MATH 19A Midterm: MATH19 Harvard Midterm1F02sol19 Fall 04
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1: consider the following di erential equation for the function y(t): dy dt. As t , y(t) : given f (x, y) = y + cos(x2y), compute the following partial derivatives: (9 points) (a) Y f (x, y) = 1 x2 sin(x2y) X2 f (x, y) = 2y sin(x2y) 4x2y2 cos(x2y: consider the following two predator-prey systems of di erential equations: 2 (i) (ii) dx dt dy dt dx dt dy dt. In one of these systems, the prey are very large animals and the predators are very small animals, such as elephants and mosquitos. Thus, it takes many predators to eat one prey, but each prey eaten is a tremendous bene t for the predator population. The other system has very large predators and very small prey, such as whales and krill. Determine which system is which and provide a justi cation for your answer. (10 points) Solution. (i) corresponds to the large predator-small prey situation.
