Math 192 Prelim 2, March 30, 2006 Calculators are not allowed. 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) Let f ( x, y, z ) = cos xy + e yz + ln xz and P = (1 ,0 , 1 / 2). i) Find the derivative of f at P in the direction toward the point Q = (2 , 2 , 5 / 2). ii) In what direction does f increase most rapidly at P ? What is the value of the derivative in this direction? 2. (12) Sketch and calculate the area of the region outside the circle r = a √ 2 and inside the lemniscate r 2 = 4 a 2 cos 2 θ . 3. (12) Find parametric equations for the line tangent to the curve of intersection of the surfaces xyz = 1 and x 2 + 2 y 2 + 3 z 2 = 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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