math 21a midterm 2 solution2

# math 21a midterm 2 solution2 - SECOND HOURLY SECOND...

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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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## This test prep was uploaded on 04/06/2008 for the course MATH 21a taught by Professor Knill during the Fall '07 term at Harvard.

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math 21a midterm 2 solution2 - SECOND HOURLY SECOND...

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