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Assignment 1 Solutions

# Assignment 1 Solutions - MAT 1330 Fall 2010 Assignment 1...

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1. Suppose you deposit \$1000 each week into a special savings account, but the bank takes 5% of the total in fees. A discrete-time system modelling your investment is x n +1 =0 . 95 x n + 1000 . (a) At the end of the ±rst week, there is x 1 =0 . 95(1500) + 1000 = \$2425 in the account. At the end of the second week, there is x 2 =0 . 95(2425) + 1000 = \$3303 . 75 in the account. At the end of the third week, there is x 3 =0 . 95(3303 . 75) + 1000 = \$4138 . 56 in the account. (b) The updating function is f ( x )=0 . 95 x + 1000 (c) The equilbrium points satisfy p =0 . 95 p + 1000 0 . 05 p = 1000 p =20 , 000 . (d) The solution is x n - 20 , 000 = 0 . 95 n ( x 0 - 20 , 000) x n = - 18 , 500 × 0 . 95 n +20 , 000 (e) See Figure 1. (f) See Figure 2. (g) The equilibrium point is stable. 2. A disease is spreading through campus. Each day, the number of people infected depends on how many were infected the day before, according to the formula y n +1 = 6 . 2976 y n 1+0 . 0112 y n . 1

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Assignment 1 Solutions - MAT 1330 Fall 2010 Assignment 1...

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