2 Tangents and normals to a curve

As we know, the relationship between an independent variable $x$ and a dependent variable $y$ is denoted by

$y=f\left(x\right)$

As we also know, the geometrical interpretation of this relation takes the form of a curve in an $xy$ plane as illustrated in Figure 4.

Figure 4

We know how to calculate a value of $y$ given a value of $x$ . We can either do this graphically (which is inaccurate) or else use the function itself. So, at an $x$ value of $x_0$ the corresponding $y$ value is $y_0$ where

$y_0=f\left(x_0\right)$

Let us examine the curve in the neighbourhood of the point $\left(x_0,y_0\right)$ . There are two important constructions of interest

• the tangent line at $\left(x_0,y_0\right)$
• the normal line at $\left(x_0,y_0\right)$

These are shown in Figure 5.

Figure 5

We note the geometrically obvious fact: the tangent and normal lines at any given point on a curve are perpendicular to each other.

Task!

The curve $y=x^2$ is drawn below. On this graph draw the tangent line and the normal line at the point $\left(x_0=1,y_0=1\right)$ :

Answer

Answer

Returning to the curve $y=f\left(x\right):$ we know, from the geometrical interpretation of the derivative that

$\frac{df}{dx}{\left|\right.}_{x_0}=tan\theta$

(the notation $\frac{df}{dx}{\left|\right.}_{x_0}$ means evaluate $\frac{df}{dx}$ at the value $x=x_0$ )

Here $\theta$ is the angle the tangent line to the curve $y=f\left(x\right)$ makes with the positive $x$ -axis. This is highlighted in Figure 6:

Figure 6