# Tangent and concavity of parametric equations

### 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.

#### Lessons

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:
$\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$
• Introduction
Tangent and Concavity of Parametric Equations Overview

• 1.
Find $\;\frac{dy}{dx}\;$ and $\;\frac{d^2y}{dx^2}$
a)
$x=t-t^2$, $y=3+t$

b)
$x=e^t$, $y=e^{-t}$

• 2.
Questions Regarding to Tangents and Concavity
Find the tangent to the cycloid $x=r(\theta - \sin \theta)$, $y=r(1-\cos \theta)$ when $\theta = \frac{\pi}{4}$ and $r$ > $0$. Determine the concavity for all values of $\theta$. (Do not eliminate the parameter)

• 3.
Find the point of the parametric curve $x=t^2+1$ and $y=t^3+t^2$, in which the tangent is horizontal.

• 4.
Find the tangent to the curve $x=3 \cos t$, $y=4 \cos t$ by:
a)
Eliminating the parameter.

b)
Without eliminating the parameter.