1-4 Write the updating function associated with each of the following
discrete-time dynamical systems and evaluate it at the given arguments.
Which are linear?
1. p t +1 = p t - 2, evaluate at p t = 5, p t = 10, and p t = 15.
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2. evaluate at ? t = 4, ? t = 8, and ? t = 12.
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3. , evaluate at x t = 0, x t = 2, and x t = 4.
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4.
, evaluate at Q t = 0, Q t = 1, and Q t = 2.
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5-8 Compose the updating function associated with each discrete-time
dynamical system with itself. Find the two-step discrete-time dynamical
system. Check that the result of applying the original discrete-time
dynamical system twice to the given initial condition matches the result
of applying the new discrete-time dynamical system to the given initial
condition once.
5. Volume follows v t +1 = 1.5 v t , with v 0 = 1220 µ m 3 .
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6. Length obeys l t +1 = l t - 1.7, with l 0 = 13.1 cm.
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7. Population size follows n t+1 = 0.5n t , with n 0 = 1200.
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8.
Medication concentration obeys M t+ 1 = 0.75 M t + 2.0 with M 0 = 16.0.
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9-12 Find the backwards discrete-time dynamical system associated with
each discrete-time dynamical system. Use it to find the value at the
previous time.
9. v t+ 1 = 1.5 v t . Find v 0 if v 1 = 1220 µ m 3 (as in Exercise 5 ).
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10. l t + 1 = l t - 1.7. Find l 0 if l 1 = 13.1cm (as in Exercise 6 ).
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11. n t +1 = 0.5 n t . Find n 0 if n 1 = 1200 (as in Exercise 7 ).
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12.
M t+ 1 = 0.75 M t + 2.0. Find M 0 if M 1 = 16.0 (as in Exercise 8 ).
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13-14 Find the composition of each of the following mathematically
elegant updating functions with itself, and find the inverse function.
13. The updating function . Put things over a common denominator to simplify the composition.
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14.
The updating function ???. Put things over a common denominator to
simplify the composition.
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15-18 Find and graph the solutions of the following discrete-time
dynamical systems for five steps with the given initial condition.
Compare the graph of the solution with the graph of the updating
function.
15. v t +1 = 1.5 v t , with v 0 = 1220 µ m 3 .
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16. l t + 1 = l t - 1.7, with l 0 = 13.1 cm.
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17. n t +1 = 0.5 n t , with n 0 = 1200.
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18.
M t +1 = 0.75 M t + 2.0 with M 0 = 16.0.
19-22 Using a formula for the solution, you can project far into the
future without computing all the intermediate values. Find the
following, and say whether the results are reasonable.
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19. Find a formula for v t for the discrete-time dynamical system in Exercise 15 , and use it to find the volume at t = 20.
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20. Find a formula for v t for the discrete-time dynamical system in Exercise 16 , and use it to find the length at t = 20.
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21. Find a formula for v t for the discrete-time dynamical system in Exercise 17 , and use it to find the number at t = 20.
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22.
Find a formula for v t for the discrete-time dynamical system in
Exercise 18 , and use it to find the concentration at t = 20 (use the
method in Example 1.5.14 after finding the value it seems to be
approaching).
23-26 Experiment with the following mathematically elegant updating
functions and try to find the solution.
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23.
Consider the updating function from Exercise 13 . Starting from an
initial condition of x 0 = 1, compute x 1 , x 2 , x 3 , and x 4 , and
try to spot the pattern.
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24. Use the updating function in Exercise 23 but start from the initial condition x 0 = 2.
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25.
Consider the updating function g ( x ) = 4 - x . Start from initial
condition of x 0 = 1, and try to spot the pattern. Experiment with a
couple of other initial conditions. How would you describe your results
in words?
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26.
Consider the updating function from Exercise 14 . Start from initial
condition of x 0 = 3, and try to spot the pattern. Experiment with a
couple of other initial conditions. How would you describe your results
in words?
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27-30 Consider the following actions. Which of them commute (produce the
same answer when done in either order)?
27.
A population doubles in size; 10 individuals are removed from a
population. Try starting with 100 individuals, and then try to figure
out what happens in general.
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28.
A population doubles in size; population size is divided by 4. Try
starting with 100 individuals, and then try to figure out what happens
in general.
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29. An organism grows by 2.0 cm; an organism shrinks by 1.0 cm.
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30. A person loses half his money. A person gains $10.
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31-34 Use the formula for the solution to find the following, and say whether the results are reasonable.
31. Using the solution for tree height h t = 10.0 + t m ( Example 1.5.13 ), find the tree height after 20 years.
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32. Using the solution for tree height h t = 10.0 + t m ( Example 1.5.13 ), find the tree height after 100 years.
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33.
Using the solution for bacterial population number b t = 2.0 t · 1.0 (
Equation 1.5.2 ), find the bacterial population after 20 hours. If an
individual bacterium weighs about 10 -12 grams, how much will the whole
population weigh?
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34.
Using the solution for bacterial population number b t = 2.0 t · 1.0 (
Equation 1.5.2 ), find the bacterial population after 40 hours. How much
would this population weigh?
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35-36 Find a formula for the solution of the given discrete-time
dynamical system.
35.
Find the pattern in the number of mites on a lizard with x 0 = 10 and
following the discrete-time dynamical system x t +1 = 2 x t + 30. (
Hint: Add 30 to the number of mites.)
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36.
Find the pattern in the number of mites on a lizard with x 0 = 10 and
following the discrete-time dynamical system x t+1 = 2 x t + 20.
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37-40 The following tables display data from four experiments: a. Cell
volume after 10 minutes in a watery bath b. Fish mass after 1 week in a
chilly tank c. Gnat population size after 3 days without food d. Yield
of several varieties of soybean before and after fertilization For each,
graph the new value as a function of the initial value, write the
discrete-time dynamical system, and fill in the missing value in the
table.
37.
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38.
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39.
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40.
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41–44 Recall the data used for Exercises 49 - 52 in Section 1.2.
These data define several discrete-time dynamical systems. For example,
between the first measurement (on day 0.5) and the second (on day 1.0),
the length increases by 1.5 cm. Between the second measurement (on day
1.0) and the third (on day 1.5), the length again increases by 1.5 cm.
41.
Graph the length at the second measurement as a function of length at
the first, the length at the third measurement as a function of length
at the second, and so forth. Find the discrete-time dynamical system
that reproduces the results.
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42. Find and graph the discrete-time dynamical system for tail length.
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43. Find and graph the discrete-time dynamical system for mass.
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44.
Find and graph the discrete-time dynamical system for age.
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45-48 Suppose students are permitted to take a test again and again
until they get a perfect score of 100. We wish to write a discrete-time
dynamical system describing these dynamics.
45. In words, what is the argument of the updating function? What is the value?
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46. What are the domain and range of the updating function? What value do you expect if the argument is 100?
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47. Sketch a possible graph of the updating function.
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48.
Based on your graph, how would a student do on her second try if she
scored 20 on her first try?
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49-50 Consider the discrete-time dynamical
system b t +1 = 2.0 b t for a bacterial population ( Example 1.5.1 ).
49. Write a discrete-time dynamical system for the total volume of bacteria (suppose each bacterium takes up 10 4 µ m 3 ).
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50.
Write a discrete-time dynamical system for the total area taken up by
the bacteria (suppose the thickness is 20 µ m).
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51-52 Recall the
equation h t +1 = h t + 1.0 for tree height.
51. Write a discrete-time dynamical system for the total volume of the cylindrical trees in Section 1.3, Exercise 27 .
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52.
Write a discrete-time dynamical system for the total volume of a
spherical tree (this is kind of tricky).
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53-54 Consider the following
data describing the level of medication in the blood of two patients
over the course of several days.
53.
Graph three points on the updating function for the first patient. Find
the discrete-time dynamical system for the first patient.
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54.
Graph three points on the updating function for the second patient and
find the discrete-time dynamical system.
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55-56 For the following discrete-time dynamical systems, compute
solutions with the given initial condition. Then find the difference
between the solutions as a function of time, and the ratio of the
solutions as a function of time. In which cases is the difference
constant, and in which cases is the ratio constant? Can you explain why?
55.
Two bacterial populations follow the discrete-time dynamical system b t
+1 = 2.0 b t , but the first starts with initial condition b 0 = 1.0 ×
10 6 and the second starts with initial condition b 0 = 3.0 × 10 5 .
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56.
Two trees follow the discrete-time dynamical system h t +1 = h t + 1.0,
but the first starts with initial condition h 0 = 10.0 m and the second
starts with initial condition h 0 = 2.0 m.
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57-60 Follow the steps to derive discrete-time dynamical systems
describing the following contrasting situations.
57.
A population of bacteria doubles every hour, but 1.0 × 10 6 individuals
are removed after reproduction to be converted into valuable biological
by-products. The population begins with b 0 = 3.0 × 10 6 bacteria. a.
Find the population after 1, 2, and 3 hours. b. How many bacteria were
harvested? c. Write the discrete-time dynamical system. d. Suppose you
waited to harvest bacteria until the end of 3 hours. How many could you
remove and still match the population b 3 found in part a ? Where did
all the extra bacteria come from?
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58.
Suppose a population of bacteria doubles every hour, but that 1.0 × 10 6
individuals are removed before reproduction to be converted into
valuable biological by-products. Suppose the population begins with b 0 =
3.0 × 10 6 bacteria. a. Find the population after 1, 2, and 3 hours. b.
Write the discrete-time dynamical system. c. How does the population
compare with that in the previous problem? Why is it doing worse?
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59.
Suppose the fraction of individuals with some superior gene increases
by 10% each generation. a. Write the discrete-time dynamical system for
the fraction of organisms with the gene (denote the fraction at time t
by f t and figure out the formula for f t +1 ). b. Write the solution
with f 0 = 0.0001. c. Will the fraction reach 1.0? Does the
discrete-time dynamical system make sense for all values of f t ?
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60.
The Weber-Fechner law describes how human beings perceive differences.
Suppose, for example, that a person first hears a tone with a frequency
of 400 hertz (cycles per second). He is then tested with higher tones
until he can hear the difference. The ratio between these values
describes how well this person can hear differences. a. Suppose the next
tone he can distinguish has a frequency of 404 hertz. What is the
ratio? b. According to the Weber-Fechner law, the next higher tone will
be greater than 404 by the same ratio. Find this tone. c. Write the
discrete-time dynamical system for this person. Find the fifth tone he
can distinguish. d. Suppose the experiment is repeated on a musician,
and she manages to distinguish 400.5 hertz from 400 hertz. What is the
fifth tone she can distinguish?
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61-62 The total mass of a population of bacteria will change if either
the number of bacteria changes, the mass per bacterium changes, or both.
The following problems derive discrete-time dynamical systems when both
change.
61. The number of bacteria doubles each hour, and the mass of each bacterium triples during the same time.
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62.
The number of bacteria doubles each hour, and the mass of each
bacterium increases by 1.0 × 10 -9 g. What seems to go wrong with this
calculation? Can you explain why?
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