Tangent and concavity of parametric equations - Parametric Equations and Polar Coordinates

Tangent and concavity of parametric equations

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 tt by using the equation:
dydx=dydtdxdt \frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \; where dxdt0\;\frac{dx}{dt} \neq0
The horizontal tangent occurs when dydt=0\;\frac{dy}{dt} =0\; given that dxdt0\;\frac{dx}{dt} \neq0.
The vertical tangent occurs when dxdt=0\;\frac{dx}{dt} =0\; given that dydt0\;\frac{dy}{dt} \neq0.

To find the concavity (or second derivative), we use the following equation:
  • 1.
    Find dydx\;\frac{dy}{dx}\; and d2ydx2\;\frac{d^2y}{dx^2}
  • 4.
    Find the tangent to the curve x=3costx=3 \cos t, y=4costy=4 \cos t by:
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Tangent and concavity of parametric equations

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