Concavity of Parametric Curves - Calculus 2

Another application of parametric derivatives is the ability to determine the concavity for plane/parametric curves. In fact, this is specifically an application of the second parametric derivative for a set of parametric equations.

You were first introduced to concavity in Calculus 1, where you learned to determine the intervals of concavity for functions (in terms of x and y) to aid in graphing. Additionally, you saw how to use concavity to help identify types of relative extrema. However for parametric curves, we are only going to be interested in determining concavity at certain values of the parameter (in most cases that is t), and intervals of t values for which the curve is concave up or concave down.

Determining concavity for parametric curves will require that we first find the second derivative for a set of parametric equations, which we saw how to find in a previous lesson. Once the second parametric derivative is found, any value of t can be plugged into the second derivative in order to determine the concavity of the curve at that specific value of t.

In Calculus 1 you learn that a function is concave up when the second derivative is positive, and the function is concave down when the second derivative is negative. These same rules will apply to parametric curves and the second parametric derivative. If the second parametric derivative is positive for a specific value of t, the curve is concave up, and if it is negative, the curve is concave down. In addition to determining concavity at specific values of t, we can use these rules to help us find t intervals of concavity for parametric curves as well.

Intervals of concavity tell us the range of values for t where a curve is concave up and concave down. Finding these intervals requires that we know where a curve will change in concavity. The concavity of a curve can only change at points where the second derivative is equal to 0 or it is undefined (from Calc 1 you might remember that these locations are sometimes called points of inflection). We will learn how to use those points to help us find the intervals of concavity for a curve as well as everything else mentioned above regarding concavity in this new lesson for Calculus 2, so let's get right to it!

In this lesson, you will learn:

• When a parametric curve is concave up or concave down
• How to find the concavity for a parametric curve at a specific value of t 
• How to find the t intervals of concavity for parametric curves

I hope you find this video to be helpful!

-Josh