Normal straight line to a curve at a point

It is the straight line that, when crossing the curved line, it is perpendicular to the curve.

Example

The following figure shows the normal straight line to the curve $y = \frac{1}{x - 1} + 1$ :

imagen

Two functions $f (x), g (x)$ will be normal in a point if, in the crossing point $a$ , it is satisfied that: $f^{'} (a) \cdot g^{'} (a) = - 1$

Example

The following table shows several values of slopes of perpendicular straight lines:

$f^{'} (a)$	$g^{'} (a)$
$1$	$- 1$
$2$	$- \frac{1}{2}$
$- 3$	$\frac{1}{3}$
$\frac{3}{8}$	$- \frac{8}{3}$

The general expression of the normal straight line to $f (x)$ at a point $a$ is: $y - f (a) = - \frac{1}{f^{'} (a)} \cdot (x - a)$

Example

Solve the figure showed previously, that is, find the normal straight line to $f (x) = \frac{1}{x - 1} + 1$ at the point $a = 2$ :

a) The slope of the curve at the crossing point is: $\begin{array}{rcl} f^{'} (x) & = & - \frac{1}{(x - 1)^{2}} \\ f^{'} (2) & = & - 1 \end{array}$ And the slope of the straight line is: $m = - \frac{1}{f^{'} (2)} = 1$

b) Using the above mentioned, the straight line will go through $(a, f (a)) = (2, 2)$

Finally, the equation of the normal straight line is: $\begin{array}{rcl} y - 2 & = & 1 \cdot (x - 2) \\ y & = & x \end{array}$ Consistent with what we can observe in the figure.

Example

Find the tangent straight line to the function $y = \sqrt{x}$ at the point $x = 0$ , as well as its normal straight line.

a) We start by looking at the derivative of the function at $x = 0$ .

However we can see that it does not exist. We then have to compute the limit as $x$ goes to zero from the right: $\begin{array}{l} y^{'} (x) = \frac{1}{2 \sqrt{x}} \\ lim_{x \to 0} y^{'} (x) = lim_{x \to 0} \frac{1}{2 \sqrt{x}} = \infty \end{array}$

b) Since the formula $y = a \cdot x + b$ is not useful when we have an infinite slope, we have to realize that it is the axis that is normal to curve. That is if we take the straight line $x = 0$ we obtain a straight line with infinite slope.

c) Finally, we should realize that the line perpendicular to the straight line defined by the $y$ axis is the $x$ axis. That is, by the line defined by $y = 0$ .