Calculus 3 Practice Test 2

Show all work and justify each step.

(1) The position vector of an object in space is

Find so that the sum of the object's tangential and normal components of acceleration equals half its speed.

(2) Determine a formula for and use it with

(3) Let where and Find

(4) Let where and Find and

(5) Let Find the directional derivative of at in the direction towards the point

(6) Let Find the directional derivative of at in the direction that makes an angle of with the positive -axis.

(7) Find the shortest distance from the origin to the surface assuming the extrema value exists.

(8) Find the minimum of subject to the constraint where and

(9) Let Show that has a minimum at on every line that passes through the origin. Then show that has no relative minimum at

(10) Suppose and Find the maximum of subject to the constraint

(11) Sketch the region of integration, exchange the order, and evaluate the integral using both orders of integration.

(12)  Sketch the region of integration, exchange the order, and evaluate the integral using both orders of integration.

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

(14) Find the volume under the surface and above by the region bounded by with and