chapter  29
Differentiation of parametric equations
Pages 5

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.