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Unformatted text preview: Math 115 Final Exam December 17, 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 8 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 TI92 (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. 9. Use the techniques of calculus to solve the problems on this exam. Problem Points Score 1 10 2 13 3 12 4 14 5 12 6 10 7 14 8 15 Total 100 Math 115 / Final (December 17, 2010) page 2 1 . [10 points] Given below is a graph of h ( x ), the derivative of a function h ( x ). 1 2 3 1 2 3 x h H x L (a) On the axes below, sketch a possible graph of h ( x ). 1 2 3 1 2 3 x h H x L (b) List the xcoordinates of all inflection points of h . x = 1 , 1 , 2. (c) Give the xcoordinate of the global minimum of h on [3,3]. x = 0. (d) Give the xcoordinate of the global maximum of h on [3,3]. x = 3. Math 115 / Final (December 17, 2010) page 3 2 . [13 points] The Uvalue of a wall of a building is a positive number related to the rate of energy transfer through the wall. Walls with a lower Uvalue keep more heat in during the winter than ones with a higher Uvalue. Consider a wall which consists of two materials, material A with Uvalue a and material B with Uvalue b . The Uvalue of the wall w is given by w = ab b + a . Considering a as a constant, we can think of w as a function of b , w = u ( b ). a . [4 points] Write the limit definition of the derivative of u ( b ). Solution: The derivative of u ( b ) is defined to be u ( b ) = lim h u ( b + h ) u ( b ) h = lim h a ( b + h ) / ( b + h + a ) ab/ ( b + a ) h b . [4 points] Calculate u ( b ). (You do not need to use the limit definition of the derivative for your calculation.) Solution: By the quotient rule, u ( b ) = ( b + a )( a ) ab ( b + a ) 2 = a 2 ( b +...
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This note was uploaded on 01/28/2012 for the course MATH 115 taught by Professor Blakelock during the Fall '08 term at University of Michigan.
 Fall '08
 BLAKELOCK
 Math

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