316 Higher Engineering Mathematics
(a) Ellipse (b) Parabola
(c) Hyperbola (d) Rectangular hyperbola
(e) Cardioid (f) Astroid
(g) Cycloid
Figure 29.1
It may be shown that this can be written as:
dy
dx
=
dy
dθ
dx
dθ
(1)
For the second differential,
d^2 y
dx^2
=
d
dx
(
dy
dx
)
=
d
dθ
(
dy
dx
)
·
dθ
dx
or
d^2 y
dx^2
=
d
dθ
(
dy
dx
)
dx
dθ
(2)
Problem 1. Givenx= 5 θ−1and
y= 2 θ(θ− 1 ), determine
dy
dx
in terms ofθ.
x= 5 θ− 1 ,hence
dy
dθ
= 5
y= 2 θ(θ− 1 )= 2 θ^2 − 2 θ,
hence
dy
dθ
= 4 θ− 2 = 2 ( 2 θ− 1 )
From equation (1),
dy
dx
=
dy
dθ
dx
dθ
=
2 ( 2 θ− 1 )
5
or
2
5
( 2 θ− 1 )
Problem 2. The parametric equations of a
function are given byy=3cos2t,x=2sint.
Determine expressions for (a)
dy
dx
(b)
d^2 y
dx^2
.
(a) y=3cos2t, hence
dy
dt
=−6sin2t
x=2sint, hence
dx
dt
=2cost
From equation (1),
dy
dx
=
dy
dt
dx
dt
=
−6sin2t
2cost
=
− 6 (2sintcost)
2cost
from double angles, Chapter 17
i.e.
dy
dx
=−6sint
(b) From equation (2),
d^2 y
dx^2
=
d
dt
(
dy
dx
)
dx
dt
=
d
dt
(−6sint)
2cost
=
−6cost
2cost
i.e.
d^2 y
dx^2
=− 3
Problem 3. The equation of a tangent drawn to a
curve at point(x 1 ,y 1 )is given by:
y−y 1 =
dy 1
dx 1
(x−x 1 )