Normal Line Calculator

The calculator will find the normal line to the explicit, polar, parametric and implicit curve at the given point, with steps shown.

It can handle horizontal and vertical normal lines as well.

The normal line is perpendicular to the tangent line.

Related calculator: Tangent Line Calculator

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Your Input

Calculate the normal line to $$$y = x^{2} + 1$$$ at $$$x = 2$$$.

Solution

We are given that $$$f{\left(x \right)} = x^{2} + 1$$$ and $$$x_{0} = 2$$$.

Find the value of the function at the given point: $$$y_{0} = f{\left(2 \right)} = 5$$$.

The slope of the normal line at $$$x = x_{0}$$$ is the negative reciprocal of the derivative of the function, evaluated at $$$x = x_{0}$$$: $$$M{\left(x_{0} \right)} = - \frac{1}{f^{\prime }\left(x_{0}\right)}$$$.

Find the derivative: $$$f^{\prime }\left(x\right) = \left(x^{2} + 1\right)^{\prime } = 2 x$$$ (for steps, see derivative calculator).

Hence, $$$M{\left(x_{0} \right)} = - \frac{1}{f^{\prime }\left(x_{0}\right)} = - \frac{1}{2 x_{0}}$$$.

Next, find the slope at the given point.

$$$m = M{\left(2 \right)} = - \frac{1}{4}$$$

Finally, the equation of the normal line is $$$y - y_{0} = m \left(x - x_{0}\right)$$$.

Plugging the found values, we get that $$$y - 5 = - \frac{x - 2}{4}$$$.

Or, more simply: $$$y = \frac{11}{2} - \frac{x}{4}$$$.

Answer

The equation of the normal line is $$$y = \frac{11}{2} - \frac{x}{4} = 5.5 - 0.25 x$$$A.