fin_2009S - f ( x,y ) = x + 6 y − 7 attains its maximum...

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MATH 210 Final Exam May 7, 2009 Name: 1. Let f ( x,y,z ) = ( x 2 + y ) z + x cos( y 2 z ). (a) Find the gradient f at the point P = (0 , 1 , 1). (b) Find the directional derivative D v f (0 , 1 , 1) where v is the unit vector from P towards Q = (2 , 3 , 0). 2. Consider the vector ±elds F = a y e xy + y 2 , xe xy + 2 xy A and G = a xe xy , ye xy A . (a) Which of the two vector ±elds is conservative and which is not? (justify) (b) Find a potential φ for the conservative among the vector ±elds. 3. Use Green’s theorem to compute c C xy 2 dx + ( x y ) dy where C traces the triangle with vertices (0 , 0), (1 , 0), (0 , 2) traversed in this order. 4. Let u = a 1 , 2 , 3 A and v = a 2 , 1 , 0 A . (a) What can be said about the angle between u and v : acute/obtuse/right? (b) Find an equation for the plane through (1 , 1 , 1) containing u and v . 5. Find the equation of the tangent plane to the level surface e xz + ( x + y ) 3 yz = 3 at the point (0 , 2 , 3). 6. Use the method of Lagrange multipliers to ±nd points where
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Unformatted text preview: f ( x,y ) = x + 6 y − 7 attains its maximum and minimum on the ellipse x 2 + 3 y 2 = 13. 7. Find all the critical values of f ( x,y ) = x 3 + 2 xy − 2 y 2 − 10 x and classify them into local maxima, local minima, and saddle points. 8. Let C be the curve parametrized by c ( t ) = a 3 t, 2 cos( t ) , 2 sin( t ) A for 0 ≤ t ≤ 2 π . (a) Find c ′ ( t ) and c ′′ ( t ). (b) Find the length of the curve. 9. Let H be the upper semi-ball x 2 + y 2 + z 2 ≤ 4, z ≥ 0. Compute i i i H z dV. 10. Change the order of integration and evaluate the iterated integral i 1 i 1 y 1 / 3 ( xy + sin( x 4 )) dxdy....
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This note was uploaded on 07/06/2011 for the course MATH 210 taught by Professor Slodowski during the Fall '08 term at Ill. Chicago.

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