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18.02 Pset 3

# 18.02 Pset 3 - B.2(After Sept 24 12 pts Let f x y:= x 2 6...

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18.02 HOMEWORK #3, DUE OCTOBER 7, 2010 BJORN POONEN 1. Part A (After Sept. 24) 2F-1b, 2F-3* (12 pts.), 2F-4* (6 pts.) (After Sept. 24) p. 887: 43 (After Sept. 24) 2G-1a* (10 pts.), 2G-5* (10 pts.) (After Sept. 24) p. 934: 13 (where do the critical points lie in the contour diagram?) (After Sept. 24) 2H-1d, 2H-4b (After Sept. 28) 2C-1b, 2C-5b (After Sept. 28) p. 904: 13, 35, 40* (10 pts.) (After Sept. 28) 2E-3c* (10 pts.), 2E-4 (After Oct. 1) p. 916: 31 (After Oct. 1) 2D-1b, 2D-2a, 2D-5, 2D-9 2. Part B B.1) (After Sept. 28, 10 pts.) A particle is moving in the plane so that its distance from the origin is increasing at a constant rate of 2 meters per second, and its argument θ is increasing at a rate of 3 radians per second. At a time when the particle is at (3 , 4), what is its velocity vector?
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Unformatted text preview: B.2) (After Sept. 24, 12 pts.) Let f ( x, y ) := x 2 + 6 xy + 9 y 2 + 5 (a) What are the critical points of f ( x, y )? (b) What does the second derivative test say about their type? (c) What is their type? (d) Describe the shape of the graph. B.3) (After Oct. 1) Suppose that two intersecting lines are both level curves for a diﬀeren-tiable function f ( x, y ). (a) (12 pts.) Prove that the point where the two lines intersect must be a critical point. (b) (8 pts.) Must it be a saddle point? (If YES, explain why; if NO, give a counterex-ample.) Reminder: Please write “Sources consulted: none” at the top of your homework, or list your (animate and inanimate) sources. See the course information sheet for details. 1...
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