Higher Engineering Mathematics, Sixth Edition

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 )