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University of British Columbia
MATH 200

April 11, 2006 MATH 200 Name Page 2 of 9 pages Marks [15] 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 2 Continued on page 3 April 11, 2006 MATH 200 Name Page 3 of 9 pages [10] 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 [10] 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 [15] 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
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