Tangent Planes in Multivariable Calculus

This video tutorial explains tangent planes, a multivariable generalization of tangent lines. It covers how to define and compute tangent planes, derive their equations using normal vectors, and make simplifying assumptions. The video also discusses linear approximation of nonlinear functions and provides a detailed example calculation of a tangent plane.

What is a tangent plane in multivariable calculus?

• A plane that approximates a surface near a point

• A plane that is parallel to the XY plane

• A line that touches a curve at a single point

• A line that intersects a surface at multiple points

How is a tangent plane related to a tangent line?

• A tangent plane is a 3D version of a tangent line

• A tangent plane is a perpendicular line to a tangent line

• A tangent plane is a 2D version of a tangent line

• A tangent plane is unrelated to a tangent line

What is the first step in defining a tangent plane?

• Determining the curvature of the surface

• Calculating the area under the curve

• Identifying a specific point on the graph

• Finding the slope of the tangent line

What is the role of the normal vector in the equation of a plane?

• It defines the direction of the plane

• It is perpendicular to the plane

• It lies within the plane

• It is parallel to the plane

How do you find the normal vector to a plane?

• By using the cross product of two vectors in the plane

• By using the dot product of two vectors in the plane

• By calculating the area of the plane

• By finding the gradient of the function

What assumption is made about the coefficient of Z in the tangent plane equation?

• It is always zero

• It is a nonzero number

• It is a negative number

• It is equal to one

What is the purpose of using partial derivatives in the tangent plane equation?

• To calculate the area under the curve

• To approximate the nonlinear function linearly

• To determine the curvature of the surface

• To find the slope of the tangent line

In the example problem, what is the function used to illustrate the tangent plane calculation?

• f(x, y) = x^2 - y^2

• f(x, y) = 2x + 2y

• f(x, y) = x^2 + y^2

• f(x, y) = 2 - x^2 - y^2

What is the value of Z at the point (1/2, 1/2) in the example problem?

• 3/2

• 2

• 1

• 1/2

What is the partial derivative of the function with respect to X in the example problem?

• -2x

• 2x

• -2y

• 2y