Aplikasi Turunan

1.

f(x) = sin (2x - π/6)

syarat stasioner → f'(x) = 0

f'(0)

2 cos (2x - π/6) = 0

cos (2x - π/6) = 0

•

π/6 = 180°/6 = 30°

2x - 30° = 90° + k.360°

x = (90° + 30° + k.360°)/2

x = 60° + k.180° ... (1)

•

2x - 30° = 270° + k.360°

x = 150° + k.180° ... (2)

Interval 0 < x < π :

stasioner saat x = 60° = π/3 dan x = 150° = 5π/6

f(60°) = sin (120° - 30°) = 1

f(150°) = sin (300° - 30°) = -1

Titik stasioner :

(π/3 , 1) dan (5π/6 , -1)

2.

Kurva y = sin x + cos x

memotong sumbu → y = 0

Interval 0 < x < π

sin x + cos x = 0

sin x = - cos x

sin x = sin (270° - x)

x = 270° - x

2x = 270°

x = 135° = 3π/4 → A(3π/4 , 0)

Gradien m = y' = cos x - sin x

m = cos 135° - sin 135°

m = 2 × (-1/2 √2)

m = - √2

PGS :

m = - √2

A(3π/4 , 0)

y = m(x - x1) + y1

y = - √2 (x - 3π/4) + 0

y = -√2 x + 3π/4 √2

3.

f(x) = (5x² - 2x + 1) /(2x - 1)²

(2x - 1)² ≠ 0

x ≠ 1/2

asimtot tegak fungsi :

x = 1/2