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exam1_solnsw10

# exam1_solnsw10 - Math 116 First Midterm February 8 2010...

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Math 116 — First Midterm February 8, 2010 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 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 00 × 5 00 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. Problem Points Score 1 11 2 13 3 8 4 12 5 10 6 12 7 7 8 6 9 6 10 15 Total 100

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Math 116 / Exam 1 (February 8, 2010) page 2 1 . [11 points] There is a classic result in mathematics, which states that the number of prime numbers less than any number x 2 is approximated by the function li( x ) = R x 2 dt ln t . a . [3 points] Is li( x ) increasing, decreasing, or neither for x 2? Provide justiﬁcation for your answer. Solution: This function is increasing. To see that it is increasing, we take its derivative d dx li( x ) = d dx Z x 2 dt ln t = 1 ln x > 0 . -OR- Since ln x is positive for x 2, we know f ( x ) = 1 ln x > 0. Since f ( x ) is the derivative of li( x ), we know li( x ) is increasing. b . [3 points] Is li( x ) concave up, concave down, or neither for x 2? Provide justiﬁcation for your answer. Solution: This function is concave down. To see it is concave down, take the second derivative: d 2 dx 2 li( x ) = d dx 1 ln x = - 1 x ln 2 x < 0 . -OR- Since ln x is an increasing function, we know f ( x ) = 1 ln x is decreasing, which means f 0 ( x ) < 0. Since f 0 ( x ) is the second derivative of li( x ), we know li( x ) is concave down. c
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exam1_solnsw10 - Math 116 First Midterm February 8 2010...

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