Calculus 1 - Assignment 2

Calculus 1 - Assignment 2 - Pawel Lorek, University of...

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Unformatted text preview: Pawel Lorek, University of Ottawa, MAT 1330D, Winter 2009 Assignment 2, due March 4, 17:30 in class Student Name Student Number Problem 1: [4 points] Consider the following DTDS: Mt+1 = 8Mt2 15 + Mt2 Analyze this DTDS, i.e., determine the biologically relevant fixed points (i.e. equilibria) and their stability. Draw the updating function and use cobwebbing to illustrate and confirm your analytical results. Problem 2: [8 points] Consider the following nonlinear DTDS for growing population with harvesting: xt+1 = f (xt ) = 4xt (3 − xt ) − hxt , where h ≥ 0 denotes the intensity of harvesting. (a) [4p] Analyze this DTDS, i.e., determine the biologically relevant fixed points and their stability. Summarize your results in the form of a little table: range of h fixed point(s) stability (b) [2p] Draw the updating function and use cobwebbing for h = 10.5 to illustrate your analysis. Start with x = 0.2 (c) [2p] Find the value of the parameter h that guarantees the highest harvest (in equilibrium). Problem 3: [4 points] Find the global minimum and the global maximum of the following function on the given interval: 1 3 2 − ≤ x ≤ 4. f (x) = ex −3x , 2 Problem 4: [6 points] Use the first and second order derivatives to sketch the graph of 2 −4 f (x) = x −4 . You have to find the critical points, inflexion points, intervals where the x function is increasing, limits at the endpoints of the domain . . . , and the vertical and horizontal asymptotes if any. Problem 5: [4 points] Find the derivative of the following function. (a) f (x) = sin(x2 ) − 1 , tan(x) (b) g(x) = cos(sin(x3 ) + ln(x)). 1 ...
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This note was uploaded on 01/16/2011 for the course MAT 1330 taught by Professor Dumitriscu during the Fall '08 term at University of Ottawa.

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