MATH 230 Final: Final winter 2013 solutions

• School: Northwestern University
• Department: Mathematics
• Course: MATH 230
• Professor: All

Document Summary

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 instead of 3. 14, 2 instead of 1. 414, and 1. 3 instead of 0. 3333 make sure that your nal answer is clearly indicated. This test has 12 questions on thirteen pages. Check to make sure you have pages 1 - ??. Calculators, course notes, textbooks, etc. are not allowed. 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. Each of the following equations (in either spherical or cylindrical coordinates) describes a sphere in r3. For each sphere, give an equation in rectangular coordinates that describes the surface. Find the radius and center of each sphere. (a) (6 points) r2 + z2 = 4 + 4r cos( ) 2r sin( )