Tangent Lines

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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.

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 $f (x)$ at a point $P$ , with coordinates $(a, f (a))$ , is the line through $P$ with slope

$m = \lim_{x \to a} \frac{f (x) - f (a)}{x - a}$

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 $(a, f (a))$ :

$(y - f (a)) = m (x - a)$

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 $P$ .

Tangent Line geometric interpretation StudySmarter 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.

Tangent Lines slope of tangent line equal to curve StudySmarter Zooming in at the point where the tangent line touches the curve - StudySmarter Original

Zooming in a bit more...

Tangent Lines slope of tangent line equal to curve StudySmarter

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

$m = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$

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 $f (x) = x^{2}$ at the point (2, 4).

As we are given $f (x)$ 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

$\begin{array}{rcl} m & = & \lim_{h \to 0} \frac{f (2 + h) - f (2)}{h} \\ = & \lim_{h \to 0} \frac{{(2 + h)}^{2} - 4}{h} \\ = & \lim_{h \to 0} \frac{h^{2} + 4 h + 4 - 4}{h} \\ = & \lim_{h \to 0} \frac{h^{2} + 4 h}{h} \\ = & \lim_{h \to 0} \frac{h (h + 4)}{h} \\ = & \lim_{h \to 0} h + 4 \\ = & 4 \end{array}$

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

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

$f' (x) = 2 x f' (2) = 4$

Tangent lines graph of tangent line and function example StudySmarter 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 $f (x) = x - x^{3}$ at the point (1, 0).

Again, as we are given $f (x)$ 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:

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

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

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

$f' (x) = 1 - 3 x^{2} f' (1) = 1 - 3 (1) = - 2$

Tangent Line graph of the line tangent to the curve example StudySmarter 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 $P$ .

Tangent Lines tangent line circle perpendicular radius StudySmarter 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 $(y - f (a)) = m (x - a)$
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.

Frequently Asked Questions about Tangent Lines

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.

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

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

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.

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

Flashcards in Tangent Lines9 Start learning

What is a tangent line?

It is a line that touches a curve at a fixed point P.

The slope of the tangent line at a point P is...

equivalent to the slope of the curve at P

The slope of a tangent line is equivalent to which important Calculus concept?

the derivative

What is the geometric definition of the tangent line?

The tangent line is a line that joins two infinitely close points on the curve.

What does it mean if the slope of the tangent line is negative?

If the slope of the tangent line is negative at a point on the curve, then the curve must be decreasing at the point.

What does it mean if the slope of the tangent line is positive?

If the slope of the tangent line is positive at a point on the curve, then the curve must be increasing at the point.

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Tangent Lines

Definition and formula of a Tangent Line

Equation of a Tangent Line

Tangent Lines on a Graph