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Unformatted text preview: Math 105 Second Midterm November 14, 2011 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 10 pages including this cover. There are 8 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor when you hand in the exam. 4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work (including appropriate explanation) for each problem, so that graders can see not only your answer but how you obtained it. Include units in your answer where that is appropriate. 6. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. 7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 8. Turn off all cell phones and pagers , and remove all headphones. 9. You must use the methods learned in this course to solve all problems. Problem Points Score 1 14 2 12 3 10 4 13 5 13 6 10 7 14 8 14 Total 100 Math 105 / Exam 2 (November 14, 2011) page 2 1 . [14 points] No work or explanation is expected on this page. a . [4 points] If the graph of the function y = k ( w ) has a vertical asymptote w = 5 and a hor- izontal asymptote y = 3, then what, if any, asymptotes does the graph of y = 2 k (4 w ) + 3 have? (Write None in the blank if no asymptotes of that type can be found from the information provided.) Solution: The graph of y = 2 k (4 w ) + 3 is obtained from the graph of y = k ( w ) by first compressing horizontally by a factor of 1 / 4 then stretching vertically by a factor of 2, and finally shifting the graph up by 3 units. These transformations send the vertical asymptote w = 5 to w = 5 / 4 and the horizontal asymptote y = 3 to y = 9. Vertical: w = 5 / 4 Horizontal: y = 9 b . [2 points] If the function f ( q ) has a zero q =- 2, find a zero of the function 4 f (3( q- 1)). Solution: The graph of 4 f (3( q- 1)) is obtained from the graph of f ( q ) by first com- pressing horizontally by a factor of 1 / 3 (which sends q =- 2 to q =- 2 / 3), then shifting right by 1 unit (taking q =- 2 / 3 to q = 1 / 3) and finally stretching vertically by a factor of 4 (which does not change the horizontal intercepts). Hence, q = 1 / 3 is a zero of 4 f (3( q- 1))....
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This note was uploaded on 01/28/2012 for the course MATH 105 taught by Professor Rhea during the Fall '08 term at University of Michigan.
- Fall '08