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fx265s2008

# fx265s2008 - 5(10 points Evaluate R 1 R 1 √ y √ x 3 1...

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Math 265 Final Exam May 7, 2008 Name: Instructor: TA: Section: Instructions: 8 questions. You have 120 minutes to complete this exam. Answer each question completely. Show all work. No credit given for mere answers with no work shown. 1. (15 points) a) Find the symmetric equations for the straight line passing through P (2 , 4 , 1) and Q ( - 1 , 0 , 3). Symmetric equation : b) Find the equation of the plane through the point (2 , 5 , 1) that is perpendicular to the line found in a). Equation : 2

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2. (10 points) A curve is given by the parametric equations x = t 3 +2 t , y = t 2 and z = 3 t 2 - t . Find the parametric equations for the tangent line through the point at t = 1. Parametric equations : 3. (10 points) Find the equation of the tangent plane to the surface x 3 - y 3 + z 2 = 1 at the point ( - 2 , 3 , 6). Equation : 3
4. (15 points) Find the minimum value of f ( x, y ) = x 2 + ( y - 2) 2 subject to the constraint x 2 - y 2 = 1 and find the point(s) at which the minimum is attained. Minimum value : Minimum attained at the point(s)

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Unformatted text preview: 5. (10 points) Evaluate R 1 R 1 √ y √ x 3 + 1 dxdy by interchanging the order of integration. R 1 R 1 √ y √ x 3 + 1 dxdy = R R dy dx = 4 6. (10 points) Find the mass of the solid inside the sphere x 2 + y 2 + z 2 = 9 and above the cone z = p x 2 + y 2 , if the density is δ ( x,y,z ) = z 2 . mass = 7. (15 points) Apply Green’s Theorem to evaluate H C ( x 2 cos x + 2 y ) dx +( y 3 + 5 x ) dy where C is the triangle with vertices (0 , 0), (1 , 0) and (0 , 2), traversed in the counterclockwise direction. H C ( x 2 cos x + 2 y ) dx +( y 3 + 5 x ) dy = 5 8. (15 points) Let S be the cylindrical region x 2 + y 2 ≤ 4, 0 ≤ z ≤ 3, and let n be the outward unit normal to the boundary ∂S of S . If F ( x,y,z ) = ( x 3 z + y 2 ) i + ( x 2 + y 3 z ) j + ( x 2 + y ) k , ﬁnd the ﬂux of F across ∂S . ﬂux = 6...
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fx265s2008 - 5(10 points Evaluate R 1 R 1 √ y √ x 3 1...

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