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 πinstead of 3.14...,√2 instead of 1.414..., and 1

3instead 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!

XSec. # 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,vare two vectors in R3such that kuk= 3, kvk= 2, and the angle

between them is θ=π

3.

(a) (3 points) Calculate u·vand ku×vk.

(b) (5 points) Calculate (u−v)·(u+ 2v).

(c) (4 points) Calculate (u+ 3v)·(3u×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)

y3+ 5x2+ 5y2

x2+y2

(b) (5 points) lim

(x,y,z)→(0,0,0)

yz

x2+y2+z2.

(c) (5 points) lim

(x,y)→(0,0)

x4+y4

px4+y4+ 1 −1