Prove d'esame Analisi matematica

Grafici esame

1 | -x² + 1 | + 1 - 2 -e^|x| + 1 - 3 | 2^x - 1 | - 4 e^|x| - 1

Integrali propri

1 ∫₁² ln(arctan x + 1) / (x² + 1)(arctan x + 1)² dx - 2 ∫₁² 1/x³ arctan 1/x dx - 3 ∫ 3x + 6 / 2x² - 3x + 3 dx

Serie esame

1 ∑ n! 1/3ⁿ - 2 ∑ (n² + 7) / (2m² + 1)ⁿ - 3 ∑ 4ⁿ / nⁿ - 1 / m

Limiti esame

1 lim_x→∞ (1 + sin 2/x)^x - 2 lim_x→π/2 (tan x)^{cos x} - 3 lim_x→∞ (1 + 2x²)^1/2x2

Integrali impropri

1 ∫₀^∞ e^x / (2x² - 1) dx - 2 ∫₀^∞ e^-x sin(e^-x) dx - 3 ∫₀¹ x / √1 - x² dx

Convessità esame + max & min

1 f(x) = x - √2 - x

2 f(x) = x(x - 1) (max & min)

f_cnl : ln(x² - 25)

3 f(x) = x²e^x

4 f(x) = 1/x + ln x

Grafici esame

1 | -x² + 1 | + 1 - 2 - e^|x| + 1 - 3 | 2^x - 1 | - 4 e^{|x| - 1}

Integrali propri

1 ∫₁³ ln(arctan x + 1) / (x² + 1)(arctan x + 1)² dx - 2 ∫₁² 1 / x³ arctan 1 / x dx - 3 ∫ 3x + 6 / 2x² - 3x + 3 dx

Serie esame

1 ∑ n! 3^√n - 2 ∑ (n² + 7) / (2m² + 1) - 3 ∑ 4ⁿ/n - 1/n

Limite esame

1 lim_x→∞ (1 + sin 2/√x)^x - 2 lim_x→π/2 (tan x)^{cos x} - 3 lim_x→0 (1 + 2x²)^1/x2

Integrali impropri

1 ∫₀^∞ e^x / e^2x - 1 dx - 2 ∫₀^∞ e^-x sin(e^-x) dx - 3 ∫ x / √1 - x² dx

Concavità esame + max & min

1 f(x) = x - ^√2x

2 f(x) = x (x - 1) ^{(max & min)}

f(x) = ln(x² - 25)

3 f(x) = x²e^x

4 f(x) = 1 / x + ln x

Grafici esame

• 1 |–x² + 1| + 1
• 2 –e^|x| + 1
• 3 |2^|x| – 1|
• 4 e^|x| – 1
• 5 log₂(x + 1)
• 6 (x – 2)³ + 1
• 7 –(½)^x + 1
• 8 log₃(x – 2)
• 9 |x + 5| + 2
• 10 |cos x| – 1
• 11 |sin x| + 1
• 12 |ln(x – 1)| + 2

8 log₃(x)log₃(x - 2)(x + 5)(x + 5)|2

10 cos x|cos x||cos x| - 1

11 sen x|sen x||sen x| + 1

12 ln(x + 1)|ln(x + 1)||ln(x + 1)| + 2

Convessità e max e min

• f(x) = x - √(2 - x)
• f(x) = x(x - 1)
• f(x) = x²e^x
• f(x) = 1/x + ln x
• f(x) = [3x² + 1] e^x2
• f(x) = x ln x - x²
• f(x) = ^x√ln x / 2
• f(x) = x ln √(2x - 1)
• f(x) = x ln (1 + x²)
• f(x) = arctan (1 + x²)
• f(x) = e^x - (x² - x)
• f(x) = e^x (x² + x)

1

f(x) = x - √2 - x

f'(x) = 1 + 1 / 2√2 - x

f''(x) = 1 / 4√2 - x √(2 - x)

1 / 4√2 - x √(2 - x) > 0

{ √x > 0 4√2 - x √(2 - x) > 0 x < 2 }

2 min e max

f(x) = x(x - 1)

f''(x) = 2x - 1

f''(x) ≠ 0 2x - 1 > 1/2

f(x) = ln (x² - 25)

f'(x) = 2x / x² - 25

f''(x) = -2x - 50 / (x² - 25)²

-2x² - 50 / (x² - 25)² > 0

2x - 50 / (x² - 25)² ≤ 0

{ 2x² - 50 < 0 (x² - 25)² > 0 }

Δf = x ± 5, x ≠ -5

{ x ≤ 5 x > 0 x > 5, x < 0 → 5 ∩ 5 }

x ∈ ℝ

3

f(x) = x²e^x

f'(x) = 2 xe^x + x²e^x

f'' = 2e^x + 2xe^x + 2xe^x + x²e^x → e^x(x² + 4x + 2)

x² + 4x + 2 ⇒_{-4 ± √18} ------------------ ₂ → -2 + √2 → -2 - √2

(x - (-2 + √2))(x - (-2 - √2)) ≥ 0

{ x + 2 - √2 > 0 x + 2 + √2 ≥ 0

{ x > -2 + √2 x ≥ -2 - √2 → [ -2 + √2, ∞ )

{ x + 2 - √2 ≤ 0 x + 2 + √2 ≤ 0

{ x

4

f(x) = 1x + ln x → f'(x) = -1x2 + 1x → f''(x) = 2x3 - 1x22 - xx3