Math 192 Prelim 2, March 30, 2006Calculators arenotallowed. Write your section number and TA’s name on the front of your workbook.This exam is worth 100 points. Point values for each problem are in parentheses.Show all your work.1. (14) Letf(x, y, z) = cosxy+eyz+ lnxzandP= (1,0,1/2).i) Find the derivative offatPin the direction toward the pointQ= (2,2,5/2).ii) In what direction doesfincrease most rapidly atP? What is the value of the derivative in this direction?2. (12) Sketch and calculate the area of the region outside the circler=a√2 and inside the lemniscater2= 4a2cos 2θ.3. (12) Find parametric equations for the line tangent to the curve of intersection of the surfacesxyz= 1andx2+ 2y2+ 3z2= 6 at the point (1,1,1).4. (12) Find the volume of the wedge cut from the first octant by the parabolic cylinder
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