Assignment2sol - MAT235-CALCULUS II, FALL 2010 ASSIGNMENT...

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Unformatted text preview: MAT235-CALCULUS II, FALL 2010 ASSIGNMENT #2, Solutions Problem 1 Identify and sketch the quadric surface consisting of all points ( x,y,z ) with x- y 2 +2 y + z 2 = 2. Justify your answer carefully. It is recommended that you try to sketch with pencil so that you can improve your sketch. Scanned solution is posted onto the next file Problem 2 Let A,B,C,D be four points in R 3 which are not necessarily coplanar. Let M be the midpoint of the segment AC , and N be the midpoint of the segment BD . If-- MB -- MD =-- NA -- NC, show that |- AC | = |-- BD | . Hint: Observe that- AC =-- NC--- NA, and-- BD =-- MD--- MB. (1) Also show that 2-- MN =-- MB +-- MD, 2-- NM =-- NA +-- NC . Explain why the last two relations imply that |-- MB +-- MD | = |-- NA +-- NC | (2) Use (1), (2) and the assumption of the exercise to show the desired result. The identity |- a | 2 =- a - a is useful. Solution : We have-- MN =-- MB +-- BN, (3) and-- MN =-- MD +-- DN. (4) Since-- BN =--- DN (because N is the midpoint of BD), by adding equations (3) and (4) we obtain that 2-- MN =-- MB +-- MD. Similarly, we obtain that 2-- NM =-- NA +-- NC. (5) The last two relations give that (-- MB +-- MD ) 2 = (-- NA +-- NC ) 2 (here by the square we mean...
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This note was uploaded on 12/05/2010 for the course MAT mat235 taught by Professor Jung during the Fall '10 term at University of Toronto- Toronto.

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Assignment2sol - MAT235-CALCULUS II, FALL 2010 ASSIGNMENT...

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