pfinal - Practice Problems for the Final Fall 2006 1 Find...

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Practice Problems for the Final, Fall 2006 1. Find the plane that contains the lines L 1 : x = 1+ t, y = 1 t, z = t and L 2 : x = 1+3 t, y = 3 t, z = 3 3 t. 2. Find and classify the critical points of the function f ( x, y ) = 4 x 2 e y 2 x 4 e 4 y . 3. Find the absolute maximum of the function f ( x, y, z ) = x 2 yz on the triangle cut by the plane x + y + z = 12 and the first octant. 4. A thin wire lies along the curve r ( t ), 0 t 1. The wire density δ (its mass per unit length) increases linearly along the wire and is δ ( r ) = s + 1, where s is the arc length parameter, and s = 0 corresponds to the t = 0 endpoint of the wire. Determine the wire mass if | v ( t ) | = t 2 +1. 5. Consider the gravitational field: F = GMm ( x i + y j + z k ) ( x 2 + y 2 + z 2 ) 3 / 2 , where G, M, m are constants. (a) Show that the field is conservative. (Hint: Set r = x 2 + y 2 + z 2 , and use that ∂r ∂x = x r .) (b) Find a potential function f ( x, y, z ) for F . (Hint: Try f of the form C/r where C is a constant.) (c) Let P and Q be points at distances s and t from the origin. Find the work done by the
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