# Tangent Lines

In Latin, the word tangent means "to touch." So then, a tangent line is a line that touches. Consider a bicycle moving along the flat pavement. The road is essentially tangent to the bicycle wheel as it touches the wheel at a point. In this article, we will further discuss the meaning of a tangent line, a tangent line's formula, and what the slope of a tangent line means.

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## Definition and formula of a Tangent Line

A tangent line is a line that "just touches" the point $P$. It can also be defined as a line joining 2 infinitely close points on a curve.

The tangent line to a curve$f\left(x\right)$ at a point $P$, with coordinates $\left(a,f\left(a\right)\right)$, is the line through $P$ with slope

$m=\underset{x\to a}{\mathrm{lim}}\frac{f\left(x\right)-f\left(a\right)}{x-a}$

if the limit exists.

## Equation of a Tangent Line

Once the slope $m$ is found, the equation of a tangent line is the same as any other line in point-slope form through a point $\left(a,f\left(a\right)\right)$:

$\left(y-f\left(a\right)\right)=m\left(x-a\right)$

## Tangent Lines on a Graph

In the graph below, we say that $y$ is a tangent line to the curve $f$ at point $P$. Or, we can say that $y$ is tangent to the curve $f$ at point $P$.

The tangent line in green merely touches the curve f at point P - StudySmarter Original

Notice how the tangent line "just touches" the curve at point P.

## The slope of a tangent line

### Geometry

The slope of the tangent line at a point on a curve is equal to the slope of the curve at that point. The assumption behind tangent lines is when looking at the graph of a curve, if you zoom in close enough to a segment of the curve, the curve will look indistinguishable from the tangent line.

For example, let's zoom in on the graph above.

Zooming in at the point where the tangent line touches the curve - StudySmarter Original

Zooming in a bit more...

Here we can see that the tangent line and the point of the curve where the tangent line touches are indistinguishable - StudySmarter Original

Notice how the tangent line comes from connecting two infinitely close points on a curve.

### Additional tangent line slope equation

There is another version of the equation for the slope of the tangent line that can be easier to work with. This equation says that the slope of the tangent line $m$ is

$m=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(a+h\right)-f\left(a\right)}{h}$

This equation sets $h=x-a$ and $x=h+a$. As $x$ approaches $a$, $h$ approaches 0. Thus, the supplementary equation of the tangent line is formed.

This equation for the slope of the tangent line should look familiar to you.... It is the equation for the derivative of a function at a point (a, f(a)). So, we can say that the slope of the tangent line on a curve at point P is equal to the derivative of the curve at point P !

## Examples of finding the Tangent Line Equation

### Example 1

Find the equation for the tangent line to $f\left(x\right)={x}^{2}$ at the point (2, 4).

As we are given $f\left(x\right)$ and a point, all we need to form the equation of the tangent line is the slope. To find the slope, we will use the supplementary equation of the tangent line.

The slope of the tangent at (2, 4) is

$\begin{array}{rcl}m& =& \underset{h\to 0}{\mathrm{lim}}\frac{f\left(2+h\right)-f\left(2\right)}{h}\\ & =& \underset{h\to 0}{\mathrm{lim}}\frac{{\left(2+h\right)}^{2}-4}{h}\\ & =& \underset{h\to 0}{\mathrm{lim}}\frac{{h}^{2}+4h+4-4}{h}\\ & =& \underset{h\to 0}{\mathrm{lim}}\frac{{h}^{2}+4h}{h}\\ & =& \underset{h\to 0}{\mathrm{lim}}\frac{h\left(h+4\right)}{h}\\ & =& \underset{h\to 0}{\mathrm{lim}}h+4\\ & =& 4\end{array}$

So, the equation of the line tangent to f(x) at (2, 4) is $y=4\left(x-2\right)+4$.

Recall how the slope of the tangent line is the same of the derivative. So, we could also simply take the derivative of $f\left(x\right)$ and plug in $x=2$ to find the slope of the tangent line at the point $\left(2,4\right)$.

$f\text{'}\left(x\right)=2x\phantom{\rule{0ex}{0ex}}f\text{'}\left(2\right)=4$

The graph of f(x) and the line tangent to f(x) at (2, 4) - StudySmarter Original

### Example 2

Find the equation of the tangent line to the curve $f\left(x\right)=x-{x}^{3}$ at the point (1, 0).

Again, as we are given $f\left(x\right)$ and a point, all we need to form the equation of the tangent line is the slope. To find the slope, we will use the supplementary equation of the tangent line.

With , the slope of the tangent at (1, 0) is:

$\begin{array}{rcl}m& =& \underset{h\to 0}{\mathrm{lim}}\frac{f\left(1+h\right)-f\left(1\right)}{h}\\ & =& \underset{h\to 0}{\mathrm{lim}}\frac{\left[\left(1+h\right)-{\left(1+h\right)}^{3}\right]-0}{h}\\ & =& \underset{h\to 0}{\mathrm{lim}}\frac{\left(1+h\right)-\left({h}^{3}+3{h}^{2}+3h+1\right)}{h}\\ & =& \underset{h\to 0}{\mathrm{lim}}\frac{{h}^{3}-3{h}^{2}-2h}{h}\\ & =& \underset{h\to 0}{\mathrm{lim}}{h}^{2}-3h-2\\ & =& -2\end{array}$

So, the equation of the line tangent to f(x) at (1, 0) is $y=-2\left(x-1\right)$.

Again, recall how the slope of the tangent line is the same of the derivative. So, we could also simply take the derivative of $f\left(x\right)$ and plug in 1 to find the slope of the tangent line at the point $\left(1,0\right)$.

$f\text{'}\left(x\right)=1-3{x}^{2}\phantom{\rule{0ex}{0ex}}f\text{'}\left(1\right)=1-3\left(1\right)=-2$

The graph of f(x) and the line tangent to f(x) at (1, 0) - StudySmarter Original

## Tangent Lines in a Circle

A line is said to be tangent to a circle if it touches the circle at exactly one point. If a line is tangent to a circle at a point $P$, then the tangent line is perpendicular to the radius drawn to point $P$.

A line that is tangent to a circle at point P is perpendicular to the radius drawn to point P - StudySmarter Originals

For more information on tangent lines in a circle, check out our article on Tangent of a Circle!

## Tangent Lines - Key takeaways

• A tangent line is a line that touches a curve at a fixed point $P$
• The equation of a tangent line in point-slope form is $\left(y-f\left(a\right)\right)=m\left(x-a\right)$
• The slope of the tangent line at a point on a curve is equal to the slope of the curve at that point
• If you zoom in close enough to a segment of a curve, the tangent line at the segment and the curve will look indistinguishable
• In a circle, the tangent line drawn at any point is perpendicular to the radius at that point.

#### Flashcards in Tangent Lines 9

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How do you find tangent line?

To find a tangent line, use the equation for the slope of a tangent line at a point and plug in the slope and the point into point-slope form.

What is tangent line of a curve?

A tangent line of a curve is a line that touches the curve at a fixed point P.

What defines a tangent line?

A tangent line is a line that touches the curve at a fixed point P. The slope of the point at the curve is equal to the slope of the tangent line at point P.

How to find the equation of line tangent?

To find the equation of a tangent line, use the equation for the slope of a tangent line at a point and plug in the slope and the point into point-slope form.

How to find slope of a tangent line?

The slope of a tangent line can be found using the limit definition of the slope of a tangent line.

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