differentiation

316 DIFFERENTIAL CALCULUS
[
(a) x = 4(θ − sin θ),
hence dx
(a) − 1 ]
1
cot θ (b) −
= 4 − 4 cos θ = 4(1 − cos θ)
4 16 cosec3 θ
dθ dy
y = 4(1 − cos θ), hence dy
4. Evaluate at θ = π radians for the
dθ = 4 sin θ
dx 6
hyperbola whose parametric equations are
x = 3 sec θ, y = 6 tan θ. [4]
From equation (1),
dy
5. The parametric equations for a rectangular
dy
dx = dθ 4 sin θ
=
dx 4(1 − cos θ) = sin θ
hyperbola are x = 2t, y = 2 dy
. Evaluate
t dx
(1 − cos θ) when t = 0.40
[−6.25]
dθ
The equation of a tangent drawn to a curve at
(b) From equation (2),
point (x
( ) ( )
1 , y 1 ) is given by:
d dy d sin θ
d 2 y
dx 2 = dθ dx dθ 1 − cos θ
y − y 1 = dy 1
(x − x 1 )
=
dx 1
dx 4(1 − cos θ)
Use this in Problems 6 and 7.
dθ
6. Determine the equation of the tangent drawn
(1 − cos θ)(cos θ) − (sin θ)(sin θ)
(1 − cos θ)
=
2
to the ellipse x = 3 cos θ, y = 2 sin θ at θ = π 6 .
[y =−1.155x + 4]
4(1 − cos θ)
= cos θ − cos2 θ − sin 2 θ
7. Determine the equation of the tangent drawn
4(1 − cos θ) 3
to the rectangular hyperbola x = 5t, y = 5
= cos θ − ( cos 2 θ + sin 2 θ )
t at
t = 2. [
4(1 − cos θ) 3
y =− 1 ]
4 x + 5
= cos θ − 1
4(1 − cos θ) 3
−(1 − cos θ)
=
4(1 − cos θ) 3 = −1 29.4 Further worked problems on
4(1 − cos θ) 2
differentiation of parametric
equations
Now try the following exercise.
Problem 5. The equation of the normal drawn
Exercise 128 Further problems on differentiation
of parametric equations
to a curve at point (x 1 , y 1 ) is given by:
1. Given x = 3t − 1 and y = t(t − 1), [ determine ]
y − y 1 =− 1 (x − x
dy
1
dy 1 )
1
dx in terms of t. 3 (2t − 1) dx 1
2. A parabola has parametric equations:
Determine the equation of the normal drawn to
x = t 2 , y = 2t. Evaluate dy
dx when t = 0.5 the astroid x = 2 cos 3 θ, y = 2 sin 3 θ at the point
θ = π
[2] 4
3. The parametric equations for an ellipse
are x = 4 cos θ, y = sin θ. Determine (a) dy x = 2 cos 3 θ, hence dx
dx
dθ =−6 cos2 θ sin θ
(b) d2 y
dx 2 y = 2 sin 3 θ, hence dy
dθ = 6 sin2 θ cos θ