Question 1 Suppose that u , v are two vectors in R 3 such...

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Northwestern University NetID: ID # : Math 230 Final Exam Fall Quarter 2012 Wednesday, December 12, 2012 Complete the front page!!!! Mark your section!! [ 3 points! ] Do not write your name on this exam! Write your NetID and your Student ID Number in the spaces provided at the top of this page. Instructions: Show ALL of your work on the pages provided! Explain your reasoning using complete sentences and correct grammar, spelling, and punctuation. All numerical answers MUST be exact; e.g., you should write π instead of 3 . 14 ... , 2 instead of 1 . 414 ... , and 1 3 instead of 0 . 3333 ... Make sure that your final answer is clearly indicated. This test has seventeen pages. Check to make sure you have pages 1 - 17. Calculators, course notes, textbooks, etc. are NOT allowed. You have 2 hours! Good luck! X Sec. # Time Instructor 21 8:00 Garton 31 9:00 Bohmann 33 9:00 Tosatti 37 9:00 Yang 41 10:00 TiM 47 10:00 Tacy 49 10:00 Yang 57 11:00 Burslem 59 11:00 Garton 61 12:00 Burns 67 12:00 Tacy 71 1:00 Drummond-Cole 81 2:00 Burns 87 2:00 Drummond-Cole Question Possible Points Score Front Page 3 1 12 2 15 3 9 4 9 5 18 6 14 7 18 8 16 9 18 10 16 11 18 12 18 13 16 Total 200
Math 230 Final Exam Fall Quarter 2012 Page 2 of 17 Question 1. Suppose that u , v are two vectors in R 3 such that k u k = 3, k v k = 2, and the angle between them is θ = π 3 . (a) (3 points) Calculate u · v and k u × v k . (b) (5 points) Calculate ( u - v ) · ( u + 2 v ). (c) (4 points) Calculate ( u + 3 v ) · (3 u × v ).
Math 230 Final Exam Fall Quarter 2012 Page 3 of 17 Question 2. For each of the following, calculate the limit or explain why the limit does not exist: (a) (5 points) lim ( x,y ) (0 , 0) y 3 + 5 x 2 + 5 y 2 x 2 + y 2 (b) (5 points) lim ( x,y,z ) (0 , 0 , 0) yz x 2 + y 2 + z 2 . (c) (5 points) lim ( x,y ) (0 , 0) x 4 + y 4 p x 4 + y 4 + 1 - 1
Math 230 Final Exam Fall Quarter 2012 Page 4 of 17

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