Find dy/dx and d^2y/dx^2 Given Parametric Equations x = theta - cos(theta), y = 1 - sin(theta) | Summary and Q&A

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October 26, 2018
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The Math Sorcerer
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Find dy/dx and d^2y/dx^2 Given Parametric Equations x = theta - cos(theta), y = 1 - sin(theta)

TL;DR

The video explains how to find the slope (dy/dx) and the concavity (second derivative) of a set of parametric equations.

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Key Insights

  • 🫡 The slope (dy/dx) of parametric equations can be found by calculating the derivatives of Y and X with respect to theta and then taking their ratio.
  • 🫡 The second derivative of Y with respect to X can be determined by differentiating the dy/dx formula with respect to theta and dividing it by DX/Dtheta.
  • 😥 Evaluating dy/dx and the second derivative at a specific value of theta provides information about the slope and concavity of the parametric equations at that point.

Transcript

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Questions & Answers

Q: How do we find dy/dx for parametric equations?

To find dy/dx, we calculate the derivatives of Y and X with respect to theta individually and then divide the derivatives of Y by the derivatives of X.

Q: What is the value of dy/dx when theta is equal to pi?

Evaluating dy/dx at theta = pi, we substitute pi into the dy/dx formula and find that the slope is equal to 1.

Q: How do we calculate the second derivative of Y with respect to X?

The second derivative of Y with respect to X can be found by taking the derivative of dy/dx with respect to theta and dividing it by DX/Dtheta.

Q: What is the concavity of the parametric equations at theta = pi?

Evaluating the second derivative at theta = pi, we plug in the value into the formula and find that the concavity is equal to 1, indicating a concave up shape.

Summary & Key Takeaways

  • The video discusses the process of finding the dy/dx of parametric equations by utilizing the formulas for derivatives.

  • The presenter demonstrates the calculation by evaluating the derivatives for the given parametric equations at a specific value of theta (pi).

  • The slope (dy/dx) is found to be 1 at theta = pi.

  • The video also explains how to find the second derivative of Y with respect to X using the quotient rule and simplifying it to 1.

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