1.5.1 Write
the updating function associated with each of the following
discrete-time dynamical systems and evaluate it at the given arguments.
Which are linear? ...
Get solution
1.5.2 Write
the updating function associated with each of the following
discrete-time dynamical systems and evaluate it at the given arguments.
Which are linear? ...
Get solution
1.5.3 Write
the updating function associated with each of the following
discrete-time dynamical systems and evaluate it at the given arguments.
Which are linear? ...
Get solution
1.5.4 Write
the updating function associated with each of the following
discrete-time dynamical systems and evaluate it at the given arguments.
Which are linear? ...
Get solution
1.5.5 Compose
the updating function associated with each discrete-time dynamical
system with itsel
f. 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 discretetime dynamical system to the given initial
condition once. Volume follows ...
Get solution
1.5.6 Compose
the updating function associated with each discrete-time dynamical
system with itsel
f. 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 discretetime dynamical system to the given initial
condition once. Length obeys ...
Get solution
1.5.7 Compose
the updating function associated with each discrete-time dynamical
system with itsel
f. 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 discretetime dynamical system to the given initial
condition once. Population size follows ...
Get solution
1.5.8 Compose
the updating function associated with each discrete-time dynamical
system with itsel
f. 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 discretetime dynamical system to the given initial
condition once. Medication concentration obeys ... with ...
Get solution
1.5.9 Find
the backwards discrete-time dynamical system associated with each
discrete-time dynamical system. Use it to find the value at the previous
time. ...
Get solution
1.5.10 Find
the backwards discrete-time dynamical system associated with each
discrete-time dynamical system. Use it to find the value at the previous
time. ...
Get solution
1.5.11 Find
the backwards discrete-time dynamical system associated with each
discrete-time dynamical system. Use it to find the value at the previous
time. ...
Get solution
1.5.12 Find
the backwards discrete-time dynamical system associated with each
discrete-time dynamical system. Use it to find the value at the previous
time. ...
Get solution
1.5.13 Find
the composition of each of the following mathematically elegant
updating functions with itself, and find the inverse function. The updating function f (x) = .... Put things over a common denominator to simplify the composition.
Get solution
1.5.14 Find
the composition of each of the following mathematically elegant
updating functions with itself, and find the inverse function. The updating function h(x) = .... Put things over a common denominator to simplify the composition.
Get solution
1.5.15 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. ...
Get solution
1.5.16 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. ...
Get solution
1.5.17 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. ...
Get solution
1.5.18 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. ...
Get solution
1.5.19 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. Find a formula for ... for the discrete-time dynamical system in Exercise 15, and use it to find the volume at t = 20. Reference Exercise 15 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. ...
Get solution
1.5.20 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. Find a formula for ... for the discrete-time dynamical system in Exercise 16, and use it to find the length at t = 20. Reference Exercise 16 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. ...
Get solution
1.5.21 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. Find a formula for ... for the discrete-time dynamical system in Exercise 17, and use it to find the number at t = 20. Reference Exercise 17 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. ...
Get solution
1.5.22 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. Find a formula for ... 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). Reference Exercise 18 ... Reference Example
1.5.14... Finding patterns in this way and translating them into
formulas can be tricky. It is much more important to be able to describe
the
behavior of solutions with a graph or in words. In this case, our
calculations quickly revealed that the solution moved closer and closer
to 2.0. In Section 1.6, we will develop a powerful graphical method to
deduce this pattern with a minimum of calculation.
Get solution
1.5.23
Experiment with the following mathematically elegant updating
functions and try to find the solution. Consider the updating
function ...
Get solution
1.5.24
Experiment with the following mathematically elegant updating
functions and try to find the solution. Use the updating function
in Exercise 23 but start from the initial condition ... = 2.
Get solution
1.5.25
Experiment with the following mathematically elegant updating
functions and try to find the solution. Consider the updating
function g ( x )=4 − x. Start from initial condition of ...
= 1, and try to spot the pattern. Experiment with a couple of other
initial conditions. How would you describe your results in words?
Get solution
1.5.26
Experiment with the following mathematically elegant updating
functions and try to find the solution. Consider the updating function
... from Exercise 14. Start from initial condition of ...=
3, and try to spot the pattern. Experiment with a couple of other
initial conditions. How would you describe your results in words?
Get solution
1.5.27 Consider the following actions. Which of them commute (produce the same answer when done in either order)? 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.
Get solution
1.5.28 Consider the following actions. Which of them commute (produce the same answer when done in either order)? 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.
Get solution
1.5.29
Consider the following actions. Which of them commute (produce the
same answer when done in either order)? An organism grows by 2.0 cm; an
organism shrinks by 1.0 cm.
Get solution
1.5.30
Consider the following actions. Which of them commute (produce the
same answer when done in either order)? An organism grows by 2.0 cm; an
organism shrinks by 1.0 cm. A person loses half his money. A person
gains $10.
Get solution
1.5.31
Use the formula for the solution to find the following, and say
whether the results are reasonable. Using the solution for tree height ... = 10.0 + t m (Example 1.5.13), find the tree height after 20 years.
Get solution
1.5.32
Use the formula for the solution to find the following, and say
whether the results are reasonable. Using the solution for tree height
...= 10.0 + t m (Example 1.5.13), find the tree height after 100 years.
Get solution
1.5.33
Use the formula for the solution to find the following, and say
whether the results are reasonable. Using the solution for bacterial
population number ... = ... (Equation 1.5.2), find the
bacterial population after 20 hours. If an individual bacterium weighs
about ...grams, how much will the whole population weigh?
Get solution
1.5.34
Use the formula for the solution to find the following, and say
whether the results are reasonable. Using the solution for bacterial
population number ... = ...· 1.0 (Equation 1.5.2), find the bacterial population after 40 hours. How much would this population weigh?
Get solution
1.5.35
Find a formula for the solution of the given discrete-time dynamical
system. Find the pattern in the number of mites on a lizard with ... =10 and following the discrete-time dynamical system ... (Hint: Add 30 to the number of mites.)
Get solution
1.5.36
Find a formula for the solution of the given discrete-time dynamical
system. Find the pattern in the number of mites on a lizard with ...
Get solution
1.5.37 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. ...
Get solution
1.5.38 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. ...
Get solution
1.5.39 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. ...
Get solution
1.5.40 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. ...
Get solution
1.5.41 ... 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. 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.
Get solution
1.5.42
... 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.
Find and graph the discrete-time dynamical system for tail length.
Get solution
1.5.43
... 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.
Find and graph the discrete-time dynamical system for mass.
Get solution
1.5.44
... 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.
Find and graph the discrete-time dynamical system for age.
Get solution
1.5.45 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. In words, what is the argument of the updating function? What is the value?
Get solution
1.5.46
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. What are the domain and range of the
updating function? What value do you expect if the argument is 100?
Get solution
1.5.47 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. Sketch a possible graph of the updating function.
Get solution
1.5.48 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. Based on your graph, how would a student do on her second try if she scored 20 on her first try?
Get solution
1.5.49 Consider the discrete-time dynamical system ...for
a bacterial population (Example 1.5.1). Write a discrete-time
dynamical system for the total volume of bacteria (suppose each
bacterium takes up ...).
Get solution
1.5.50 Consider the discrete-time dynamical system ...for
a bacterial population (Example 1.5.1). Write a discrete-time
dynamical system for the total area taken up by the bacteria (suppose
the thickness is 20 μm).
Get solution
1.5.51
Recall the equation ... for tree height. Write a discrete-time
dynamical system for the total volume of the cylindrical trees in
Section 1.3, Exercise 27.
Get solution
1.5.52
Recall the equation ... for tree height. Write a discrete-time
dynamical system for the total volume of a spherical tree (this is kind
of tricky).
Get solution
1.5.53
Consider the following data describing the level of medication in the
blood of two patients over the course of several days. ...
Graph
three points on the updating function for the first patient. Find the
discrete-time dynamical system for the first patient.
Get solution
1.5.54
Consider the following data describing the level of medication in the
blood of two patients over the course of several days. ...
Graph three points on the updating function for the second patient and
find the discrete-time dynamical system.
Get solution
1.5.55 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? Tw o bacterial populations follow the discrete-time dynamical system ..., but the first starts with initial condition ...and the second starts with initial condition ....
Get solution
1.5.56 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? Two trees follow the discrete-time dynamical system ...= ...but the first starts with initial condition ... m and the second starts with initial condition ...
Get solution
1.5.57
Follow the steps to derive discrete-time dynamical systems describing
the following contrasting situations. A population of bacteria doubles
every hour, but 1.0×... individuals are removed after
reproduction to be converted into valuable biological by-products. The
population begins with b0 = 3.0 ×... 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 ... found in part a? Where did all the extra bacteria come from?
Get solution
1.5.58
Follow the steps to derive discrete-time dynamical systems describing
the following contrasting situations. Suppose a population of bacteria
doubles every hour, but that 1.0 ×... individuals are removed
before reproduction to be converted into valuable biological
by-products. Suppose the population begins with ...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?
Get solution
1.5.59
Follow the steps to derive discrete-time dynamical systems describing
the following contrasting situations. 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 ... and figure out the formula for ...) .
b. Write the solution with...= 0.0001.
c. Will the fraction reach 1.0? Does the discrete-time dynamical system make sense for all values of ...?
Get solution
1.5.60 Follow the steps to derive discrete-time dynamical systems describing the following contrasting situations. 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?
Get solution
1.5.61 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. The number of bacteria doubles each hour, and the mass of each bacterium triples during the same time.
Get solution
1.5.62 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. The number of bacteria doubles each hour, and the mass of each bacterium increases by 1.0×.... What seems to go wrong with this calculation? Can you explain why?
Get solution
