ABSTRACT
Problem 3. The equation of a tangent drawn to a curve at point (x1, y1) is given by:
y − y1 = dy1dx1 (x − x1)
Determine the equation of the tangent drawn to the parabola x = 2t2, y = 4t at the point t.
At point t, x1 = 2t2, hence dx1dt = 4t
and y1 = 4t, hence dy1dt = 4 From equation (1),
dy dx
= dy dt dx dt
= 4 4t
= 1 t
Hence, the equation of the tangent is: y − 4t = 1
t
( x − 2t2)
Problem 4. The parametric equations of a cycloid are x = 4(θ − sin θ), y = 4(1 − cos θ). Determine (a) dy
dx (b) d
(a) x = 4(θ − sin θ), hence
dx dθ
= 4 − 4 cos θ = 4(1 − cos θ)
y = 4(1 − cos θ), hence dy dθ
= 4 sin θ From equation (1),
dy dx
= dy dθ dx dθ
= 4 sin θ 4(1 − cos θ) =
sin θ (1 − cos θ)
(b) From equation (2),
d2y dx2
= d dθ
( dy dx
)
dx dθ
= d dθ
( sin θ
1 − cos θ )
4(1 − cos θ)
= (1 − cos θ)(cos θ) − (sin θ)(sin θ)
(1 − cos θ)2 4(1 − cos θ)
= cos θ − cos 2 θ − sin2 θ
4(1 − cos θ)3
= cos θ − ( cos2 θ + sin2 θ)
4(1 − cos θ)3
= cos θ − 1 4(1 − cos θ)3
= −(1 − cos θ) 4(1 − cos θ)3 =
−1 4(1 − cos θ)2
Now try the following exercise.
