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HARVARD MATH 1A - Midterm #2

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;l,lLast Name:First Name:Mathematics laMidterm # 2April 8, 1998.Your section (choose one):Otto Bretscher Bing Cheng Greg WarringtonMWF 11-12 T@ 1G-11:30 MWF 10-11QuestionScore1zJi5TotalCalculators are not allowed.No justification is necessary for question 1.Write your answers in the spaces provided.Each question on this exam is worth 10 points.In questions 2,3, 4 and,S, unsubstantiated answers cannot be given credit.Good luck.I-1. (10 points). True or false, no explanation necessary.a)T FIf (c' f (c)) is a critical point of the function f, then f necessarily has a local extremumat c.b)rFlimr--.roJ4 +x- 2c)T Felog2(x) _ Xllln(2)d) T FIf f '(-3) : 4 and f(x) is always concave up, then lim*-- f (x) : qo.e)T F1-9x2t(x) : 3i _ 4x + 5,* has a horizontal asymptote ata :1/3.I,aJ2. (10 points). A girl is flying a kite at a constant height of 300 ft. When the girl has letout 500 ft. of string, the wind is moving the kite horizontally at 25 ft./sec. How fast mustshe let out the kite string at this time?3. (10 points).a) Graph the function g - (x-1) /x2 in the space provided. Make sure to label carefullyall occurrences of relative extrema, inflection points, x and g-intercepts, and asymptotes.b) How many solutions does the equation (x - l)/xz = k have for an arbitrary constantk? (You may have to distinguish several cases.)o54. (i0 points). A bottle of sparkling cider is put in the snow (at 0" Celsius) to chill inanticipation of the end of classes. The bottle is initially at 30' Celsius. If its temperaturedrops at an instantaneous rate of 3% per minute, how long must our revelers wait untilthe bottle is at 5' Celsius? You may use the following table of values to help you in youranswer.x1/2 1/31/4 1/51/61/71/8 1/e 1/10lnx-0.69 -l.10 -1.39-l .51-1.79 -r.95 -2.08-2.3065. (10 points). A farmer wishes to build an E-shaped fence (see picture) along a straightriver bank so as to create two identical rectangulir pastures. witl aoo feet or fencingmaterial, what is the largest area he can encloseiRJVER B.q,NK\,FENCE\l


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