Polar Coordinates: Example 9: Cardioid: Part 3: Question B

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Published on Oct 15, 2022

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In this video I go over Part 3 of Example 9 on Polar Coordinates, and this time look at Question B which looks at determining the points at which the tangents lines are either horizontal or vertical. I use the formula for the derivative or slope of a tangent line to the polar cardioid (i.e. heart shaped) curve r = 1 + sinϴ in determining the vertical and horizontal tangents. The horizontal tangents are when the derivative is equal to 0, while the vertical tangents occur when the derivative approaches infinity. When we look at the derivative in terms of its parametric equivalent, in which ϴ is the parameter, we notice that the conditions that need to be met are when dy/dϴ = 0 for horizontal tangents and dx/dϴ = 0 for vertical tangents. But it is very important to realize the special case when both dy/dϴ and dx/dϴ = 0, in which we will get an undefined 0/0 result. But luckily we can use the L’Hospital’s Rule to evaluate these cases to determine if the denominator or numerator is approaching 0 “faster”, which governs whether the result approaches 0 or approaches infinity. In this video, this special case arises and I show that it does in fact approach infinity. This is an extremely detailed and thorough video that takes many different concepts from L’Hospital’s Rule and Trigonometry to Polar and Parametric calculus in order to solve for vertical and horizontal tangent lines, so make sure to watch this whole video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvQCHR0c727P_-7rSg

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-polar-coordinates-example-9-cardioid-part-3-question-b

Related Videos:

Polar Coordinates: Example 9: Cardioid: Part 2: Question A: https://youtu.be/gUYmzZ8Cfhg
Polar Coordinates: Example 9: Cardioid: Part 1: Slope Formula & Trigonometric Algebra: https://youtu.be/MlIhueP36EE
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