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).

Transcript

• [Instructor] Consider the closed curve in the xy-plane given by this expression here. Find the coordinates of the two points on the curve where the line tangent to the curve is vertical. So pause this video, and see if you could have a go at it. So I don't know what the exact shape of this closed curve is. But if I were to draw some type of a clo... Read More

Key Insights

• 😚 The task is to find points on a closed curve with a vertical tangent line.
• 😀 Implicit differentiation is used to find the derivative of y with respect to x.
• 🏙️ Setting the denominator of the derivative equal to zero gives the y values for the points with a vertical tangent line.
• ❣️ Substituting the y values back into the original equation solves for the corresponding x values.

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.