Chapter 1
22
Introduction to Discrete-Time Dynamical Systems
l line test to check whether a function
different result. Finally, we used the horizonta
used to compute the input from the output.
has an inverse. If it does, the inverse can be
and whose
A function whose argument is the name of a bird
bird.
that
of
value is the length
16.
Mathematical Techniques
ing situ
1—2. Identify the variables and parameters in the follow
an
choose
and
ations, give the units they might be measured in,
each.
appropriate letter or symbol to represent
of
1. A scientist measures the mass of fish over the course
levels
nt
differe
three
at
ent
100 days, and repeats the experim
of salinity: 0%, 2% and 5%.
bandicoots
2. A scientist measures the body temperature of
different
three
at
so
does
and
,
winter
the
every day during
altitudes: 500 m, 750 m, and 1000 m.
at the points
3—6. Compute the values of the following functions
indicated and sketch a graph of the function.
4.
f(x)=x+5atx=0,x=1,andx=4.
g(v)=5yaty=0,y=l,andy=r4.
5.
h(z)=-— atz=l,z=2,andz:=4.
3.
r4.
dr=r
=
l,an
,r=)
(r
5atr=0
2
F
+
does not seem
7—10 • Graph the given points and say which point
n.
to fall on the graph of a simple functio
6.
7.
8.
9.
10.
(0, —1), (1, 1), (2, 1), (3, 5), (4,7).
(0, 5), (1, 10), (2, 8), (3, 6), (4,4).
(0, 2), (1, 3), (2,6), (3, 11), (4, 10).
(0, 45), (1,25), (2, 12), (3, 12.5), (4, 10).
algebraic
11—14 • Evaluate the following functions at the given
arguments.
11.
f(x)=zx+Satx=a,x=a+1,andx=z4a.
aty=x y=2x + 1, andy=2 —x.
,
12. N g(y)—5v 2
5
c
1
,z=—,andz=c+1.
13. h(z)=—atz=—
c
5
5
14.
.
F(r)=r2+5atr=x+1,r=3x,andr=
a more
15—16 • Sketch graphs of the following relations. Is there
ents?
convenient order for the argum
and whose
15. A function whose argument is the name of a state
state.
that
in
e
t
value is the highes altitud
State
Highest Altitude (ft)
California
14,491
Idaho
12,662
Nevada
13,143
Oregon
11,239
Utah
13,528
Washington
14,410
Length
Bird
Cooper’s hawk
50 cm
Goshawk
66 cm
Sharp-shinned hawk
35 cm
ns, graph each
17—20 • For each of the following sums of functio
= —1, x =0,
x
—2,
=
x
at
values
the
te
Compu
component piece.
sum.
the
plot
and
2
=
x
and
x = 1,
17.
f(x)=2x+3andg(x)=3x —5.
18.
f(x)=2x+3andh(x)=—3x— 12.
F(x)_—x + 1 and G(x)=x + 1.
2
19.
1.
x
=—x+
)=
dH(x)
(x
lan
2
F
+
ns, graph each
21—24’ For each of the following products of functio
t at x = —2,
produc
the
of
value
the
te
component piece. Compu
result.
the
graph
and
2
=
x
and
1,
=
x
x —1, .r = 0,
20.
21.
f(x)=2x+3andg(x)=3x—5.
22.
f(x) =2x +3 and h(x)
23.
.
x
=x+1
)=
dG(x)
(x
lan
2
F
+
l.
x
=—x+
)=
dH(x)
(x
lan
F
+
2
24.
=
—3x
—
12.
functions. In
25—28 • Find the inverses of each of the following
n at an input
functio
l
origina
the
each case, compute the output of
the function.
of
action
the
undoes
inverse
the
of 1.0, and show that
25.
f(x)=2x+3.
26.
g(x)=3x—5.
27.
G(’)z= l/(2+v) fory0.
28.
=y + 1 fory0.
2
F(v)
inverse. Mark
29—32. Graph each of the following functions and its
n.
the given point on the graph of each functio
raphsoff
29. f(x)=2x+3.Markthepoint(l,f(l))ontheg
and f (based on Exercise 25).
graphs of g
g(x) = 3x —5. Mark the point (1, g(1)) on the
26).
se
and g (based on Exerci
graphsof
31. G(y)=1/(2+y).Markthepoint(1,G(1))onthe
27).
se
Exerci
on
G and G (based
on the
l for y0. Mark the point (1, F(1))
32. 2
F(y)=v
+
28).
se
Exerci
on
(based
F
and
graphs of F
30.
Find the compositions of the given functions. Which pairs
of functions commute?
33—36 •
2x + 3 and g(x)
f(x)
34.
f(x)=2x+3andh(x)=—3x— 12.
35.
2 + I and G(x) =x + 1.
F( —r
36.
F(x)=x- + I and H(x)= —x ±. 1.
=
A plot of an Internet stock price against time.
160
140
120
3x —5.
33.
=
40.
23
Exercises
1.2
ioo
“
40
20
0
0
10
5
15
20
25
Time (days)
Applications
Describe what is happening in the graphs shown.
37—40
•
37.
A plot of cell volume against time in days.
•
41.
A population of birds begins at a large value, decreases to a
tiny value, and then increases again to an intermediate value.
42.
The amount of DNA in an experiment increases rapidly from
a very small value and then levels out at a large value before
declining rapidly to 0.
43.
Body temperature oscillates between high values during the
day and low values at night.
44.
Soil is wet at dawn, quickly dries out and stays dry dur
ing the day, and then becomes gradually wetter again during
the night.
2
45—48 • Evaluate the following functions over the suggested range,
sketch a graph of the function, and answer the biological question.
Time (days)
38.
Draw graphs based on the following descriptions.
41—44
45.
The number of bees b found on a plant is given by b = 2f ± I
where f is the number of flowers, ranging from 0 to about
20. Explain what might be happening when f = 0.
46.
The number of cancerous cells c as a function of radiation
dose r (measured in rads) is
A plot of a Pacific salmon population against time in years.
1000
c = r —4
800
for r greater than or equal to 5, and is zero for r less than 5.
r ranges from 0 to 10. What is happening at r = 5 rads?
600
400
47.
Insect development time A (in days) obeys A = 40
where T represents temperature in C for 10 T 40.
Which temperature leads to the more rapid development?
48.
Tree height Ii (in meters) follows the formula
200
0
1950
1960
1970
1980
—
1990
Year
39.
Ii
A plot of the average height of a population of trees plotted
against age in years.
10
8
6
-
0
10
20
Age (years)
30
40
=
lOOa
100 + a
where a represents the age of the tree in years for 0
1000. How tall would this tree get if it lived forever?
a
49—52 • Consider the following data describing the growth of a
tadpole.
Age, a Length, L Tail Length, T Mass, M
(g)
(cm)
(days)
(cm)
0.5
1.5
1.0
1.5
1.0
3.0
0.9
3.0
1.5
4.5
0.8
6.0
2.0
6.0
0.7
12.0
2.5
7.5
0.6
24.0
3.0
9.0
0.5
48.0
-
24
49.
—
----
-
-
—
Time Dynamical Systems
Chapter 1 Introduction to Discreteg a plant.
Consider the following data describin
ose Production, G
Age, a Mass, M Volume, V Gluc
(mg)
)
3
(cm
(g)
(days)
0.0
5.1
1.5
0.5
3.4
6.2
3.0
1.0
6.8
7.2
4.3
1.5
8.2
8.1
5.1
2.0
9.4
8.9
5.6
2.5
8.2
9.6
5.6
3.0
59—62
Graph length as a function of age.
.
age.
50. Graph tail length as a function of
of length.
51. Graph tail length as a function
th and then graph length as
52. Graph mass as a function of leng
hs compare?
a function of mass. How do the two grap
l compositions describe
53—56 • The following series of functiona
ents.
urem
connections between several meas
end up in a room is a func
53. The number of mosquitos (M) that
) according to
2
(W, in cm
open
is
ow
tion of how far the wind
(B) depends on the
M(W)5W + 2. The number of bites
this function have an
) = 0.5M. Find the
59. Graph M as a function of a. Does
number of mosquitos according to B(M
e out the age of the
far the window is open.
inverse? Could we use mass to figur
number of bites as a function of how
2
cm
window was 10
plant?
How many bites would you get if the
this function have an
open?
60. Graph V as a function of a. Does
e out the age of the
figur
to
me
inverse? Could we use volu
is a function of how far
54. The temperature of a room (T)
.
plant?
to T(W) =40 0.2W
the window is open (W) according
tion
func
a
is
s)
hour
in
ured
function have an in
meas
(S.
How long you sleep
61. Graph G as a function of a. Does this
how
to figure out the ag
n
Find
uctio
prod
14
ose
=
gluc
verse? Could we use
of the temperature according to 5(T)
far the window is
of the plant?
long you sleep as a function of how
2
cm
10
was
ow
wind
the
if
sleep
this function have ai
open. How long would you
62. Graph G as a function of M. Does
use glucose pro
open?
inverse? What is strange about it? Could we
t?
plan
the
duction to figure out the mass of
in trillions or 1012) that
55. The number of viruses (V, measured
s
ppre
nosu
mmu
i
of
ee
degr
kg) as a function of th
infect a person is a function of the
63—66 • The total mass of a population (in
off
d
turne
is
that
m
syste
iduals, P(t), and th
une
indiv
of
imm
ber
sion (1, the fraction of the
time, t, is the product of the num
512. The fever (F, measured
following exercise
the
of
V(1)
each
In
to
kg).
by stress) according
mass per person, WQ) (in
ber
num
the
of
tion
func
h graphs of P(t), W(t
in °C) associated with an infection is a
find the formula for the total mass, sketc
a
as
r
feve
Find
.
0.4V
for 0 < t < 100, and de
of viruses according to F(V) 37 +
and the total mass as functions of time
be
r
feve
the
will
high
How
n.
essio
function of immunosuppr
cribe the results in words.
= 1)?
if immunosuppression is complete (1
2.0 x 106 + 2.0 x io
63. The population of people is P(t) =
tem
the
of
is W(t) = 80— 0.5t.
kg)
tion
(in
func
a
W(t)
is
and the mass per person
56. The length of an insect (L, in mm)
rding
acco
°C)
in
ured
meas
(T,
t, and tI
4
x 106 —2.Ox 10
perature during development
64. The population is P(t)z=2.O
c
The volume of the insect (V, in cubi
0.5t.
80
=
+
W(t)
to L(T) = 10 +
is
mass per person W(t)
.
3
rding to V(L) = 2L
mm) is a function of the length acco
106 + 2
1000t and the ma
,
nds on volume accord
65. The population is P(t) = 2.0 x
The mass (M in milligrams) depe
0.5t.
80—
era
W(t)
is
per person W(t)
a function of temp
ing to M(V) = 1.3V. Find mass as
loped at
deve
that
h
weig
106 + 2.0 x lO
t
t, and
1
insec
an
ld
wou
h
ture. How muc
66. The population is P(t) = 2.0 x
5t
0.00
.
2
80
=
W(t)
is
W(t)
n
?
25°C
mass per perso
two
of
sum
the
is
ents
urem
meas
57—58 • Each of the following
. Sketch a graph of each
Computer Exercises
components. Find the formula for the sum
< t
3. Describe
forO
time
of
tions
func
as
puter plot the folio
total
the
component and
67—70 • Have your graphics calculator or com
s.
word
in
sum
in words?
the
and
them
nt
ribe
pone
desc
com
you
each
ing functions. How would
.
x
x<20
)=
of two types a and b. The
r0<
x
’fo
f(
e
67. a. 2
57. A population of bacteria consists
=
b(t)
ws
follo
nd
seco
, and the
2
first follows a(r) = 1 + t
0 sin(x) for 0 <x <20.
b. g(x) = 1.5 + e
ured in millions and
2 where populations are meas
1 2t + t
<5.
population is P(t) =
c. h(s) = sin(5s) cos(7s) for 0 <x
time is measured in hours. The total
functions in p
a(t) + b(t).
d. f(s) + li(s) for0x 20 (using the
a and c).
and leaves) of a plant
58. The above-ground volume (stem
functions in par
me
and the below-ground volu
e. g(x) h(s) for 0 x 20 (using the
is Va(t)3.Ot + 20.0+
and c).
is measured in days
(roots) is Vb(t) —lOt +40.0 where t
is
me
volu
total
The
.
3
cm
function in pan
in
ured
and volumes are meas
f. h(x) h(x) for 0 x 20 (using the
(t).
6
V
V (t) V (r ) +
—
—
.
.
—
—
—
,
-
-