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Unformatted text preview: M ATH 115 –F IRST M IDTERM E XAM February 6, 2007 NAME: SOLUTIONS INSTRUCTOR: SECTION NUMBER: 1. Do not open this exam until you are told to begin. 2. This exam has 9 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 show how you arrived at your solution. 8. Please turn off all cell phones and pagers and remove all headphones. PROBLEM POINTS SCORE 1 9 2 12 3 12 4 10 5 8 6 12 7 11 8 12 9 14 TOTAL 100 2 1. According to a survey by the UM Transportation Research Institute, gasoline prices are projected to reach $5.00 a gallon by the year 2020. (a) (5 points) Assuming that the average gas price in 2007 is $2.00 per gallon (yes, we know that is wishful thinking), find an exponential function, P , that models the average gas price t years after 2007. Show either an “exact” answer or at least 4 decimal places in your answer. Since the function we are looking for is exponential, we are looking for a function of the form P = ab t , where a and b are constants, passing through the points (0 , 2) and (13 , 5) . Using the point (0 , 2) we have 0 = ab ⇒ a = 2 . Now, using the point (13 , 5) , combined with what we just showed we have 5 = 2 b 13 ⇒ b = ( 5 2 ) 1 13 . Thus the function we are looking for is P = 2( 5 2 ) t 13 or approximating to four decimal places P = 1 . 0730 t . (b) (2 points) What is the annual percent change in the average gas price according to this model? (Show to at least one decimal place.) To find the annual percent change in gas prices we note that the value b we calculated in (a) was approximately 1 . 0730 . Thus the annual percent change in gas prices is (to one decimal place) 7 . 3% . (c) (2 points) What is the yearly continuous percent rate of change for this model? (Show to two decimal places.) To find the continuous percent rate of change for the model we have to express P ( t ) in the form P ( t ) = ae kt , for constants a and k....
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This note was uploaded on 11/09/2010 for the course MATH CALC 115 taught by Professor Reit during the Spring '09 term at University of Michigan.
 Spring '09
 REIT
 Calculus

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