test 2A soln - Calculus for the Life Sciences I MAT1330...

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Calculus for the Life Sciences I MAT1330 Test 2 Instructor: Jing Li , Catalin Rada , Frithjof Lutscher Question 1 Show that the equation 4 points 2 x + sin( x ) = e x has at least one solution in the interval [0 , π ]. State the theorem (hypothesis and conclusion) from class that you are using. Solution: Let f ( x ) = 2 x + sin( x ) e x . Since f (0) = 1 < 0 and f ( π ) = 2 π e π = 6 . 23997 . . . > 0, and since f is continuous on the interval [0 , π ], we may use the Intermediate Value Theorem to conclude that there exists c ]0 , π [ such that f ( c ) = 0. Thus, 2 c + sin ( c ) = e c . Question 2 Consider the discrete dynamical system 6 points x t +1 = ax t 1 + x t 4 x t , t = 0 , 1 , 2 , 3 , . . ., where x t is the number between of individual after t periods. Consider only a > 0. a ) [2 points] Find all the equilibrium points of this discrete dynamical system. At least one of them will depend on the parameter a . b ) [1 point] Find the values of a for which the equilibrium points in ( a ) are biologically meaningful. c ) [3 points] Determine the stability of the equilibrium points in function of the possible values of the parameter a determined in ( b ). Solution: The updating function of the system is f ( x ) = ax 1 + x 4 x . a ) The equilibrium points are the solutions of p = f ( p ) = ap 1 + p 4 p . p = 0 is a possible solution. If p negationslash = 0, we may divide both sides of the equality by p to get 1 = a 1 + p 4 5 = a 1 + p 5(1 + p ) = a 5 p = a 5 p = a 5 5 .
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