Tangent Plane and Partial Derivatives

The video tutorial explains how to find the equation of a tangent plane to a surface at a given point. It begins with a graphical representation and then formulates the surface equation in terms of x, y, and z. The tutorial discusses the role of the gradient in deriving the tangent plane equation and demonstrates how to calculate partial derivatives at the point of tangency. Finally, it derives the tangent plane equation using these derivatives and simplifies it to solve for z.

What is the main objective of the tutorial?

• To find the equation of a tangent line.

• To find the equation of a tangent plane.

• To find the area of a surface.

• To find the volume of a surface.

What is the first step in finding the tangent plane equation?

• Identify the point of tangency.

• Graph the surface.

• Write the surface equation in a specific form.

• Calculate the gradient.

What is the significance of the point of tangency in the context of the tangent plane?

• It is the point where the plane is parallel to the surface.

• It is the point where the plane is tangent to the surface.

• It is the point where the plane is perpendicular to the surface.

• It is the point where the plane intersects the surface.

What role does the gradient of the function play in finding the tangent plane?

• It is tangent to the surface.

• It is perpendicular to the tangent plane.

• It is parallel to the tangent plane.

• It is irrelevant to the tangent plane.

How is the partial derivative with respect to x calculated?

• By treating y and z as constants.

• By treating x and z as variables.

• By treating y and z as variables.

• By treating x and y as constants.

What is the value of the partial derivative with respect to z at any point?

• Zero

• One

• Negative one

• Two

What is the standard form of the tangent plane equation derived in the tutorial?

• ax - by + cz = d

• ax + by - cz = d

• ax + by + cz = d

• ax - by - cz = d

What is the general form of the tangent plane equation after simplification?

• 8x + y - z = 0

• -8x + y - z = 0

• -8x + y - z = 4

• 8x - y + z = 4

How is the equation of the tangent plane solved for z?

• By subtracting z from both sides.

• By adding x to both sides.

• By subtracting y from both sides.

• By adding z to both sides.

What is the final form of the tangent plane equation solved for z?

• z = -8x + y + 4

• z = 8x + y - 4

• z = -8x + y - 4

• z = 8x - y + 4