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Final_Solutions - M ATH 115 F INAL E XAM N AME I NSTRUCTOR...

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M ATH 115 –F INAL E XAM April 23, 2009 N AME : S OLUTIONS I NSTRUCTOR : S ECTION N UMBER : 1. Do not open this exam until you are told to begin. 2. This exam has 10 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. P ROBLEM P OINTS S CORE 1 14 2 8 3 9 4 10 5 8 6 14 7 16 8 10 9 11 T OTAL 100
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2 1. (2 points each) Suppose f is a twice-differentiable function. Use the graph of the derivative f , shown below, to answer the following questions. No explanations are required. y = f ( x ) A B C D E F (a) At which of the marked x -values does f attain a global minimum on the interval [A,F]? B (b) At which of the marked x -values does f attain a global maximum on the interval [A,F]? F (c) At which of the marked x -values does f attain a global minimum on the interval [A,F]? A (d) At which of the marked x -values does f attain a global maximum on the interval [A,F]? F (e) At which of the marked x -values does f ′′ attain a global maximum on the interval [A,F]? C (f) For which of the marked x -values does integraldisplay x A f ( t ) dt attain a global minimum on the interval [A,F]? B (g) For which of the marked x -values does integraldisplay x A f ( t ) dt attain a global maximum on the interval [A,F]? F
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3 2. (2 points each) Next to each of the functions graphed on the left below, identify which one of the inequalities on the right below best describes the situation. Here, L is the left Riemann sum for integraltext 6 0 f ( x ) dx using three equal subdivisions, and R is the right Riemann sum using three equal subdivisions. [ You may find it helpful to compute L , R , and the integral for each graph. ] 2 4 6 1 2 3 4 - 1 Best described by (f) 2 4 6 1 2 3 4 - 1 Best described by (g) 2 4 6 1 2 3 4 - 1 - 2 Best described by (h) 2 4 6 1 2 3 4 - 1 - 2 Best described by (i) (a) L < R < integraldisplay 6 0 f ( x ) dx (b) L = R < integraldisplay 6 0 f ( x ) dx (c) L < R = integraldisplay 6 0 f ( x ) dx (d) L < integraldisplay 6 0 f ( x ) dx < R (e) L = integraldisplay 6 0 f ( x ) dx < R (f) R < L < integraldisplay 6 0 f ( x ) dx (g) R < L = integraldisplay 6 0 f ( x ) dx (h) R < integraldisplay 6 0 f ( x ) dx < L (i) R = integraldisplay 6 0 f ( x ) dx < L
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4 3. The figure below shows a differentiable function f and the line tangent to the graph at the point (2 , 5) : ( picture not drawn to scale ) (2 , 5) (2 . 5 , 6) f ( x )
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