Slope & Tangent Lines of Polar Curves - Calculus 2

One of the first topics that you learn about in calculus is how to determine the slope of a tangent line at a particular point along a curve. This is known as "the tangent line problem" and we solve it with the concept of a derivative. We have used the derivative in the past to find slope of tangent lines for rectangular equations and parametric equations within the rectangular coordinate system. However, we can also apply the derivative to polar curves in the polar coordinate system.

Since the definition of slope is "rise over run," or the change in y over the change in x, it does not make sense that this definition would change when determining slope within the polar coordinate system. What I mean by this is that slope in polar coordinates is not magically going to be defined in terms of r and θ. Instead, we will continue to define slope as the change in y over the change in x, by first expressing polar curves with parametric equations and making use of the parametric form of a derivative.

If you remember the formulas that are used to convert between rectangular equations and polar equations, two of them relate x and y individually in terms of r and θ. Specifically, these equations are the two below:

x=rcosθ, y=rsinθ

Since r is defined as a function of θ (r=f(θ)), we can rewrite these equations to be a pair of parametric equations where the parameter is θ as follows:

x=f(θ)cosθ, y=f(θ)sinθ

These parametric equations define x and y in terms of a parameter θ, and will represent a polar curve that corresponds to f(θ). By defining a polar curve in this way, we can find the parametric derivative of the parametric equations and use that parametric derivative to determine slope at particular angles of θ for a polar curve, as well as locate points where a curve has horizontal and vertical tangent lines.

We discuss all of this and more in this new lesson of Calculus 2. Let's get to it!

In this lesson, you will learn:

• How to find the slope of a tangent line for a polar curve
• How to locate points of horizontal tangency for a polar curve
• How to locate points of vertical tangency for a polar curve

I hope you find this video to be helpful!

-Josh