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Final sample(2)

# Final sample(2) - Final sample exam If you have any...

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Final sample exam If you have any questions, or think you’ve spotted an error / typo, please get in touch with me. Solutions will appear by the end of Tuesday (hopefully before, but that’s probably excessively optimistic). 1. Find an equation describing the collection of points that is equidistant between the sphere center the origin and radius 2, and the origin itself. (6 points) 2. Find a scalar equation for the plane that is perpendicular to the line segment from P = (1 , 2 , 3) to Q = (3 , 2 , 1) and contains Q . (6 points) 3. Find the cosine of the angle between the plane containing the points (1 , 2 , 3), (3 , 2 , 1) and (0 , 0 , 0), and the xy -plane. (6 points) 4. Are the lines ~ r 1 ( t ) = h 1 + t, 2 , 1 - t i and ~ r 2 ( t ) = h 3 t, t, - 2 t i parallel, skew, or do they intersect at a unique point? If the last, find the point. (8 points) 5. Let P denote the parallelepiped formed from the vectors ~a = h 1 , 0 , 0 i , ~ b = h 0 , 1 , 0 i and ~ c = h 0 , 1 , 1 i . (a) Use a triple integral to write down an expression for the volume of P . (10 points) (b) Using your answer to part (a), or otherwise, compute the volume of P . (5 points) 6. The force due to gravity coming from a planet at the origin is given by ~ F = f , where f ( x, y, z ) = 1 p x 2 + y 2 + z 2 . A spaceship flies along a (clockwise) circular path in the yz -plane at dis- tance 2 from the planet, starting at the point (0 , 2 , 0). (a) Show that the direction of motion of the spaceship is always perpen- dicular to the gravitational forcefield. (7 points) (b) What can you conclude about the integral R C ~ F · d~ r , where C is the path followed by the spaceship? (2 points) (c) Say the spaceship flies instead around the (complete) circle C 0 , center (0 , 0 , 4), radius 3 and parallel to the xy -plane. What can you say about R C 0 ~ F · d~ r ? Justify your answer (in no more than 30 words). (5 points) 7. Say ~ r ( t ) = h 3+ t, 2 , 1 - t i , 1 t 4 represents the path of a sloth through the jungle (... a particularly clear bit of jungle where he can go in a straight line, so not terribly realistic). 1

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(a) Find the arclength function s ( t ). What does this represent physi- cally? (5 points) (b) Find ~ r 00 ( t ). What does this tell you about the Sloth’s motion? (3 points) 8. Let f ( x, y ) = xy x 2 + y 2 . Is f continuous at the origin? Justify your answer. (5 points) 9. Let f ( x, y ) be a smooth (i.e. infinitely differentiable) function of two vari- ables, which is defined on all of R 2 and has at least two critical points. For each of the following, label T (true for all such f), F (false for all such f), or M (true for some such f and not for others). (2 points each - partial credit for giving a sensible reason for a wrong answer). (a) f xyx = f yxy . (b) The graph of f is unbounded in the z direction. (c) The graph of f is unbounded in the x direction.
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