Online Mathematics Book

Tangents & Normals

In this section we learn how to find the equation of the tangent and the normal to a curve at a point along its length.

For each we learn a twp-step method as well as view a tutorial and work our way through exercises to consolidate our knowlede.

The following graph illustrates the tangent and the normal to the curve \(y=x^2-1\) at the point \(\begin{pmatrix}2,3\end{pmatrix}\):


Method : Tangent to a Curve

Given a function \(f(x)\), described by a curve \(y=f(x)\), we find the equation of the tangent to the curve at a point \(\begin{pmatrix}a,b\end{pmatrix}\) along its length, in two steps:

  • Step 1: find the gradient \(m\).

    The tangent's gradient equals to the curve's gradient at the point \(\begin{pmatrix}a,b\end{pmatrix}\), which equals to the derivative \(f'(x)\) evaluated at \(x=a\): \[m = f'(a)\]
  • Step 2: find the tangent's equation, \(y=mx+c\), by making \(y\) the subject in the formula: \[y-b = m \begin{pmatrix}x - a \end{pmatrix}\]

Tutorial

Example

Find the equation of the tangent to the curve \[y = \frac{x^2}{2}+x + \frac{3}{2}\] at the point along its length with coordinates \(\begin{pmatrix}1,3\end{pmatrix}\).

Solution

We follow our two-step method:

  • Step 1:
  • Step 2:

Exercise

Answers Without Working