MATH 2374 Final: MATH 2374 UMN Spring 06Fall Final Exam
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135 : (8, 6, 6) (x 2, y 3, z 1) = 0, 8 . (a) 3 + 1. 2 ( 3(x 2)2 2(y 8)2 + 2(x 2)(y 8)). (note that f. Y are 0 at (2, 8) because it is a critical point. ) (b) local max at (2, 8). (d = 5 and fxx = 3. : 0. 7. (a) 0. 6. (use the chain rule. ) (b) 0. 18. (find the directional derivative. : 1. 2 sin3 d d d . (b) 32 . 9 . (use pythagorean theorem for the -integral. : this is the same as showing that the integral along c1 plus the integral along the opposite orientation of c2 is equal to 0. But this is the boundary of the cylinder s with outward-pointing normal, parametrized by ( , z) = (cos , sin , z), for 0 2 and 1 z 2. By stokes" theorem, the sum of line integrals equals.
