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Midterm2_practice - MAT 1330 Practice Midterm 2 Question 1...

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MAT 1330 Practice Midterm 2 Question 1. Evaluate the following limits: (a) lim x 0 ln( x 2 + 1) x (b) lim x →∞ e x x ln x (c) lim x →∞ 3 x + e - 2 x 7 x 2 Question 2. Consider a population of size x t with per capita production r , x t +1 = rx t . If r = x t 1 + x 2 t (a) Write the associated discrete-time dynamical system and find the equilibrium (b) Determine the stability of the equilibrium (c) Find the global minimum and the global maximum of the associated updating function, f on the interval [0 , 1] (d) Compute lim x →∞ f ( x ), where f is the associated updating function Question 3. Let f ( x ) = x 3 - 3 x e x (a) Find the minimum value of f on [0 , 2] (b) Find the maximum value of f on [0 , 2] Question 4. The Taylor polynomial of degree 4 with base point a = 2 is given by P 4 ( x ) = 1 - 2( x - 2) + 3( x - 2) 2 - 4( x - 2) 3 + 5( x - 2) 4 . (a) Find f 00 (2) (b) Find f (4) (2) 1
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Question 5. (a) Use the Intermediate Value Theorem to show that the equation e x + x 2 - 2 = x has a solution for 0 x 1 (b) Find the point guaranteed by the Mean Value Theorem for the function f ( x ) = x 2
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