Slope & Tangent Lines of Parametric Curves - Calculus 2

One of the applications of parametric derivatives is the ability to determine the slope at a particular value of t (the parameter) along a plane/ parametric curve, and use that slope to find the equation of the tangent line for that value of t. This should be a familiar process, as finding slope and tangent lines was one of the first applications that is covered in Calculus 1 after introducing basic derivatives.

If you find the parametric derivative for a set of parametric equations, you can plug values of t into that derivative to determine the slope at those values of t for the curve represented by the parametric equations. Once you find the slope at a particular value of t, you can create the equation of a tangent line at that value of t by then finding the corresponding coordinate point (x,y) and using the point-slope formula from algebra (to find the coordinate point, the value of t should be plugged into each parametric equation for x and y to get the respective x- and y-coordinates).

There are also two special types of tangent lines that we can find for a plane/ parametric curve with the help of a parametric derivative. These two types of tangent line are horizontal tangent lines, where the slope is 0, and vertical tangent lines, where the slope is undefined. We can find the location of horizontal and vertical tangent lines by setting dy/dt equal to 0 and dx/dt equal to 0 respectively, and solving for any values of t. These values of t will correspond to coordinate points which will be the location of these special tangent lines.

So altogether, we look at the general process of finding slope, equations of tangent lines, and points of horizontal and vertical tangency for parametric curves in this lesson of Calculus 2, as well as use the processes in example problems. So let's get into it!

In this lesson, you will learn:

• How to find the slope for a curve given a set of parametric equations
• How to use the slope and a point for a parametric curve to find the equation of a tangent line
• How to find points of vertical and horizontal tangent lines for parametric curves

I hope you find this video to be helpful!

-Josh