Tangent and Normals

Definitions

Tangent Line - A straight line that passes through a point on a function and has the same slope as that function at that point.

Normal Line - A straight line that passes through a point on a function is perpendicular to the function at that point.

A common question is to find an equation for the normal or tangent line at a given point on a function. Since the equation is linear the answer will take the form of:

(1)
$$y=mx+b$$

If you are looking for a tangent line then m will simply be the value of the derivative at the given point on the function.

If you are looking for a normal line then m will be the negative reciprocal of the value of the derivative at the given point on the function

The y-intercept can then be found using the coordinates of the given point on the function.

Example

Find equations for both the normal and the tangent lines on the function $f(x)=2x^2+x$ at the point where $x=2$.

Let's work on the tangent line first:

The derivative of $f(x)$ is $f'(x)=4x+1$ so $f'(x=2)=9$. This gives us an equation for the tangent line:

(2)
$$y=9x+b$$

We want the line to be tangent when $x=2$ which occurs at the point $(2,10)$. Use those values in the equation above.

(3)
$$10=9(2)+b$$

Solving for b we get a final equation for the tangent line of $y=9x-8$

Now on to the normal line.

The slope of the normal line will be the negative reciprocal of the slope of the tangent line. So we can write an equation for the tangent line as:

(4)
\begin{align} y=\frac{-1}{9}x+b \end{align}

Again this will go through the point $(2,10)$. Using these point to solve for b we can write an equation for the normal line as:

(5)
\begin{align} y=\frac{-1}{9}x+\frac{46}{45} \end{align}