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.

Related concepts:
Critical number & maximum and minimum values

Lessons

Notes:

We can find the tangent (or derivative) without having to eliminate the parameter $t$ by using the equation:

$\frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \;$ where $\;\frac{dx}{dt} \neq0$
The horizontal tangent occurs when $\;\frac{dy}{dt} =0\;$ given that $\;\frac{dx}{dt} \neq0$ .
The vertical tangent occurs when $\;\frac{dx}{dt} =0\;$ given that $\;\frac{dy}{dt} \neq0$ .

To find the concavity (or second derivative), we use the following equation:

$\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$

1.
Find $\;\frac{dy}{dx}\;$ and $\;\frac{d^2y}{dx^2}$
4.
Find the tangent to the curve $x=3 \cos t$ , $y=4 \cos t$ by: