Formulario Analisi 1

LIMITI NOTEVOLI

^sin(x)/_x → 1 se x → 0ln(1 + x)/x → 1 se x → 0loga(1 + x)/x → 1/sin(a) se x → 0(e^x - 1)/x → 1 se x → 0(a^x - 1)/x → ln(a) se x → 0(1 + (a/x))^x → e x → ±∞(1 + x/c)^c - 1/x → c se x → ±∞(1 - cos x)/x² → 1/2 se x → 0tan(x)/x → 1 se x → 0arcsin x/x → 1 se x → 0arctg(x)/x → 1 se x → 0sinh x/x → 1 se x → 0cosh x - 1/x² → 1/2 se x → 0tanh(x)/x → 1 se x → 0

ASINTOTICI e "O PICCOLI"

^{sin x} ~ x1 - cos x ~ 1/2 x²arctg x ~ xarcsin x ~ xe^x - 1 ~ xa^x - 1 ~ log(a) xlog(1 + x) ~ xloga(1 + x) ~ 1/log a x(1 + x)^c - 1 ~ cxcosh x - 1 ~ 1/2 x²sinh x ~ xsin x ~ x + o(x)cos x ~1 - 1/2 x² + o(x)arctg x ~ x + o(x)arcsin x ~ x + o(x)e^x = 1 + x + o(x)a^x = 1 + (log a)x + o(x)log(1 + x) = x + o(x)loga(1 + x) = 1/log a x + o(x)(1 + x)^a = 1 + ax + o(x)cosh x = 1 + 1/2 x² + o(x²)sinh x = x + o(x)

DOMINIO FUNZIONI

① tg x : ]-^π /₂, ^π /₂[ → ℝ② arctg y : ℝ → [-^π /₂, ^π /₂]③ arcsin x : [-1, 1] → [-^π /₂, ^π /₂]④ arccos x : [-1, 1] → [0, π]

NOTAlim_x→0 (a)^b = lim_x→0 e^b·loga

LIMITI NOTEVOLI

^sin(x)⁄_x → 1 se x → 0

^ln(1+x)⁄_x → 1 se x → 0

^log_a(1+x)⁄_x → ¹⁄_ln(a) se x → 0

^ex-1⁄_x → 1 se x → 0

^ax-1⁄_x → ln(a) se x → 0

^(1+a⁄_x)x → e se x → ∞

^(1+x)c-1⁄_x → c se x → 0

^1-cos(x)⁄_x² → ¹⁄₂ se x → 0

^tan(x)⁄_x → 1 se x → 0

^arcsin(x)⁄_x → 1 se x → 0

arctg(x) → 0 se x → 0

sinh_x⁄_x → 1 se x → 0

^{cosh x - 1}⁄_x² → ¹⁄₂ se x → 0

^{tanh (x)}⁄_x → 1 se x → 0

ASINTOTICI e "O PICCOLI"

• sin x ≈ x
• 1 - cos x ≈ ¹⁄₂ x²
• arctg x ≈ x
• arcsin x ≈ x
• e^x - 1 ≈ x
• a^x - 1, ≈ (log a) x
• log(1+x) ≈ x
• ^log_a(1+x)⁄_{ln a} ≈ x
• (1+x)ⁿ - 1 ≈ nx
• cosh x - 1 ≈ ¹⁄₂ x²
• sinh x ≈ x

DOMINIO FUNZIONI

• ^{tg x} : ↠ &R
• arctg y: ↠ [-^π⁄₂, ^π⁄₂]
• arcsin x: [-1, 1] ↠ [-^π⁄₂, ^π⁄₂]
• arccos x: [-1, 1] ↠ [0, π]

NOTA

lim_x→0 (a)^b = lim_x→0 e^b⋅loga

FORMULARIO

SVILUPPI IN SERIE di TAYLOR-McLAURIN

e^x = 1 + x + ^x2/_2! + ^x3/_3! + ... + ^xn/_n! + o(xⁿ)

sin x = x - ^x3/_3! + ^x5/_5! - ^x7/_7! + ... + (-1)^{n x2n+1}/_(2n+1)! + o(x²ⁿ⁺¹)

cos x = 1 - ¹/_2! x² + ^x4/_4! - ^x6/_6! + ... + (-1)^{n x2n}/_(2n)! + o(x²ⁿ)

(1+x)^α = 1 + αx + ^{α (α-1)}/_2! x² + ^{α (α-1) (α-2)}/_3! x³ + ... + ^{α (α-1) ... (α-n+1)}/_n! xⁿ + o(xⁿ)

log (1+x) = x - ^x2/₂ + ^x3/₃ + ... + (-1)^{n+1 xn}/_n + o(xⁿ