Summer 2008

# Summer 2008 - MATH 53 - FINAL EXAM LECTURE 1, SUMMER 2008...

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MATH 53 - FINAL EXAM LECTURE 1, SUMMER 2008 August 15, 2008 Name : 1. Consider the following curve: x = t 2 1 ,y = t 3 3 t +2 , 2 t 2 . (a) (4 points) Show that the curve intersects itself at (2 , 2). (b) (4 points) Find the points where the curve has either a horizontal or vertical tangent. (c) (2 points) Sketch the curve.

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2. (a) (5 points) Show that the line integral of F ( x, y, z )= yze xyz i +( xze xyz + z ) j +( xye xyz + y + 1) k over a closed curve is zero. (b) (5 points) Compute the line integral of the above vector Feld over the line segment from (0 , 1 , 1) to (2 , 0 , 4).
3. (10 points) Compute the surface integral of the curl of F ( x, y, z )= y i +( x +( z 1) x x sin x ) j +( x 2 + y 2 ) k over the piece of the sphere x 2 + y 2 + z 2 = 2 above z = 1 with downward orientation.

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4. Let f ( x, y )= xe x +( y 1) 3 . (a) (5 points) Find all points where the rate of change of f in the direction of i is zero. (b) (5 points) Find all points where the rate of change of
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## Summer 2008 - MATH 53 - FINAL EXAM LECTURE 1, SUMMER 2008...

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