Math Insight

Discrete dynamical system introduction

Math 1241, Fall 2020
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Due date: Sept. 18, 2020, 11:59 p.m.
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Total points: 3
  1. A simple railway from one city to another is being constructed. The track for the railway is laid down at a rate of 10 meters per day. There is no track before the first day of construction and the entire line of track must be laid down before further construction may continue. Determine a mathematical model for the amount of track that is laid down over time by considering the following steps.
    1. The first step is to determine the state variable of the system. The variable that changes each day is the amount of
      and this variable is measured in
      (unit). Chose a variable (i.e., a letter) to represent this quantity:
      .
    2. In this course, we'll just use $n$ or $t$ to represent time, so you can pick one of the two to represent time, being sure to avoid the letter you picked above. Choose a variable for time:
      . With this choice, the state variable at a particular time can be written as $＿_＿$. (Online, this expression will be automatically filled in.)

      Putting this all together, the state variable $＿_＿$ represents the amount of
      in
      (unit) that is laid down after $＿$
      .

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    3. Now, the actual math. Specify the rule for going from one time step to the next. The rule gives the value of the state variable at the next time point $(＿_{ ＿+1})$ as a function of the value at the previous time point $(＿_{ ＿ })$.

      $＿_{ ＿+1} =$
      .

      (When entering $_＿^{＿}$ online, you need to type it as ＿_＿.)

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    4. Specify the initial conditions, which is the given value of the state variable at the initial time point $n=0$.

      $＿_0 = $

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    5. Given all the information you have defined above write down the completed discrete dynamical system. (You are repeating answers here, but we want it all together. Remember, for online answers to use ＿_＿ for $_＿^{＿}$.)

      $＿_{ ＿ + 1} =$

      $＿_0 =$

      Now. starting with the initial condition, calculate the next four points of the solution.

      $＿_0 =$

      $＿_1 =$

      $＿_2 =$

      $＿_3 =$

      $＿_4 =$

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  2. A colony of bacteria doubles in population every hour. The population initially contains $10^5$ bacteria. Describe growth of the colony by considering the following steps.
    1. Use the notation $b$ for bacteria for the state variable of this dynamical system.

      Use the notation $t$ for time, so that the state variable at a particular time is denoted $b_t$. Describe in words what $b_t$ represents.

      The state variable $b_t$ represents the
      of
      in the colony during the $t$th
      .

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    2. Specify the rule for going from one time step to the next. In this case, we will write it as a function in the form: $$b_{t+1} = f(b_t) \qquad \text{for $t=0,1,2,\ldots$}$$

      For the above equation to be the correct rule, the function $f(b)$ must be:

      $f(b) = $

      In this case, we are using the argument $b$ for the function $f(b)$ rather than using $b_t$, just to make it easier. So, be sure to write your answer for $f(b)$ using $b$ rather than $b_t$.

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    3. Specify the initial condition.

      $b_0 = $
      .

      (When typing your answer online, you can use ^ for an exponent, i.e., to write $a^b$, you can type a^b.)

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    4. Given the initial colony size, calculate the number of bacteria for the next five hours.

      $b_0=$

      $b_1=$

      $b_2=$

      $b_3=$

      $b_4=$

      $b_5=$

      (If you are entering your answer using scientific notation online, you can enter a number like $3.6 \times 10^3$ as either 3.6*10^3 or as 3.6E3.)

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  3. After an injection of penicillin, its concentration in your blood decreases by 20% every 5 minutes. Immediately after the injection, the penicillin concentration is 200 μg/ml. Describe the decay of the penicillin by considering the following steps.
    1. Denote the state variable by $p$.

      Using $n$ for the time point, we can denote the value of the state variable at a particular time point as $p_n$. Given the statement of the problem, define $p_n$ in words. Be sure to give units.

      The state variable $p_n$ is the
      of
      in your blood measured in
      (unit) that occurred $n$ intervals of 5
      (unit) each after the injection of pencillin. In other words, $p_n$ is the concentration after
      minutes.

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    2. Specify the rule for going from one time step to the next.

      $p_{n+1} =$
         for $n=0,1,2, \ldots$

      (Online, enter p_n for $p_n$.)

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    3. Specify the initial conditions.

      $p_0=$

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    4. Summarize by writing down the complete dynamical system, with evolution rule and initial condition.

      $p_{n+1} = $
         for $n=0,1,2, \ldots$
      $p_0 = $

      Evolve the dynamical system for four time steps

      $p_0=$

      $p_1=$

      $p_2=$

      $p_3=$

      $p_4=$

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    5. What is the concentration of penicillin 10 minutes after the injection?


      After 20 minutes?

      Be sure to include units. Online, put units in second box.

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Math 1241, Fall 2020