exam1_solnFall08 - Math 116 - First Midterm October 14,...

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Math 116 — First Midterm October 14, 2008 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 9 pages including this cover. There are 9 problems. Note that the problems are not of equal diFculty, and it may be to your advantage to skip over and come back 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 ±nd 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 of all cell phones and pagers , and remove all headphones. Problem Points Score 1 10 2 12 3 14 4 16 5 6 6 6 7 12 8 12 9 12 Total 100
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page 2 Math 116 / Exam 1 (October 14, 2008) 1 . [10 points] A model 1 for the extinction rate of marine species during the Phanerozoic period (which extends from approximately 545 million years before the present until now) stipulates that this extinction rate, r ( t ), in numbers of marine animal families per million years, is r ( t ) = 3130 t + 262 , where t is the number of million years after the start of the Phanerozoic period. a . [5 points] Find an expression for E ( t ), the number of extinctions that occurred between the start of the Phanerozoic period and t million years thereafter. Solution: Using the fundamental theorem of calculus, this is just E ( t ) = Z t 0 3130 x + 262 dx = (3130 ln | x + 262 | ) ± ± ± t 0 = 3130 ln( t + 262) - 3130 ln(262) . (units are marine animal families). b . [5 points] Find an expression for the average rate of extinctions between the start of the Phanerozoic period and t million years thereafter. Solution: The average rate is A ( t ) = 1 t - 0 Z t 0 3130 x + 262 dx = 1 t (3130 ln( t + 262) - 3130 ln(262)) . (units are marine animal families/million years). 1 Decline in Extinction Rates and Scale Invariance in the Fossil Record , Paleobiology 25 :234–39 (1999)
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page 3 Math 116 / Exam 1 (October 14, 2008) 2 . [12 points] Let f ( x ) be a positive, continuous and diferentiable, non-constant Function. Let F ( x ) be an antiderivative oF f that passes through the origin. ±or each oF the Following, ²nd
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exam1_solnFall08 - Math 116 - First Midterm October 14,...

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