ABSTRACT
Problem 4. The parametric equations of a cycloid are x=4(θ− sinθ), y=4(1− cosθ). Determine (a) dy
dx (b) d
(a) x=4(θ− sinθ),
hence dx dθ =4−4cosθ=4(1− cosθ)
y=4(1− cosθ), hence dy dθ =4sin θ
From equation (1),
dy dx
= dy dθ dx dθ
= 4sin θ 4(1− cosθ) =
sinθ (1− cosθ)
(b) From equation (2),
d2y dx2
= d
dθ
( dy dx
)
dx dθ
= d
dθ
( sinθ
1− cosθ )
4(1− cosθ)
= (1− cosθ)(cosθ)− (sinθ)(sinθ)
(1− cosθ)2 4(1− cosθ)
= cosθ − cos 2 θ − sin2 θ
4(1− cosθ)3
= cosθ − ( cos2 θ + sin2 θ)
4(1− cosθ)3
= cosθ−1 4(1− cosθ)3
= −(1− cosθ) 4(1− cosθ)3 =
−1 4(1− cosθ)2
Now try the following Practice Exercise
Practice Exercise 140 Differentiation of parametric equations (Answers on page 851)
1. Given x=3t−1 and y= t (t−1), determine dy dx
in terms of t
2. A parabola has parametric equations: x= t2, y=2t . Evaluate dy
dx when t=0.5
3. The parametric equations for an ellipse are x=4cosθ , y= sinθ . Determine (a) dy
dx (b) d
4. Evaluate dy dx
at θ= π 6
radians for the hyperbola whose parametric equations are x=3secθ , y=6tanθ
5. The parametric equations for a rectangular hyperbola are x=2t , y= 2
t . Evaluate
dy dx
when t=0.40
The equation of a tangent drawn to a curve at point (x1, y1) is given by:
y− y1= dy1dx1 (x− x1)
Use this in Problems 6 and 7.
