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Homework 8 Solutions Total points: 16 March 11, 2019 Chapter 7.1 Solution: When the slope is zero, the equilibrium point can have any stability (stable, unstable, semistable). Thus, the fact that dX 0 dX X * = 0 tells us nothing about the stability of X * . In other words, linear stability analysis fails in this case. Solution: As you learned in LS 30A, you find the equilibrium points by setting the change equation equal to zero and solving for X : X 0 = X 3 - X = 0 X ( X 2 - 1) = 0 X ( X - 1)( X + 1) = 0 X = - 1 , 0 , 1 Thus, the equilibrium points are - 1 , 0 , and 1. On the other hand, dX 0 dX = 2 X 2 - 1 . Thus, linear stability analysis tells us that dX 0 dX X = - 1 = 1 > 0 = ⇒ - 1 is unstable dX 0 dX X =0 = - 1 < 0 = 0 is stable dX 0 dX X =1 = 1 > 0 = 1 is unstable 1
Chapter 7.2 Solution: By definition, a function is a rule that takes each element of its domain to exactly one element of its codomain. Hence, for any ( X, Y ) R 2 , there is single value f ( X, Y ). This means that the graph of f : R 2 R has just a single point above (or below) each ( X, Y ). Otherwise said, the graph of a function cannot have two points
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