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Unformatted text preview: Math 115 First Midterm October 11, 2011 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 12 pages including this cover. There are 10 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. You are also allowed two sides of a 3 5 note card. 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 12 2 12 3 12 4 12 5 6 6 5 7 10 8 9 9 12 10 10 Total 100 Math 115 / Exam 1 (October 11, 2011) page 2 1 . [12 points] For each part below, give an explicit formula for a function which satisfies the given properties, if one exists. If such a function does not exist, explain why. Be sure to clearly indicate your final answer for each part. a . [3 points] A continuous function, f , which is not differentiable. Solution: The function f ( x ) = | x | is continuous, but not differentiable. The function is continuous as it can be drawn without picking up ones pencil, but not differentiable because there is a corner on the graph at the point (0 , 0). b . [3 points] A cubic polynomial, p , with two x-intercepts. Solution: The function p ( x ) = x 2 ( x- 1) = x 3- x 2 is a cubic polynomial with x-intercepts at x = 0 , 1. c . [3 points] A continuous function, c , satisfying lim x + c ( x ) =- 1 and lim x - c ( x ) = 1. Solution: The function described here does not exist. If lim x + c ( x ) =- 1 and lim x - c ( x ) = 1, then lim x c ( x ) does not exist since the right and left hand limits are not equal. The func- tion c ( x ) is continuous at zero means the limit as x 0 exists and equals c (0). If the right hand limit and the left hand limit are not the same, lim x c ( x ) does not exist, and so c ( x ) cannot be continuous at x = 0....
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