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Test 3 sample(1)

# Test 3 sample(1) - Test 3 sample exam If you have any...

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Test 3 sample exam If you have any questions, or think you’ve spotted an error / typo, please get in touch with me. Some of the problems are a little tricky, but its good practice (and better I get ‘rid of’ those problems before the actual test). 1. (a) Find and classify the critical points of f ( x, y ) = x 3 - 3 x + y 2 . (6 points) (b) Use your answer to part (a) to find the global minimum and max- imum of this f on the closed disk, radius 1 and center (1 , 0). (8 points) 2. Find the volume of the region above the square with vertices (1 , 1), ( - 1 , 1), (1 , - 1), ( - 1 , - 1) and below the graph of the function f ( x, y ) = 3 x 2 +6 y 2 . (5 points) 3. Evaluate Z 1 0 Z 1 y 3 cos( x 3 ) dxdy. (8 points) 4. (a) Show that Z -∞ Z -∞ e - x 2 - y 2 dxdy = Z -∞ e - x 2 dx 2 . (3 points) (b) Use your answer to part (a) (and a shift to polar coordinates on the left hand side) to show that Z -∞ e - x 2 dx = π. (7 points) 5. Let R be the region in the positive octant, inside the cylinder x 2 + y 2 = 9 and below the paraboloid z = x 2 + y 2 . Using whichever coordinate system will give the simplest result, write down (a) a double integral (4 points), and (b) a triple integral (4 points) giving the volume of R . Find the volume of R . (3 points) 6. Let f ( x, y, z ) be a function of three variables. Write down the integral of f over the region in the positive octant bounded by the parabolic cylinder x = 1 - y 2 and the plane z = y + 2 in the orders (a) dzdydx (4 points), and 1

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(b) dxdzdy (4 points). 7. Let R be the region consisting of all points in the first octant which are at distance at most 2 from the origin. Write down triple integrals giving the volume of R in: (a) Cartesian coordinates (4 points); (b) cylindrical coordinates (4 points); (c) spherical coordinates (4 points).
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Test 3 sample(1) - Test 3 sample exam If you have any...

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