MTH 206 Final

Lebanese American University Summer I Byblos 2007 Calculus IV Final exam Date: 30/07/2007 Duration: 2h 1. We consider the velocity field F = (zex + ey) i + (xey + ez) j + (yez + ex) k. (a) Prove the F is conservative. (b) Find a potential function for the field F. (c) Find the flow of F over the curve r(t) = ln(1+t2) i+ln(2+t) j+ln(3+2t2) k from t = 0 to t = 1. 2. We denote by V the volume of the region bounded below by the the paraboloid z = x2 + y2 and above by the sphere x2 + y2 + z2 = 2. (a) Express, without evaluating, the volume V as a triple integral using: i. the rectangular coordinates, ii. the cylindrical coordinates, iii. the spherical coordinates. (b) Find the volume V by evaluating one of the three triple integrals. 3. Find the extreme values of the function f(x, y, z) = x − y + z on the sphere unit sphere x2 + y2 + z2 = 1. 4. We consider the function f(x, y) = x4 + y4 − 4(x − y)2. (a) Test the function f(x, y) for local maxima and minima and saddle points. (b) Prove that the point where the test fails in (a) is in fact a saddle point. (Hint: Find f(x, x) and f(x, 0) and conclude.....) 5. Find the integral of f(x, y, z) = x2 − y + z over the curve C = C1 ∪ C2 where • C1 is the part of the curve x2 + y2 = 1 in the y ≥ 0 part of the xy-plane from (−1, 0, 0) to (1, 0, 0) • C2 is the line segment in xz-plane from (1, 0, 0) to (0, 0, 1) 6. Find the area of the band cut from the paraboloid x2 + y2 − z = 0 by the planes z = 2 and z = 6. 7. Bonus question. Find the largest volume of a rectangular box if its surface is known to be 24. MARKS : 1. [15] 2. [30] 3. [15] 4. [15] 5. [10] 6. [15] 7. [10] 1 LSF