DGD 03 - Cobwebbing DTDS\' PDF

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Cobwebbing DTDS'...

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DGD 03 for MAT1330-D Winter 2021 Hung-Chang Liao Problem 1. Consider the DTDS xt+1 =

2xt . xt − 1

a) Find the fixed points algebraically. b) Using cobwebbing, determine if the nonzero fixed point is stable.

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Problem 2. A population is modelled by the DTDS xt+1 = rxt , where r is a positive real number. Suppose we know that x1 = 60 and x3 = 6000. Find the value of r and the initial condition x0 .

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Problem 3. (This is basically Example 3.3.6 from the book) Let’s say a drink amounts to 14g of alcohol. Suppose someone had a drink just now, and then consumes half a drink every hour. The amount of alcohol xt left in the body at hour t can be modelled by the DTDS xt = xt −

10.1xt +7 4.2 + xt

with initial condition x0 = 14. a) Find the fixed points algebraically. b) Determine the long-term behaviour of xt by cobwebbing

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Problem 4. Evaluate the following limits. a) x2 − 3x x→0 x3 − 9x lim

b)

√ x−2 x→4 4 − x lim

c) (x + 1)2 − 2 x→−1 x+1 lim

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Problem 5. Consider the piecewise-defined function ( |x + 1| + 1 f (x) = |x − 2| a) Compute limx→0+ f (x) and limx→0− f (x). b) Does the limit limx→0 f (x) exist? If so, find its value.

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if x ≤ 0, if x > 0....