Question

Diketahui garis singgung kurva f(x) = sin x / 2+ cos x , dititik dengan absis π/2 memotong sumbu y di titik (0,k) nilai k adalah?
a. 1/2
b. π/8
c. π/8
d. 1/2 - π/8
e. 1/2 - 3π/8
Mohon jawab dengan caranya :)

Answer 1

Y= sin x / ( 2+ cos x)
x = π/2 --> y = sin (π/2) / ( 2 + cos (π/2))
y = 1/(2+0) = 1/2
titik singgung (x1,y1) = ( π/2 , 1/2)

m = y' = (u'v -uv')/(v²)
m = { cos x (2+cos x) - sin x(-sinx)} / {2+cos x)²
m = { 2 cosx + cos² x + sin² x) / (2+cos x)²
m = ( 2 cos x + 1) / (2 + cos x)²
x = π/2 --> m = (0+1)/(2 +0)² = 1/4

pers. garis singgung y = m(x-x1) + y1
y = 1/4 (x - π/2) + 1/2
memotong sumbu y --> x = 0
y = 1/4(0 -π/2) + 1/2
y = - π/8 + 1/2
atau
k = 1/2 - π/8