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Unformatted text preview: Math 116 Final Exam SOLUTIONS Name: Instructor: Section Number: 1. Do not open this exam until you are told to begin. 2. This exam has 11 pages including this cover. There are 9 questions. 3. Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you turn in the exam. 4. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 6. You may use your calculator. You are also allowed two sides of a 3 by 5 notecard. 7. If you use graphs or tables to obtain an answer, be certain to provide an explanation and sketch of the graph to make clear how you arrived at your solution. 8. Please turn off all cell phones and remove all headphones. Problem Points Score 1 12 2 10 3 9 4 8 5 12 6 18 7 12 8 9 9 10 Total 100 2 1. (12 points) The world shrimp production can be represented by the differential equation dP dt = . 1 P ( P 7) , where t is the number of years since 1982 and P ( t ) is the quantity of shrimp farmed in the world during year t in hundreds of thousands of metric tons. In 1982 the world shrimp production was 100,000 metric tons. (a) (3 pts.) Determine all of the equilibrium solutions of the differential equation given above. Classify each as either stable or unstable. No explanation required. P = 0 (unstable equilibrium;) P = 7 (stable equilibrium.) (b) (4 pts.) Sketch a graph of the solution to the given initial value problem. Be sure to indicate clearly on your graph where the solution curve is increasing/decreasing and where it is concave up/concave down. Clearly mark the value of any asymptotes. x P 1 3 . 5 7 inflection point (c) (3 pts.) Use Eulers method with t = 0 . 5 to estimate the world shrimp production in the year 1984 ( t = 2). t i P i slope at ( t i , P i ) P i = t i (slope at ( t i , P i )) 1 0.5 1 0.6 0.3 1 1.3 0.741 0.371 1.5 1.671 0.89 0.445 2 2.1155 So, the worlds shrimp production in 1984 was 211,550 metric tons. (d) (2 pts.) Is the estimate of world shrimp production in part (c) bigger or smaller than the exact solution to the initial value problem at t = 2? Explain in one sentence. The estimate is smaller than the actual value. The exact solution curve is concave up when P = 2 . 116, so a tangent linebased approximation to the actual solution curve yields an underestimate to the actual values. 3 2. (10 points) An apple is placed in a room whose air temperature is fixed at 50 F. Let T ( t ) be the temperature of the apple at time t , which is measured in hours. According to Newtons Law of, which is measured in hours....
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This note was uploaded on 04/12/2008 for the course MATH 116 taught by Professor Irena during the Spring '07 term at University of Michigan.
 Spring '07
 Irena
 Math

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