Formulario completo analisi 1

SVILUPPI ASINTOTICI x→0

e^x - 1 x + x²/2

(1+ x)^α 1 + α x + o(x)

sen x x - x³/6 + o(x⁵)

tanh x x - x³/3 + o(x⁵)

cos x 1 - x²/2 + o(x⁴)

tan x x + x³/3 + o(x⁵)

ex ⁻¹ x - x²/2 + o(x³)

(tgx) 1

cosh x 1 + x²/2 + o(x⁴)

log(1+x) x - x²/2 + o(x³)

arcsen x x + x³/6 + o(x⁵)

arccos x π/2 - x - x³/6 + o(x⁵)

xsen x x²/2

cos2x – 1 4x² + o(x⁴)

sh x x + x³/6 + o(x⁵)

cox x 1 - x²/2 + o(x⁴)

a^x - 1 x log a + o(x)

a^x - 1 A log a

tan⁻¹ x 1 - x²/3 + o(x⁴)

log(1 + x) 1/x

sen(ax) a x + a³x³/3 + o(x⁵)

cos (ax) 1 - a²x²/2 + o(x⁴)

(1+x)^t 1 + tx + t(t-1)x²/2! + o(x³)

INTEGRALE IMPROPRIO O GENERALIZZATO

dom illim. ∫_a^∞ f(x) dx

∞ ∫_a^bf(x) dx diverg.

∫ f(x) dx ∈ (a,b) non limit.

e ∈ E (0,∞)

Se ∫ f(x) dx div.>f(x)g(x) diverge

<e ∈ E (0,∞) ∫ f(x)dx>f(x)g(x) converge

f(x) ∈ (a,b)

t ∈ (0,1) |∫ f(x)dx| <+∞ >∫f(x) dx diverge

>T ∈ (a,b)∫ g(x) dx> ∫ f(x) dx diverge

LIMITI NOTEVOLI

lim_x→0 sin x / x = 1

lim_x→0 tan x / x = 1

lim_x→0 ln(1 + x) / x = 1

lim_x→0 (1 + x)^1/x = e

lim_x→0 (sin x) / x = 1

lim_x→0 (cos x) - 1 / x² = -1/2

lim_x→∞ (1 + 1/x)^x = e

lim_x→a (1 + x/x)^x = 1

a^1/x α∀x ∈<0,+∞>

lim_x→0 tg(x) / x = 1

lim_x→0 1-cos x / x² = 1/2

lim_x→∞ (1 + x)^1/x = e^α

lim_x→∞ x / log(a^x) = 0

DERIVATE ELEMENTARI

D(x^α) = αx^α-1

D(e^x) = e^x

D(log_ax) = 1 / (x log a), x≠0

D(sin x) = cos x

D(cos x) = -sin x

D(tan x) = sec²x

D(cot x) = -csc²x

D(sec x) = sec x tan x

D(csc x) = -csc x cot x