final265fa2004

# final265fa2004 - z = 2-x 2-y 2 and z = x 2 y 2 if the...

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Math 265 Final Exam 15 Dec 2004 NAME: Instructor: Section: Instructions: Answer each question completely. Show all work. No credit allowed for mere answers with no work shown. Show the steps of calculations. State the reasons that justify conclusions. 1. (a) Find parametric equations for the line through (2 , - 3 , 1) parallel to the vector from h 2 , 0 , 0 i to h 4 , 3 , 1 i . (b) Find an equation for the plane through the origin that is perpendicular to the line of part (a). 2. (a) Compute the gradient of F ( x, y, z ) = x 2 + e x sin y - z at (0 , 0 , 0). (b) Find the equation of the tangent plane to the surface z = x 2 + e x sin y at (0 , 0 , 0).

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Math 265 Final Exam Page 2 3. Evaluate I C (2 x 2 + 1) dx + ( xy ) dy where C is the circle of radius 2 with center at the origin, oriented counterclockwise. 4. Find a vector in the direction in which f ( x, y, z ) = x 2 y 3 z + 3 z 2 increases most rapidly at p = ( - 2 , 1 , - 1). What is the rate of change in this direction?
Math 265 Final Exam Page 3 5. Set up (but do not evaluate) an integral in cylindrical coordinates for the mass of the solid lying between the paraboloids

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Unformatted text preview: z = 2-x 2-y 2 and z = x 2 + y 2 if the density function is δ ( x, y, z ) = z . 6. Reverse the order of integration in the integral Z 1 Z y 2 f ( x, y ) dx dy . Math 265 Final Exam Page 4 7. For the vector feld F ( x, y ) = h 3 x 2 y 2 + 2 y, 2 x 3 y + 2 x i , either fnd a potential ±unction f so that F = ∇ f or explain why no such potential ±unction exists. 8. Find and classi±y the critical points o± f ( x, y ) = x 2 + y 2 + x 2 y + 4. Math 265 Final Exam Page 5 9. Given the planes x + y-z = 2 and 3 x-4 y + 5 z = 6: (a) Find the angle between them. (b) Find symmetric equations for their line of intersection. Math 265 Final Exam Page 6 10. Let S be the surface z = 4-x 2-y 2 with z ≥ 0. (a) Compute the surface area of S . (b) Let F = h-y, x, xyz 4 i and n the upward normal vector on S . Compute the surface integral RR S ∇ × F · n dS ....
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final265fa2004 - z = 2-x 2-y 2 and z = x 2 y 2 if the...

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