In this lesson, we will focus on finding the tangent and concavity of parametric equations. Just like how we can take derivatives of Cartesian equations, we can also do it for parametric equations. First, we will learn to take the derivatives of parametric equations. Then we will look at an application which involves finding the tangents and concavity of a cycloid. After, we will look at special cases of finding a point with a horizontal tangent. Lastly, we will compare the difference of finding tangents by eliminating and without eliminating the parameter.
We can find the tangent (or derivative) without having to eliminate the parameter t by using the equation: dxdy=dtdxdtdy where dtdx≠0
The horizontal tangent occurs when dtdy=0 given that dtdx≠0.
The vertical tangent occurs when dtdx=0 given that dtdy≠0.
To find the concavity (or second derivative), we use the following equation: dx2d2y=dtdxdtd(dxdy)
Find dxdy and dx2d2y
Find the tangent to the curve x=3cost, y=4cost by:
Tangent and concavity of parametric equations
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