = sin x
xo = 0
0
Polinomi di Taylor
x)x(fT1 x)x(fT2
6
xx)x(fT
3
3 6
xx)x(fT
3
4
120
x
6
xx)x(fT
53
5
ko
o)k(
)xx(!k
)x(f
!3
x3
0
x
0
!5
x5
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