For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.
The domain of is all real numbers, and
If the function is continuous everywhere, then
The linear approximation to the function of at is given by
is a critical point of
For the following exercises, sketch the function in one graph and, in a second, sketch several level curves.
For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it.
For the following exercises, find the largest region of continuity for the function.
For the following exercises, find all first derivatives, full or partial, as appropriate.
For the following exercises, find all second partial derivatives.
For the following exercises, find the equation of the tangent plane to the specified surface at the given point.
Approximate at Write down your linear approximation function How accurate is the approximation to the exact answer, rounded to four digits?
Find the differential of and approximate at the point Let and
Find the directional derivative of in the direction
Find the maximal directional derivative magnitude and direction for the function at point
For the following exercises, find the gradient.
For the following exercises, find and classify the critical points.
For the following exercises, use Lagrange multipliers to find the maximum and minimum values for the functions with the given constraints.
A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of in height and in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height cm and radius cm.
A trash compactor is in the shape of a cuboid. Assume the trash compactor is filled with incompressible liquid. The length and width are decreasing at rates of ft/sec and ft/sec, respectively. Find the rate at which the liquid level is rising when the length is ft, the width is ft, and the height is ft.