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Unformatted text preview: df = f x ( x,y ) 4 x + f y ( x,y ) 4 y D u f ( x ) = ∇ f ( x )  u If f ( x,y ) ≥ 0 on a region R , then R R R f ( x,y ) dx dy ≥ 0. If a and b are parallel, then a × b = 0. (0,1) is an interior point of the ball B = { ( x,y ) : x 2 + y 2 ≤ 1. 7) Find the curvature κ for y = cos( x ) at (0,1). Use the formula for plane curves if you know it. If not, you may be able to work with the formula for space curves, κ =  r × r 00  /v 3 . 8) Use integration in polar coordinates to ﬁnd the volume of the solid bounded by z = 8r 2 and z = r 2 . 3 9) Use LaGrange multipliers to ﬁnd the max and min values for f ( x,y,z ) = xy +2 z subject to the constraint x 2 + y 2 + z 2 = 36. 10) Choose ONE. A) State and prove the Divergence Theorem. B) State and prove the Fundamental Theorem for Line Integrals. Bonus (5pts)) Compute the integral R + ∞ ex 2 dx , as done in class, showing all work. 4...
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 Fall '06
 GRANTCHAROV
 Derivative, Fundamental Theorem Of Calculus, Vector Space, Force, cylinder x2

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