math 21a midterm 2 solution2

math 21a midterm 2 solution2 - 11/15/2007 SECOND HOURLY...

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Unformatted text preview: 11/15/2007 SECOND HOURLY SECOND PRACTICE Math 21a, Fall 2007 Name: MWF 9 Chen-Yu Chi MWF 10 Oliver Knill MWF 10 Corina Tarnita MWF 11 Veronique Godin MWF 11 Stefan Hornet MWF 11 Jay Pottharst MWF 12 Chen-Yu Chi MWF 12 Ming-Tao Chuan TTH 10 Thomas Barnet-Lamb TTH 10 Rehana Patel TTH 11:30 Thomas Barnet-Lamb TTH 11:30 Thomas Lam Start by printing your name in the above box and check your section in the box to the left. Do not detach pages from this exam packet or unstaple the packet. Please write neatly. Answers which are illeg- ible for the grader can not be given credit. No notes, books, calculators, computers, or other electronic aids can be allowed. You have 90 minutes time to complete your work. The hourly exam itself will have space for work on each page. This space is excluded here in order to save printing resources. 1 20 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total: 110 1 Problem 1) TF questions (30 points) Mark for each of the 20 questions the correct letter. No justifications are needed. 1) T F f ( x, y ) and g ( x, y ) = f ( x 2 , y 2 ) have the same critical points. Solution: The function g has always (0 , 0) as a critical point, even if f has not. 2) T F If a function f ( x, y ) = ax + by has a critical point, then f ( x, y ) = 0 for all ( x, y ). Solution: At a critical point the gradient is ( a, b ) = (0 , 0), which implies f = 0. 3) T F Given 2 arbitrary points in the plane, there is a function f ( x, y ) which has these points as critical points and no other critical points. Solution: Connect the two points with a line and take this height as the x-axes, centered at the midpoint and with units such that the two points have coordinates (-1,0),(1,0). The function f ( x, y ) = y 2 ( x 3 1) has the two points as critical points. One is a local max, the other is a saddle point. 4) T F If ( x , y ) is the maximum of f ( x, y ) on the disc x 2 + y 2 1 then x 2 + y 2 < 1. Solution: The maximum could be on the boundary. 5) T F There are no functions f ( x, y ) for which every point on the unit circle is a critical point. Solution: There are many rotationally symmetric functions with this property. 2 6) T F An absolute maximum ( x , y ) of f ( x, y ) is also an absolute maximum of f ( x, y ) constrained to a curve g ( x, y ) = c that goes through the point ( x , y ). Solution: The Lagrange multiplier vanishes in this case. 7) T F If f ( x, y ) has two local maxima on the plane, then f must have a local minimum on the plane. Solution: Look at a camel type surface. It has a saddle between the local maxima. 8) T F There exists a function f ( x, y ) of two variables which has no critical points at all. Solution: True. Every non-constant linear function for example....
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math 21a midterm 2 solution2 - 11/15/2007 SECOND HOURLY...

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