What is the tangent line?
Tangent lines are absolutely critical to calculus; you can’t get through Calc 1 without them! In this video, we’re talking all about the tangent line: what it is, how to find it, and where to look for vertical and horizontal tangent lines.
The tangent line is useful because it allows us to find the slope of a curved function at a particular point on the curve. We learned a long, long time ago in a math class far, far away that we could find the slope of a line, but we’ve never learned how to find the slope of a curved function. Since the slope of a curved function is always changing, the best we can do is find the slope of the curved function at one particular point on the function. And to do this, we actually don’t look at the function at all. Instead, we look at the tangent line to the curve that passes through the particular point we’re interested in, and we find the slope of the line instead.
To find the slope of the curve, all we have to do is take the derivative of the curve (because the derivative represents the slope), and then find the line with the correct slope that passes through the point of tangency. That’ll give us the tangent line, and the tangent line will have the same slope as the slope of the curve at the point of tangency.
We’ll also look at where to find vertical tangent lines, and where to find horizontal tangent lines, since that’s something you’ll be asked to do often. Horizontal tangent lines exist where the derivative of the function is equal to 0, and vertical tangent lines exist where the derivative of the function is undefined.
0:24 // The definition of the tangent line
1:16 // How to find the equation of the tangent line
3:10 // Where the tangent line is horizontal and vertical