Studi di funzioni goniometriche

1.2 Esercizio 72

f(x) = cosx + ¹⁄₂cos2x in [0; 2π]

1) Dominio: D=[0; 2π]

2) Int. con gli assi:

x=0 → (0,1) ∈ f(x)

f(x)=1

f(x)=0 → cosx + sinx cosx = 0 → cosx (sinx+1)=0

cosx=0 → x=π⁄2

x=3⁄2π   (π⁄2,0) ∈ ∈ f(x)

sinx=-1 → x=3⁄2π   (3⁄2π,0) ∈ f(x)

Inoltre f(2π)=1 → (2π,1) ∈ f(x)

3) Segno:

f(x)>0   cosx(sinx+1)>0

F₁>0 → cosx>0 → (0;π⁄2)

∪(3⁄2π,2π)

F₂>0 → sinx>-1   ∀x∈ D

       0   π⁄2      3⁄2π    2π

F₁  |++++ −− + 

F₂  |+++++++++ 

f(x) |+ —  +

4) Derivate:

1.2 Esercizio 72

f(x) = cos x + ¹/₂ sen 2x in [0 ; 2π]

1. Dominio: D = [0 ; 2π]

2. Int. con gli assi:

x = 0 → (0, 1) ∈ f(x)

f(x) = 1

f(x) = 0 → cos x + sen x cos x = 0 → cos x (sen x + 1) = 0

cos x = 0 → x = π/2 ∣ x = 3π/2 (π/2, 0) ∈ f(x)

sen x = -1 → x = 3π/2 (3π/2, 0) ∈ f(x)

Inoltre f(2π) = 1 → (2π, 1) ∈ f(x)

3. Segno:

f(x) > 0 → cos x (sen x + 1) > 0

F₁ > 0 → cos x > 0 → (0 ; π/2) ∪ (3π/2, 2π) ∈ D

F₂ > 0 → sen x > -1 ∀x ∈ D

F₁ 0 ----- π/2 ----- 3π/2 ----- 2π

F₂ | + | + | + |

f(x) | + | - | + |

4. Derivate:

f'(x) = -senx + cos2x

f'(x) ≥ 0 → -senx + (1 - 2sen²x) ≥ 0

-senx + 1 - 2sen²x > 0

senx = t

f'(x) ≤ 0 → 2t² + t - 1 ≥ 0

t_1,2 = ^{-1 ± √9}/₄

t ≤ -1 ∨ t ≥ 1/2

senx ≤ -1 ∨ senx ≥ 1/2

x = ³/₂π

π/6 ≤ x ≤ 5/6π

π/6 5/6π

π 2π

f'(x)

f(x)

MAX MIN F

π/2

IN (3/2π, 0) FLESSO A TG ORIZZONTALE

f’’(x) = -cosx - 2sen2x

f’’(x) ≥ 0 → cosx + 4senxcosx ≤ 0 → cosx(4senx+1) ≤ 0

f’’(x) ≥ 0 → cosx ≥ 0 → [0, π/2] ∪ [3/2π,2π]

f’’(x) ≥ 0 → senx ≥ -1/4 → [0, π + arcsen1/4] ∪ [2π - arcsen1/4, 2π]

π/2 π + 3/2 (arcsen1/4)

2π

f_’’(x)

f_’’’(x)

f(x)

FLESSI A TG

OBLIQUA x = π/4

MAX

MIN

O

F₂

F₃

F₁

x

y

6

1.3 Esercizio 73

f(x) = ^{cos x}/_{1 - cos x} in [0 ; 2π]

1. Dominio:
   • 1 - cos x ≠ 0 → cos x ≠ 1
   • x ≠ 0, x ≠ 2π
   • D = (0 ; 2π)
2. Int. con gli assi:
   • x = 0 f(x), f(x) = 0 → cos x = 0
   • x = π/2
   • x = 3π/2
   • (π/2 , 0) ∈ f(x), (3π/2 , 0) ∈ f(x)
3. Segno:
   • f(x) > 0
   • D > 0 → cos x < 1 ∀ x ∈ D
4. Limiti:
   • lim_x→0⁺ f(x) = lim_x→2π⁻ f(x) = +∞
   • x = 0, x = 2π
   • Asintoti verticali
5. Derivate:

f'(x) = -sin x (1 - cos x) - cos x sin x

(1 - cos x)²

= -sin x

(1 - cos x)²

f'(x) > 0 → -sin x > 0

→ sin x ≤ 0 → [π, 2π]

f(x)

MIN

f(π) = 1/2 → (π, -1/2) MIN

f''(x) = -cos x (1 + cos²x - 2 cos x) + sin x (2)(1 - cos x) sin x

(1 - cos x)⁴

O PI MIN 2PI

1.4 Esercizio 74

f(x) = ^{1 + cos x}/_{cos x - sen x} in [0; 2π]

1. Dominio:

cos x - sen x ≠ 0 → 1 - tg x ≠ 0 → tg x ≠ 1

→ x ≠ π/4, x ≠ 5π/4 → D = [0; π/4) ∪ (π/4; 5π/4) ∪ (5π/4; 2π]

2. Int. con gli assi:

x = 0 → (0, 2) ∈ f(x)

y = 2 → (2π, 2) ∈ f(x)

f(x) = 0 → y = 0 → (0π, 0) ∈ f(x)

f(x) cos x = 0 → cos x = 1 x∈ π

3. Segno:

f(x) > 0

N > 0 → cos x > 1 → x ∈ π

D > 0 → cos x > sen x → [0; π/4) ∪ (5π/4; 2π)

f(x)

+ | – | +

π/4 5π/4 2π

4. Limiti:

lim_x→π/4 f(x) = lim_x→π– f(x) = +^√2/_{0⊃+ = ±∞}

x = π/4 , x = 5π/4

Asintoti verticali

5. Derivate:

f'(x) = [-sin x (cos x - sen x) + (1 + cos x) (cos x + sen x)] / (cos x - sen x)²

= -sin x cos x + sen² x + cos x sen x + sen x cos² x / (cos x - sen x)²

= cos x + sen x + 1 / (cos x - sen x)²

f'(x) > 0 → cos x + sen x + 1 > 0 → [0; π] ∪ [3/2 π; 2 π]

^x[0 π | π 3/2 π | 2 π]

^f'(x)[+ | - | +]

^f(x)[↑ MAX | ↓ MIN | ↑]

f(π) = 0 → (π, 0) MAX

f(3/2 π) = 1 → (3/2 π, 1) MIN

x = π/4

x = 5/4 π

10

1.5 Esercizio 75

f(x) = senx-cosx / 3senx-cosx in [0;2π]

1. Dominio: √3 senx - cosx ≠ 0 → x ≠ π/6 x ≠ 7π/6
2. D = [0; π/6) ∪ (π/6; 7π/6) ∪ (7π/6; 2π]

2) Intersezioni con gli assi:

• x = 0 → (0,1) ∈ f(x)
• y = 1 → (2π,1) ∈ f(x)
• f(x) = 0 → x = π/4 ∨ x = 5π/4
• (π/4,0) ∈ f(x)
• (5π/4,0) ∈ f(x)

3) Segno:

f(x) > 0 → N > 0 → √3 senx > cosx → π/4 < x < 5π/4

D > 0 → (3senx-cosx) > 0 → x ∈ (0; π/6)

_________________|_____|π/6___|_____|7π/6__|_____|2π

N | + | - | - | +

D | + | - | - | -

f(x) | + | + | - | +

4) Limiti:

lim f(x) = lim f(x) = ∞

x → π/6⁺ x → 2π/6⁻

5) DERIVATE

f'(x) = [(√3cosx+sinx)(√3sinx−cosx)−(sinx−cosx)(√3sinx+sinx)]

/(√3sinx−cosx)²

f'(x) > 0

−√3−1 > 0 ∀x ∈ D

f'(x) = √3−1 / (√3sinx−cosx)²

f''(x) = −2(√3−1)/ (√3sinx−cosx)³ ⋅ (√3cosx + sinx)

f''(x) ≷ 0

D > 0 → √3cosx+sinx ≷ 0 → π/6 < x < 5π/3

FLESSI IN x = 2π/3 E IN x = 5π/3

y

x = pi/6

x = 7/6 pi

F₁

F₂

O

x

13

1.6 Esercizio 76

f(x) = 3sen²x - 2sen³x

in [0; 2π]

1. Dominio: D = [0; 2π]
2. Intersezione con gli assi:
   • x = 0 x = 2π
   • y = 0 y = 0
   • (0,0), (2π,0) ∈ f(x)
   • f(x) = 0 → sen x (3-2sen x) = 0
   • x = 0 x = 2π
   • sen x = 3/2 → ∅
3. Segno:
   • f(x) > 0 → sen²x (3-2sen x) > 0
   • sen x > 0 ∀ x ∈ (0; 2π)
   • 3-2sen x > 0 → sen x < 3/2 ∀ x ∈ D
   • f(x) > 0 ∀ x ∈ (0; 2π)
4. Derivate:
   • f'(x) = 6sen x cos x - 6sen²x cos x
   • f'(x) > 0 → 6sen x cos x (1-sen x) > 0
   • F₁) sen x > 0 → 0 ≤ x ≤ π
   • F₂) cos x > 0 → 0 ≤ x ≤ π/2 ∨ 3π/2 ≤ x ≤ 2π
   • F₃) 1-sen x > 0 → sen x ≤ 1 ∀ x ∈ D

F₁

F₂

F₃

f(x)

MAX MIN MAX

f(^π⁄₂) = 1 → (^π⁄₂, 1) MAX

f(π) = 0 → (π, 0) MIN

f(^3π⁄₂) = 5 → (^3π⁄₂, 5) MAX

O MIN

15

1.7 Esercizio 77

\(f(x) = \frac{\cos^2 x}{1+2\sin x}\) in \([0;2\pi]\)

1) Dominio: \(1+2\sin x \neq 0 \Rightarrow \sin x \neq -\frac{1}{2}\) → \(x \neq \frac{7\pi}{6}, x \neq \frac{11\pi}{6}\) → \(D = [0;\frac{7\pi}{6}) \cup (\frac{7\pi}{6};\frac{11\pi}{6}) \cup (\frac{11\pi}{6};2\pi]\)

2) Int. con gli assi:

• \(x=0\) | \(x=2\pi\) → \((0,1), (2\pi,1) \in f(x)\)
• \(y=1\) | \(y=1\)
• \(f(x)=0\) → \(x= \frac{\pi}{2}\) → \((\frac{\pi}{2},0), (\frac{3\pi}{2},0) \in f(x)\)
• \(\cos^2 x = 0\) → \(x=\frac{3\pi}{2}\)

3) Segno:

• \(f(x) > 0\)
• \(N > 0 \Rightarrow \cos^2 x > 0 \Rightarrow x \neq \frac{\pi}{2} \land x \neq \frac{3\pi}{2}\)
• \(D > 0 \Rightarrow \sin x > -\frac{1}{2} \Rightarrow [0;\frac{7\pi}{6}) \cup (\frac{11\pi}{6};2\pi]\)

\(\begin{array}{c|c|c|c|c|c|c|c|c}0 & \frac{\pi}{2} & \frac{7\pi}{6} & \pi & \frac{11\pi}{6} & \frac{3\pi}{2} & 2\pi \\\hlineN & + & 0 & + & + & + & 0 & + \\D & + & + & + & - & - & - & - \\f(x) & + & 0 & + & 0 & - & 0 & + \\\end{array}\)

4) Limiti:

\(\lim_{x \to \frac{7\pi}{6}^-} f(x) = \lim_{x \to \frac{7\pi}{6}^+} f(x) = \frac{3}{0^-} = ±\infty\) → \(x=\frac{7\pi}{6}\)

\(x=\frac{11\pi}{6}\) Asintoti Verticali

5) DERIVATE:

f'(x) = -^{2sinx cosx (1+2sinx) - 2cos3x}/_(1+2sinx)²

= -^{2(sinx cosx + 2sin2x cosx + cos3x)}/_(1+2sinx)²

f''(x) > 0 → sinx cosx + 2sin²x cosx + cos³x ≤ 0

→ cosx (sinx + 2sin²x + cos²x) ≤ 0

cosx (sinx + sin²x + sin²x + cos²x) ≤ 0

cosx (sin²x + sinx + 1) ≤ 0

F₁) cosx > 0, x ≤ ^π/₂ x ≤ ^3π/₂

F₂) sin²x + sin x + 1 > 0 Δ < 0  ∀ x ∈ ℝ

f'(x)

• 0
• ^π/₂
• ^3π/₂
• 2π

f(x)

• MIN
• MAX

f(^π/₂) = f(^3π/₂) = 0

(^π/₂ ,0) MIN

(^3π/₂ ,0) MAX