Math 281                                                                      Name: _________________

Test #3: Sections 15.6, 15.7, 16.2 – 16.4                         Date  : _________________

 Directions:  Show all work for full credit. Use only Calculus III techniques to complete problems. No TI-Voyage200, 89, or 92 graphing calculators allowed. You may not use notes, textbooks, or any other reference on this test. Be sure to show all of your supporting work carefully, clearly, and in proper mathematical form. Solutions offered without such supporting work will not receive any credit.

1. Given  where R is the semicircle bounded by and

                  y = 0.

1. (2 pts) Sketch region R.

2. (3 pts) Set up the double integral where dA = dx dy

3. (3 pts) Set up the double integral where dA = dy dx

4. (3 pts) Set up the double integral where dA =

5. (3 pts) Evaluate the easier integral.

2. (12 pts) Find the volume under the plane  and over the rectangle given by . Include a sketch of region R.

3. a.   (6 pts) Switch the order of integration and then evaluate. Include a sketch of the region of integration.

b.       (6 pts) Convert to polar coordinates and evaluate. Include a sketch of the region of

integration.

4. (12 pts) Find the volume of the solid bounded by  and the planes z = 0 and

      z = 3 – x. Include a sketch of the base of the solid.

5. (12 pts) Set up but do not evaluate the following double integral for both orders of integration where R is the triangle bounded by y = x, y = 2x, and x = 2. Include a sketch of R.

6. (14 pts) Find the absolute extrema of on the triangle bounded by x = 0, y = 0, and y = 9 – x.

7. (12 pts) Locate all local extrema and saddle points of the graph of the function .

8. Given the function , complete parts ‘a’ – ‘d’:

a.       (3 pts) Find .

b.       (3 pts) Find the direction of maximum change in f(x,y) at (-2, 0).

c.       (3 pts) Find the maximum value of the directional derivative of f(x,y) at (-2, 0).

d.       (3 pts) Compute the directional derivative of f(x, y) at the point (-2, 0) in the direction of the unit vector that makes an angle of  with the positive x-axis.

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