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Differential Equations Test Review

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1 Differential Equations Test Review
BC Calculus Differential Equations Test Review

2 As part of his summer job at a restaurant, Jim learned to cook up a big pot of soup late at night, just before closing time, so that there would be plenty of soup to feed customers the next day. He also found out that, while refrigeration was essential to preserve the soup overnight, the soup was too hot to be put directly into the fridge when it was ready. (The soup had just boiled at 100ºC, and the fridge was not powerful enough to accommodate a big pot of soup if it was any warmer than 20ºC). Jim discovered that by cooling the pot in a sink full of cold water, (kept running, so that its temperature was roughly constant at 5º C) and stirring occasionally, he could bring the temperature of the soup to 60ºC in ten minutes. How long before closing time should the soup be ready so that Jim could put it in the fridge and leave on time ? HINT: Use Newton’s Law of Cooling minutes

3 Oil is being pumped continuously from a certain oil well at a rate proportional to the amount of oil left in the well. Initially there were 1,000,000 gallons of oil in the well, and 6 years later there were 500,000 gallons remaining. It will no longer be profitable to pump oil when there are fewer than 50,000 gallons remaining. Write an equation for the amount of oil remaining in the well, y, at any time t. At what rate is the amount of oil in the well decreasing when there are 600,000 gallons of oil remaining? In order not to lose money, at what time t should oil no longer be pumped from the well?

4 Consider the differential equation
Sketch a slopefield for the given differential equation at the nine points indicated. There is a horizontal line with equation y = c that satisfies this differential equation. Find the value of c. Find the particular solution y = f(x) to the differential equation with the initial condition f(1) = 0.




8 At each point (x, y) on a certain curve, the slope of the curve is 4xy
At each point (x, y) on a certain curve, the slope of the curve is 4xy. If the curve contains the point (0,4), then its equation is __

9 What are the horizontal asymptotes of all solutions of the differential equation
Y = 0 and y = 8000

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