Formulario Analisi Matematica 1

Formulario Analisi Matematica 1

Proprietà delle potenze

• x⁰ = 1        ∀ x ∈ ℝ - {0}
• 1^d = 1        ∀ d ∈ ℝ
• x^d · x^p = x^d+p        ∀ d,p ∈ ℝ
• x^d : x^p = x^d-p        ∀ d,p ∈ ℝ, x ≠ 0
• (x^d) ^p = x^d·p        ∀ d,p ∈ ℝ
• x^-d = 1/x^d        ∀ d ∈ ℝ
• (x^d) ^β = x^d·β        ∀ d, β ∈ ℝ, x > 0
• x^d/n = ⁿ√x^d        ∀ n ∈ ℕ
• ^m√(ⁿ√x) = ^m·n√x        ∀ m,n ∈ ℕ
• x^d > 0        ∀ d ∈ ℝ

x < y, α > 0 => x^α < y^α

x < y, α < 0 => x^α > y^α

x < 1, d > p > 0 => x^d < x^p

x > 1, d > p > 0 => x^d > x^p

• monotonia
• confronto

Proprietà dei logaritmi

• log_a1 = 0
• a^log_ab = b
• log_a(b · c) = log_ab + log_ac
• log_a(b^d) = d · log_ab
• log_a(b/c) = log_ab - log_ac
• log_ab = log_cb / log_ca

• se 0 < x < y allora log_ax < log_ay se a ∈ (0; 1)
• log_ax > log_ay se a > 1 monotonia
• se 0 < a < b e a,b ≠ 1 allora log_ax > log_by per x > 1
• log_ax < log_by per 0 < x < 1 confronto

Segno delle funzioni elementari

• log(x) { log(x) > 0 x ≥ 1
• { log(x) < 0 0 < x < 0
• e^x { e^x > 0 ∀x ∈ ℝ
• { D: ℝ
• arctg(x) { arctg(x) > 0 x > 0
• { D: ℝ
• arcsen(x) { arcsen(x) > 0 x > 0
• { D: -1 < x < 1
• arccos(x) { arccos(x) > 0 ∀x
• { D: -1 ≤ x ≤ 1
• √x { √x > 0 ∀x
• { D: x > 0

Formulario Analisi Matematica 1

proprietà delle potenze

• x⁰ = 1   ∀ x ∈ ℝ - {0}
• 1^x = 1   ∀ x ∈ ℝ
• x^−α = 1/x^α   ∀α, β ∈ ℝ   ∀ x > 0
• x^α (y)^α = (xy)^α   ∀ x, y ∈ ℝ
• x^α/x^α = x^α−β   ∀α, β ∈ ℝ
• (x^α)^β = x^αβ
• xⁿ = √ⁿx   ∀ n ∈ ℕ
• x^m/n = √ⁿx^m   ∀ m, n ∈ ℕ
• x⁰ > 0   ∀ x ∈ ℝ

• If x < y, α > 0 then x^α < y^α
• If x < y, α < 0 then x^α > y^α
• If x < 1, α < β then x^α < x^β
• If x > 1, α < β then x^α < x^β

proprietà dei logaritmi

• log_a1 = 0
• a^log_ab = b
• log_a(bc) = log_ab + log_ac
• log_a(b/c) = log_ab − log_ac
• log_ab = log_cb/log_ca

• If 0 < x < y then log_ax < log_ay if a ∈ (0, 1)
• log_ax > log_ay if a > 1

segno delle funzioni elementari

• log(x) {^{log(x) > 0  x > 1}_{log(x) < 0  0 < x < 0}}
• e^x {^{ex > 0  ∀ x ∈ ℝ}_{D: ℝ}}
• arctan(x) {^{arctan(x) > 0  x > 0}_{D: ℝ}}
• arcsen(x) {^{arcsen(x) > 0  x > 0}_{D: −1 < x < 1}}
• arccos(x) {^{arccos(x) > 0  ∀ x}_{D: −1 < x < 1}}
• √x {^{√x > 0  ∀ x}_{D: x > 0}}

- Proprietà del modulo

f(x) = |x| = \[ \begin{cases} x & \text{se } x > 0 \\ -x & \text{se } x < 0 \end{cases} \]

• |x| ≥ 0 ∀ x ∈ ℝ
• |x| = 0 ⟺ x = 0
• -|x| ≤ x ≤ |x| ∀ x ∈ ℝ
• |x| = √x² ∀ x ∈ ℝ
• |x| ≥ x ∀ x ∈ ℝ
• |xy| = |x||y| ∀ x,y ∈ ℝ
• |x/y| = |x|/|y| ∀ x,y ∈ ℝ, y ≠ 0
• |x+y| ≤ |x| +