Exam 15: Differentiation in Several Variables

Given that , compute the partial derivatives , , and .

Find the global extrema of on the unit square .

Find the linearization of at the point and use it to approximate

The plane best fits the points and if the sum of at these points is minimized by and Find the plane that best fits the points , and .

Find the distance from the origin to the surface that is, the minimum distance between the origin and a point on the surface.

Find the global minimum of on the domain .

Find the critical points of the function , and analyze them using the second derivative test.

Find the range of the function .

Find the distance from the origin to the surface ; that is, the minimum distance between the origin and a point on the surface.

Let . A) Is differentiable at the origin? B) Does have a tangent plane at the origin? If so, find its equation.

The temperature at on a metal plate is . A) A heat seeking insect is placed at the point . In what direction should it move in order to feel the greatest increase in heat? What is the rate of change of temperature in this direction? B) The insect is placed at the point and is very hot. In what direction should it move in order to cool off at the fastest rate? What is the rate of change of temperature in this direction?

Evaluate the limit or state that it does not exist.

Find the global minimum and maximum of in the region .

Find and if .

Let be differentiable, and let be the function . Compute the partial derivatives , , and .

Determine whether the following function is continuous.

Find the minimum of if is restricted to the planes and .

Which of the following functions satisfies the differential equation

Let . Find the critical points of and analyze them using the second derivatives.

Find the critical points of for and analyze them using the second derivative test.