Math 1B Quiz #10Thursday, 8 November 2007GSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojfName:1. (3 pts) Show that f(t) = Cet− t − 1 is a one-parameter family of solutions to thedifferential equationdfdt= f(t) + t.If f(0) = 1, what is f(t)?12. (3 pts) Which of the three direction fields corresponds to the differential equationdfdt= (f + 1)(f − 2)?Each is graphed with f and t ranging from −3 to 3. Based on the graph (or any otherway), specify which values of f are equilibrium values. Which of these equilibria arestable and which are unstable as t → ∞? Sketch the solution to this differentialequation that satisfies the initial value condition that f = 0 when t = 0.23. (4 pts) Set up, but do not solve, a differential equation modeling the following ratherunlikely situation. Be sure to specify the meanings of your symbols: you shouldprovide a “dictionary” that translates the interesting quantities into your chosenvariables (“h = height,” for instance). Also be sure to specify the initial conditionfor the problem.Dr. Gregory House’s evil twin Dr. Geoffrey Condo breaks into the set ofGrey’s Anatomy, hoping to murder his brother’s arch rival Dr. MeredithGrey. He successfully restrains her to a hospital bed, and implants her withan intravenous drug drip, which he programs to administer an increasingsupply of painkillers: 100 miligrams this hour, then 200 mg, and so on, sothat the input rate of painkiller is proportional to the amount of time sinceshe was incapacitated. Dr. Condo hopes that by doing this, Dr. Grey willoverdose on painkillers.But Dr. Grey’s kidneys decide to try to keep her from dying, and fil-ter out the painkiller (assuming the IV also administers plenty of water atthe right salinity). In particular, in an hour, her kidneys can filter out onetenth of the total painkillers in her body. How much painkiller is in Dr.Grey’s body after a given amount of
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