Tangents and Normals

Tangents

A tangent is the linear function at a given point on the curve. It is used to find the instantaneous rate of change of a curve at a certain point.

To find the gradient of the tangent, substitute the given x-value into the derivative.

Normals

A normal is the linear function perpendicular to the tangent at any given point on the curve.

To find the equation of a normal, use the perpendicular gradient formula (𝑚𝑇×𝑚𝑁=1) to find 𝑚𝑁.

Example: 𝑓(𝑥)=𝑥2

To find the equation to the normal:
(1) Find the tangent gradient.

Let tangent be at 𝑥=2,𝑦=4𝑓(2)=2(2)=4𝑚𝑇=𝑓(𝑥)𝑚𝑇=𝑓(2)=4𝑚𝑇=4

(2) Find the normal function.

𝑚𝑇×𝑚𝑁=14×𝑚𝑁=1𝑚𝑁=14𝑦𝑁=14𝑥+𝑐Sub(2,4)4=14×2+𝑐4=12+𝑐4.5=𝑐𝑦𝑁=14𝑥+4.5