Consider
Figure 2.2.2. The line through points A and B intersects the curve in at least those two points. But as A approaches B those intersections are close together. In the limit, points A and B overlap (become one point). That means the line intersects this curve at only one point. This is a common description of a tangent line. They are sometimes known as
osculating curves. Osculate is Latin for barely touch (also used to mean “kiss”). In context the line barely touches the curve.
Finally we note that we can use either of these perspectives at every point on the curve. This means that while a derivative is defined at a point, we can expand the concept to refer to the derivative as a function that accepts the x value and outputs the derivative at that point. This is why we have the notation
for
The slope of the tangent line at
is
and the slope of the tangent line at
is
Next we use this idea to write the equation of tangent lines. This has various applications and is common in a part of multi-variable calculus.
Example 2.3.1.
Consider the function
We will write an equation for the line tangent to
at
To write the equation of a line we can use the slope and a point.
Because of the meaning of a derivative (limit on the slopes is the tangent slope), the slope of the tangent line will be
Next we need a point. Because this is the tangent line when
we want the point
Finally we put these together using the point slope form of a line equation.
That can be re-arranged into
To find an equation of the line tangent at another point, we simply repeat the process with a different x value.
