MATH 599 Midterm: MATH 5B UCSB 5B Fall 06 5bMidterm 1 Sample1
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1: (22 points) 1)(15 points) find the di erential dw of the function w = f (x, y, z) = x2yz + xy3ez. Solution 1) we have fx = 2xyz + y3ez, fy = x2z + 3xy2ez, fz = x2y + xy3ez. Hence dw = fxdx+fydy+fzdz = (2xyz+y3ez)dx+(x2z+3xy2ez)dy+(x2y+xy3ez)dz: at x0 = 1, y0 = 1, z0 = 0 we have by the above result dw = (2 1 1 0+13 e0)dx+(12 0+3 1 12 e0)dy+(12 1+1 13 e0)dz = dx+3dy+2dz. We have f (1, 1, 0) = 1. By 2) we then deduce f (1. 1, 1. 1, 0. 2) 1 + 0. 8 = 1. 8. 2. problems will be graded if you do three: (15 points) find the equation of the plane which contains the points. A = (3, 1, 1), b = (1, 2, 3) and c = (2, 0, 2). Solution 1) let v denote the vector from a to b, and w the vector from.
