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Tangent and concavity of parametric equations
- Intro Lesson3:39
- Lesson: 1a5:13
- Lesson: 1b3:49
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- Lesson: 4a2:50
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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.
Basic Concepts: Power rule, Derivative of trigonometric functions , Derivative of exponential functions, Defining curves with parametric equations
Related Concepts: Critical number & maximum and minimum values
Lessons
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)
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)
- IntroductionTangent and Concavity of Parametric Equations Overview
- 1.Find dxdy and dx2d2ya)x=t−t2, y=3+tb)x=et, y=e−t
- 2.Questions Regarding to Tangents and Concavity
Find the tangent to the cycloid x=r(θ−sinθ), y=r(1−cosθ) when θ=4π and r > 0. Determine the concavity for all values of θ. (Do not eliminate the parameter) - 3.Find the point of the parametric curve x=t2+1 and y=t3+t2, in which the tangent is horizontal.
- 4.Find the tangent to the curve x=3cost, y=4cost by:a)Eliminating the parameter.b)Without eliminating the parameter.
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6.
Parametric Equations and Polar Coordinates
6.1
Defining curves with parametric equations
6.2
Tangent and concavity of parametric equations
6.3
Area of parametric equations
6.4
Arc length and surface area of parametric equations
6.5
Polar coordinates
6.6
Tangents of polar curves
6.7
Area of polar curves
6.8
Arc length of polar curves