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S09fnlan-2

S09fnlan-2 - ρ ≤ 2 b cos φ if the density at each point...

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SAMPLE FINAL EXAM Answer all questions. You must show your working to obtain full credit. Total point score is 200 points. 1. (a) Find the equation of the line through the point (1 , 4 , 6) perpendicular to the plane 3 x 2 y + z = 7. [6 points] (b) Find the distance between the planes 3 x 2 y + z = 7 and 3 x 2 y + z = 2. [6 points] 2. Find the equation of the tangent line to the curve r ( t ) = ln(1 + t ) i + e t j +(2+ t ) k at the point given by t = 0. Where does this tangent line meet the plane z = 0? [12 points] 3. Let f ( x, y, z )= xy 2 + y 2 z 3 + xz 3 . (a) Find the directional derivative of f at (2 , 1 , 1) in the direction of the greatest rate of increase of f . [8 points] (b) Find the equation of the tangent plane to the surface f ( x, y, z ) = 5 at the point (2 , 1 , 1). [8 points] 4. Find the maximum value of xyz 3 on the unit sphere x 2 + y 2 + z 2 = 1. [20 points] 5. Sketch the region D which gives rise to the repeated integral ° 1 1 / 3 ° x x 2 f ( x, y ) dy dx and change the order of integration. [16 points] 6. Use polar coordinates to evaluate °° D x 3 dA where D is the region in the ±rst quadrant between the circles x 2 + y 2 = 1 and x 2 + y 2 = 2. [16 points] 7. This question uses spherical coordinates. Find the mass of the ball
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Unformatted text preview: ρ ≤ 2 b cos φ if the density at each point ( ρ, θ, φ ) (using spherical coordinates) is ρ . [20 points] 8. Let F ( x, y ) = (2 xy + y 3 ) i + ( x 2 + 3 xy 2 + 3 y ) j . (a) Find a function f such that ∇ f = F . [10 points] (b) Evaluate ° C F · d r along the path C : r ( t ) = (1+2 t 2 ) i +( t 3 +2) j , 0 ≤ t ≤ 1. [8 points] 9. Use Green’s theorem to evaluate ° C ( x 4 − x 2 y 2 + 3 y 2 ) dx + ( x + 2 y ) 2 dy where C is the rectangle with vertices (0 , 0), (1 , 0), (1 , 2), (0 , 2) (traversed in an anticlockwise direction). [18 points] 10. Find div F , curl F , and grad(div F ) for the vector ±eld F ( x, y, z ) = xz 3 i + y 2 j + xyz k . [12 points] 11. Evaluate ° ° S xy dS where S is the surface given by z = 1 − ( x 2 + y 2 ) / 2, 0 ≤ x ≤ 1, ≤ y ≤ 1. [20 points] 12. Use the divergence theorem to ±nd the total ²ux of F ( x, y, z ) = (2 x 2 z + x ) i − y 2 z 2 j + y 2 z k out of the region x 2 + y 2 ≤ 1, 0 ≤ z ≤ 2. [20 points]...
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