Formulario Analisi matematica 1

Tavole e Formulari Formule notevoli

2

2 ∀x ∈

cos x + sin x = 1, R π

∀k ∈ + kπ

sin x = 0 se x = kπ , , cos x = 0 se x =

Z 2

π

sin x = 1 se x = + 2kπ , cos x = 1 se x = 2kπ

2 π

−1 − −1

sin x = se x = + 2kπ , cos x = se x = π + 2kπ

2

± ±

sin(α β) = sin α cos β cos α sin β

± ∓

cos(α β) = cos α cos β sin α sin β 2 −

sin 2x = 2 sin x cos x , cos 2x = 2 cos x 1

−

x y x + y

−

sin x sin y = 2 sin cos

2 2

− x + y

x y

− −2

cos x cos y = sin sin

2 2

− −

sin(x + π) = sin x , cos(x + π) = cos x

π π −

sin(x + ) = sin x

) = cos x , cos(x + 2

2 x

a y

x+y x y x−y x xy

a = a a , a = , (a ) = a

y

a

∀x,

log (xy) = log x + log y , y > 0

a a a

x − ∀x,

= log x log y, y > 0

log a a

a y y ∀x ∀y ∈

log (x ) = y log x, > 0 , R

a a

498 Tavole e formulari Limiti notevoli

α α

lim x = +∞ , lim x = 0 , α > 0

x→+∞ +

x→0

α α

lim x = 0 , lim x = +∞ , α < 0

x→+∞ +

x→0

n

a x + . . . + a x + a a

n 1 0 n n−m

lim = lim x

m

b x + . . . + b x + b b

x→±∞ x→±∞

m 1 0 m

x x

lim a = +∞ , lim a = 0 , a> 1

x→+∞ x→−∞

x x

lim a = 0 , lim a = +∞ , a< 1

x→+∞ x→−∞ −∞

lim log x = +∞ , lim log x = , a> 1

a a

x→+∞ +

x→0

−∞

lim log x = , lim log x = +∞ , a< 1

a a

x→+∞ +

x→0

lim sin x , lim cos x , lim tan x non esistono

x→±∞ x→±∞ x→±∞ π

∓∞ ∀k ∈ ±

lim tan x = , , lim arctan x =

Z 2

x→±∞

±

( )

π

x→ +kπ

2 π

±

lim arcsin x = = arcsin(±1)

2

x→±1

lim arccos x = 0 = arccos 1 , lim arccos x = π = arccos(−1)

x→+1 x→−1 − 1

1 cos x

sin x = 1 , lim =

lim 2

x x 2

x→0

x→0 "

! x

a 1

a ∈

= e , a , lim (1 + x)

lim 1 + = e

R x

x

x→±∞ x→0

log (1 + x) 1 log(1 + x)

a

lim = , a > 0; in particolare, lim =1

x log a x

x→0 x→0 x

x −

− e 1

a 1 = log a , a > 0; in particolare, lim =1

lim x x

x→0

x→0 α −

(1 + x) 1 ∈

lim = α , α R

x

x→0 Tavole e formulari 499

Tavola delle derivate di funzioni elementari

$

f (x) f (x)

α α−1 ∀α ∈

x αx , R

sin x cos x

−

cos x sin x 1

2

tan x 1 + tan x = 2

cos x

1

√

arcsin x 2

−

1 x

1

√

−

arccos x 2

−

1 x

1

arctan x 2

1 + x x

x (log a) a

a 1

|x|

log a (log a) x

sinh x cosh x

cosh x sinh x

Regole di derivazione

! " $ $ $

αf (x) + βg(x) = αf (x) + βg (x)

! " $ $ $

f (x)g(x) = f (x)g(x) + f (x)g (x)

$

# $ $ $

−

f (x)g(x) f (x)g (x)

f (x) = % &

2

g(x) g(x)

! " $ $ $

g(f (x)) = g (f (x))f (x)