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EGET 3171

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**Differential Equations & Partial Derivatives**

1. Carbon-12, one of the 3 isotopes of carbon, is radioactive and decays at a rate proportional to the amount present. Its half life is 5730 years (that is, it takes 5730 years for a given amount to decay to half its original size). If 10 grams were present originally, how much will be left after 2000 years? [7.85 grams]

2. Under normal conditions, the rate of bacteria in a given sample grows at a rate proportional to the number present. In a certain sample, there were 10^{5} present at a given time, and there were 3 times that present after 10 hours. How many were present after 20 hours? [9 x 10^{5}]

3. Find any relative minimum, maximum or saddle points of the equation:

z = 3x – x^{2 }