Find all Points at which the Tangent Plane is Horizontal | Summary and Q&A

TL;DR

Find the points on a surface where the tangent plane is horizontal by equating the normal vector to the gradient vector of the surface.

Key Insights

• ✈️ The normal vector of a tangent plane is perpendicular to the plane and can be found by taking the gradient of the surface function.
• 🚥 Equating the normal vector to a vector representing a horizontal plane helps us find the points where the tangent plane is horizontal.
• 🤪 The x and y coordinates can be easily determined, whereas the z coordinate is obtained by plugging the values back into the surface equation.

Transcript

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Questions & Answers

Q: What does it mean for a tangent plane to be horizontal?

A tangent plane is horizontal when it is flat and its normal vector points straight up, perpendicular to the tangent plane itself.

Q: How do we find the normal vector of a tangent plane?

The normal vector of a tangent plane can be found by taking the gradient of the function representing the surface and evaluating it at a point on the surface.

Q: How do we determine the points where the tangent plane is horizontal?

By equating the normal vector of the tangent plane to a vector representing a horizontal plane and solving for the coordinates, we can find the points where the tangent plane is horizontal.

Q: What is the significance of the gradient in finding the points of a horizontal tangent plane?

The gradient represents the direction of steepest ascent of the surface function, and its orthogonality to the tangent plane's normal vector allows us to find the points where the tangent plane is horizontal.

Summary & Key Takeaways

• To find the points where the tangent plane is horizontal, we need to find the coordinates that make the normal vector of the plane straight up.

• The normal vector can be found by taking the gradient of the surface function.

• Setting the normal vector equal to the vector representing a horizontal plane, we can solve for the coordinates of the points.