Finding points with vertical tangents | Summary and Q&A

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May 8, 2018
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Khan Academy
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Finding points with vertical tangents

TL;DR

Use implicit differentiation to find the derivative of y with respect to x, then solve for dy/dx. Determine the y values that make the denominator equal zero to find the corresponding x values. The two points on the curve with a vertical tangent line are (-4, -1) and (2, -1).

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Questions & Answers

Q: How can we find the coordinates of the points on the closed curve with a vertical tangent line?

To find the points, we can use implicit differentiation to obtain the derivative of y with respect to x. By setting the denominator of the derivative equal to zero, we get the y values. Substituting these y values back into the original equation will solve for the corresponding x values.

Q: What is the process of implicit differentiation?

Implicit differentiation involves differentiating both sides of an equation with respect to x while treating y as an implicit function of x. Apply the chain rule as necessary, then solve for the derivative of y with respect to x.

Q: How do we determine the y values that make the denominator of the derivative equal to zero?

By setting the denominator of the derivative expression equal to zero, we can solve for y. These y values correspond to the points on the curve with a vertical tangent line.

Q: What are the coordinates of the points on the curve with a vertical tangent line?

The two points are (-4, -1) and (2, -1). These are the x and y coordinates that satisfy the equation and have a vertical tangent line.

Summary & Key Takeaways

  • The task is to find the coordinates of the two points on a closed curve where the tangent line is vertical.

  • Implicit differentiation is used to find the derivative of y with respect to x.

  • By setting the denominator of the derivative equal to zero, the y values that correspond to the points with a vertical tangent line are found.

  • Substitute these y values back into the original equation to solve for the corresponding x values.

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