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m23oldfin,s03

# m23oldfin,s03 - the boundary of the region and a clear...

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MATH 23 Practice Exam Questions August, 2001 (last exam in summer course) 1. (10 points ) Let -→ r ( t ) be the vector function -→ r ( t ) = 3 ti - t 2 j - 2 t 3 k. Find an equation for the tangent line at the point P (3 , - 1 , - 2) . 2. (15 points ) Find all critical points of the function h ( x, y ) = x 2 + y 2 + xy 2 + 2 . For each critical point determine whether it is a local maximum, local minimum or a saddle point. 3. (10 points ) Evaluate ZZZ E xz dV where E is bounded by the planes z = 0 , z = y and the cylinder x 2 + y 2 = 4 in the half-space y 0 . 4. (10 points ) Compute Z C y dx + xy dy if C goes from (0 , 0) to (1 , 2) along the line parameterized as x = t, y = 2 t. 5. (10 points ) (a) Find a function f ( x, y ) so that F ( x, y ) = e 3 y i + (2 + 3 xe 3 y ) j is the gradient of f. (b) Use your answer to part (a) to evaluate the Z C F · dr along the path r ( t ) = te t i + (1 + t ) j, 0 t 2 . 6. (15 points ) (a) Let C be the triangle with vertices (1 , 0) , (3 , 0) and (3 , 2) traveled in the counter-clockwise direction. Use Green’s Theorem to evaluate the integral
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Unformatted text preview: the boundary of the region and a clear statement giving the double integral that results from applying Green’s Theorem. (Partial credit: For a line integral with P dx + Q dy, start by ﬁnding Q x-P y . ) 7. (15 points ) Use the Divergence Theorem to ﬁnd Z S ± F · ±n dS when ± F = x 2 ± i + (2 y + z ) ± j + (3 z + x ) ± k, S is the surface of the cube bounded by x = 0 , x = 1 , y = , y = 2 , z = 1 and z = 3 , and ±n is the outer normal. 8. (15 points ) Use Stokes’ Theorem to evaluate the line integral Z C ± F · d± r, when ± F = xz ± i +3 xy ± j +3 xy ± k, and C is the boundary of the part of the plane 3 x + y + z = 3 in the ﬁrst octant (oriented counterclockwise when viewed from above)....
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