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061Exam4soln - EXAM 4 Section 024 NAME Make sure to read...

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EXAM 4 Section 024 April 27, 2006 NAME: Make sure to read the question carefully and answer the question that is asked. Show ALL relevant work so that partial credit may be given and indicate where the solution is. Lack of sufficient work may result in a loss of credit, even if a correct answer is given. Good luck!! “On my honor, as a University of Colorado at Boulder student, I have neither given nor received unautho- rized assistance on this work.” YOUR SIGNATURE: 1. True or False (1 point each) Determine whether each of the following statements is True of False and circle the appropriate response. Note that no work is required and no partial credit is possible. (a) integraldisplay 2 0 (1 - x ) dx gives the area of the region between the graph of f ( x ) = 1 - x and the x -axis on the interval [0 , 2]. True False Solution: False. integraldisplay 2 0 (1 - x ) dx = x - x 2 2 vextendsingle vextendsingle vextendsingle 2 0 = 0 whereas the area of the region between the graph of f ( x ) and the x -axis is clearly nonzero. The area is in fact 1. (b) When approximating the area under a curve with rectangles, using right-hand endpoints to de- termine the height of the rectangles will always result in an overestimate of the actual area of the region. True False Solution: False. The estimated area will depend on the slope of the graph. For example, consider an interval of a continuous function f ( x ) in which f ( x ) < 0 and f ′′ ( x ) < 0. (c) If a continuous function f ( x ) > 0 for all x in the interval [0 , 4] (that is, the curve is above the x -axis on that entire interval), then integraldisplay 3 0 f ( x ) dx < integraldisplay
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