April 11, 2006 MATH 200 Name Page 2 of 9 pages
 1. If two resistors of resistanc1 Rd R 2re wired in parallel, then the resulting
1 1 1
resistance R satisﬁes the equation = + . Use the linear or diﬀerential
R R 1 R 2
approximation to estimate the change in R 1f Rcreases from 2 to 1.9 ohms and
R increases from 8 to hm.o
Continued on page 3 April 11, 2006 MATH 200 Name Page 3 of 9 pages
 2. Assume that the directional derivative of w = f(x,y,z)a tapoit P is a maximum
in the direction o√ the vector 2i − j + k, and the value of the directional derivative in
that direction is 3 6.
(a) Find the gradient vector of w = f(x,y,z)a t P.[%]
(b) Find the directional derivative of w = f(x,y,z)at P in the direction of the vector
i + j
Continued on page 4 April 11, 2006 MATH 200 Name Page 4 of 9 pages
 3. Use the Second Derivative Test to ﬁnd all values of the constant c for which the
function z = x2 + cxy + y has a saddle point at (0,0).
Continued on page 5 April 11, 2006 MATH 200 Name Page 5 of 9 pages
 4. Use the Method of Lagrange Multipliers (no credit will be given for any other method)
to ﬁnd the radius of the base and the height of a right circular cylinder of maximum
2 2 2
volume which can be ﬁt inside the unit sphere x + y + z =1.
Continued on page 6 April 11, 2006 MATH 200 Name