Exam 14: Partial Derivatives

Use differentials to estimate the amount of metal in a closed cylindrical can that is 12 cm high and 8 cm in diameter if the metal in the top and bottom is 0.09 cm thick and the metal in the sides is 0.01 cm thick. (rounded to the nearest hundredth.)

Find the second partial derivatives of the function

Use the definition of partial derivatives as limits to find if .

Describe the level surfaces of the function .

Find all the saddle points of the function.

Find and classify the relative extrema and saddle points of the function for and .

Find the indicated partial derivative.

Sketch the graph of the function

Evaluate the limit.

Find all the second partial derivatives.

Find the equation of the tangent plane to the given surface at the specified point.

Find the absolute extrema of the function on the region bounded by the disk defined by .

Use Lagrange multipliers to find the maximum value of the function subject to the given constraints.

Use the Chain Rule to find .

Consider evaluating the limit (a) Evaluate the limit along the path (b) Evaluate the limit along the path (c) What can you conclude about the existence of ?

Find the maximum rate of change of at the point (2,1). In what direction does it occur?

Determine where the function is continuous.

Find the limit.

Use the linearization L(x, y) of the function. at to approximate .

Find the indicated partial derivative.