Calculus 3 Practice Test 4

Show all work and justify each step.

(1) Write the number 12 as the sum of three positive numbers in such a way that the product is a maximum.

(2) Use iterated integration to compute calculus 3 practice test 4 _gr_3.gif] over the region

(3) Find the volume under the plane and above the region bounded by the lines and

(4) Evaluate the integral

.

(5) Find the surface area of the portion of the sphere that lies above the plane

(6) Find the volume of the solid region bounded by the sphere and the paraboloid

(7) Evaluate calculus 3 practice test 4 _gr_15.gif] where is the solid sphere

(8) Show that the function is a solution of the equation

(9) Find an equation of the tangent plane to the surface at the point

(10) Use differentials to find an approximate value for

(11) Determine whether the function is differentiable at either or Explain why.

(12) If where show that

(13) Let be defined by Find the directional derivative of at in the direction toward the origin.

(14) Find the maximum rate of change of at the point and the direction in which it occurs.

(15) Find the absolute extrema of on the closed bounded set in the plane described by:   is the square region with vertices

(16) Find the maximum of subject to and

(17) Verify that the function is a solution of the three-dimensional Laplace equation

(18) Find an equation of the tangent plane to the surface at the point

(19) Use differentials to find an approximate value for

(20) Determine whether the function is differentiable at either or Explain why.

(21) If where and show that

(22) Let be defined by Find the directional derivative of at in the direction toward the origin.

(23) Find the maximum rate of change of at the point and the direction in which it occurs.

(24) Find the hottest and coldest points on the metal plate whose temperature is given by

(25) Use Lagrange multipliers to find the point on the line of intersection of the planes and that is closest to the origin.