MATH 222 Name_________________________
Calculus and Analytic Geometry III Student#_____________________
Summer 2003

In-Class Examination II

Show all work. If you must continue a problem outside of the space provided, leave clear instructions where to find the additional work. Work outside the area provided will be ignored in assigning (partial) credit if instructions to its location are not found in the space provided. Correct answers without sufficient supporting work will be worth at most 1 point for problems or parts of problems with 5 points or less, and at most 2 points for other problems. Point values of problems follow the question number. Vector or numerical calculations done using a calculator must be documented (“using a calculator we obtain. . .”). If a graph is used (e.g. to decide the direction of a principal normal vector) the graph must be shown as part of the work. Provide exact solutions (e.g. 2p not 6.28..., or arccos(1/3)-1 not 0.23).

Vectors are denoted in bold-face (e.g. u, v, i, j, k), scalars in italic (e.g. v, x, k).

1. [8] Suppose f(x,y,z) is a differentiable function, and we have
substituted differentiable functions x = X (u,v), y = Y (t,v),
and z = Z (t). Use the chain rule to give the partial derivatives of f with respect to the new variables t, u, and v. Avoid writing any terms which are certain to be zero for any situation of this sort.

2. [12] Use the following steps to find and classify the critical points of

f (x,y ) = x 3 + 3xy - 6y 2

a) Find the partial derivatives fx and fy.







b) Find the second partial derivatives fxx , fxy=fyx and fyy.








c) Find all critical points.

2. (continued)
d) Classify each point found in c) as a local minimum, local maximum or saddle point, or indicate that the second derivative test fails to yield any information.









3. [10] a) Find the gradient of the function

f (x,y,z) = xz + xy + yz


at the point (1,2,2)









b) Find the directional derivative of f at (1,2,2) in the direction (12/13, -4/13, -3/13).


4. [10] Find

ÚÚ y dA , where R is the region bounded by the line y = x,
R
the y -axis, and the circle x2 + y2 = 4, and lying in the first quadrant.
5. [12] Write, but do not compute , iterated integrals to express
 ÚÚÚ x2 + y2 + z2 dV , where R is the region bounded by the
R
unit sphere centered at the origin and lying in the first octant (all rectangular coordinates non-negative),

a) in rectangular coordinates







b) in cylindrical coordinates








c) in spherical coordinates
6. [10] Find the centroid of the plane lamina covering the unit disk {(x,y) | x2 + y2 £ 1}, with density given by

d(x,y) = x + y + 2.

(Hints: Reflection in the line x = y is a symmetry of both the disk and the density function, so you should only need to do two double integrals. I’d use polar coordinates, but rectangular could work too.)

7. [10] Find the volume of the solid inside both the sphere


and the cylinder

.

8. [8] Find an equation for the line normal to the surface


at the point (1,-3,-2).

9. [10] Find the surface area of the portion of the circular
paraboloid given in cylindrical coordinates by z = 2r2 and
lying between the planes z = 2 and z = 10. (Hint: use two
of the three cylindrical coordinates of parameters.)
10. [10] Use Lagrange multipliers to find the maximum and minimum values of the function f (x,y ) = x + y 2 on the curve
x 2+ y 2 = 25.

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