Calculus 1 - Assignment 1 Solutions 2009

Calculus 1 - Assignment 1 Solutions 2009 - Pawel Lorek,...

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Unformatted text preview: Pawel Lorek, University of Ottawa, MAT 1330, Winter 2009 Assignment 1, due FEB 4, 17:30 in class Student Name Student Number Problem 1: [5 points] Suppose that at time t the population size of some individual is x t . Then, at time t + 1 each individual will produce 2 (1+0 . 01 · x t ) new ones (this quantity is called per capita production ). (a) [1p] Write the linear DTDS for the population size, x t +1 = f ( x t ) and find x 2 when x = 2. (b) [2p] Draw the updating function and start the cobwebbing process at 50 . (c) [2p] Find the equlibria explicitly. Solution (a) The population size at time t + 1 is ( per capita production ) × (current population size), i.e. DTDS is the following: f ( x t ) = 2 1 + 0 . 01 · x t · x t . We have x = 2 x 1 = f ( x ) = 2 x 1 + 0 . 01 · x = 2 · 2 1 + 0 . 01 · 2 = 4 1 . 02 = 400 102 x 2 = f ( x 1 ) = 2 x 1 1 + 0 . 01 · x 1 = 2 · 400 102 1 + 0 . 01 · 400 102 = 800 102 1 + 4 102 = 800 102 106 102 = 800 102 · 102 106 = 800 106 ≈ 7 . 547169811 1 (b) Cobwebbing: 100 75 50 25 xt 125 25 50 100 125 75 (c) Equilibrium x ∗ must fulfill: x ∗ = f ( x ∗ ), i.e. x ∗ = 2 x * 1+0 . 01 x * x ∗- 2 x * 1+0 . 01 x * = 0 x ∗ parenleftBig 1- 2 1+0 . 01 x * parenrightBig = 0 and there are two solutions (two equilibria): either x ∗ = 0 or 1- 2 1+0 . 01 x * = 0 1 = 2 1+0 . 01 x * 1 + 0 .....
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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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Calculus 1 - Assignment 1 Solutions 2009 - Pawel Lorek,...

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