Higher Engineering Mathematics

Differential calculus

29

Differentiation of parametric equations

29.1 Introduction to parametric

equations

Certain mathematical functions can be expressed

more simply by expressing, say,xandyseparately

in terms of a third variable. For example,y=rsinθ,

x=rcosθ. Then, any value given toθwill produce

a pair of values forxandy, which may be plotted to

provide a curve ofy=f(x).

The third variable,θ, is called aparameterand the

two expressions foryandxare calledparametric

equations.

The above example ofy=rsinθandx=rcosθ

are the parametric equations for a circle. The equa-

tion of any point on a circle, centre at the origin and

of radiusris given by:x^2 +y^2 =r^2 , as shown in

Chapter 14.

To show thaty=rsinθandx=rcosθare suitable

parametric equations for such a circle:

Left hand side of equation

=x^2 +y^2

=(rcosθ)^2 +(rsinθ)^2

=r^2 cos^2 θ+r^2 sin^2 θ

=r^2

(

cos^2 θ+sin^2 θ

)

=r^2 =right hand side

(since cos^2 θ+sin^2 θ=1, as shown in

Chapter 16)

29.2 Some common parametric

equations

The following are some of the most common param-

etric equations, and Figure 29.1 shows typical shapes

of these curves.

(a) Ellipse x=acosθ,y=bsinθ

(b) Parabola x=at^2 ,y= 2 at

(c) Hyperbola x=asecθ, y=btanθ

(d) Rectangular x=ct, y=

c

t

hyperbola

(e) Cardioid x=a(2 cosθ−cos 2θ),

y=a(2 sinθ−sin 2θ)

(f) Astroid x=acos^3 θ, y=asin^3 θ

(g) Cycloid x=a(θ−sinθ),y=a( 1 −cosθ)

(a) Ellipse

(c) Hyperbola

(e) Cardioid

(b) Parabola

(d) Rectangular hyperbola

(f) Astroid

(g) Cycloid

Figure 29.1

29.3 Differentiation in parameters

Whenxandyare given in terms of a parameter,

sayθ, then by the function of a function rule of