Understanding Engineering Mathematics

10.2.2 Tangent and normal to a curve

➤

291 310➤

The derivative at a point (a,b)onacurvey=f(x)will give us the slope,m,ofthe
tangentto the curve at that point. This tangent is a straight line with gradientmpassing
through the point (a,b) and therefore has an equation (212

➤

)

y−b=m(x−a)

Thenormalto the curvey=f(x)at the point (a,b) is the line through (a,b) perpendicular
to the tangent – see Figure 10.1.

x

y

y = f(x)

0

(a,b)

Normal

Tangent

Figure 10.1Tangent and normal to a curve.

In mechanics, the normal is important because it defines the direction of the reaction to
a force applied to a smooth surface represented by the curve. If the gradient ofy=f(x)

at (a,b)ismthen the gradient of the normal through (a,b) will be−

1

m

(214

➤

). So the

equation of the normal is

y−b=−

1

m

(x−a)

Solution to review question 10.1.2

For the curvey=x^2 −x−1wehave

dy

dx

= 2 x−1.

So at (2, 1), the gradient ism= 2 × 2 − 1 =3.

The equation of the tangent at (2, 1) is therefore

y− 1 = 3 (x− 2 )

or

y= 3 x− 5

The gradient of the normal at (2, 1) is

−

1

m

=−

1

3