MATH 222 Name___________________________
Calculus and Analytic Geometry III Student#______________________
Summer 2003

Final Examination
Take Home Part

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).

On the take home part, you may used your notes, text and other references, but you may not consult with other individuals, whether student or faculty.

The take home part is due at the beginning of the in class final on 1 August 2003.


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

1. [15] Find the work done by the force field F = y i + z j + x k
in moving a particle along the curve C given by r(t) = (t2,t 3,t 4)
from (0,0,0) to (1,1,1).
2. [18] Find the equations of the following planes.
a) the plane containing the points (1,0,-1), (0,-1,1) and (-1,1,1)









b) the plane containing (2,5,-1) and parallel to the plane
x + 2y - 4z = 1











c) the plane tangent to x 2 + y 2 - z 2 = 4 at (3,2,-3)








3. [25] A particle is moving in 3 space subject to a constant acceleration of a = i - 5k, with an initial velocity of v0 = j and an initial position of r0 = 100k.
a) Find its equation of motion (that is position as a function of time)






b) Find its speed as a function of time





c) Find the tangential component of its acceleration at time t = 1.





d) Find the curvature of its trajectory at time t = 1.





e) Find the normal component of its acceleration at time t = 1.




4. [18] Find the maximum and minimum values of the function f (x, y ) = x 2 - y 3 + 3y on the region bounded by the circle of radius 4 about the origin by using the following steps

a) Find all critical points of f (x, y ), and eliminate any which lie outside the region.






b) Use Lagrange multipliers to identify possible locations of maxima or minima on the boundary.










c) Find the value of f (x, y ) at each point found in a) and b) and identify the maximum and minimum values.









5. [15] A body occupies the region bounded below by the cone
 and above by the sphere of radius 4 about the origin. If the density is given by , calculate the total mass by choosing a convenient coordinate system (not rectangular) and calculating a triple integral.