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273s09practice-final - MATH 273 PRACTICE FINAL EXAM NAME...

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MATH 273 PRACTICE FINAL EXAM NAME: 1. (9 points). Find the length of the curve r ( t ) = 1 2 t 2 i - 1 3 t 3 j + k , 0 t 1. Hint: to integrate, factor out t 2 and use substitution. 1
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2. (9 points). Use Stokes’ theorem to find C ¯ F · d r where ¯ F = xy i + ( x + y ) j + z k , and C is the boundary of the triangle with vertices (1 , 0 , 0) , (0 , 1 , 0) and (0 , 0 , 1). NOTE : This triangle lies in the plane z = 1 - x - y . The projection of this triangle onto the ( xy )-plane is another triangle, bounded by the x -axis, y -axis, and the line y = - x + 1. 2
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3. (9 points). Determine the critical points of f ( x, y ) = 1 3 x 3 + 1 3 y 3 - x 2 y - x . Determine whether they are points of a local extremum or saddle points. HINT : use difference of squares formula ( a 2 - b 2 = ( a - b )( a + b )) to factor the equation f y = 0. Then solve for y in terms of x , and substitute into f x = 0. This should give you two critical points. 3
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4. (9 points). Find the equation of the tangent plane to the surface z = f ( x, y ) = x 3 y + xy 2 + 2 at the point P (1 , 1 , 4).
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