Leaning to deal with growing, decaying, and oscillating processes

1
Leaning to deal with growing, decaying, and
oscillating processes

• The Exponential Function
• The Number e
• The Slope and graph of the Exponential Function
• Positive and Negative Growth and Decay Graphs.
  Rate of Increase
• Terminal Speed
• The Logarithms
• Base Change
• Half-Life t1/2. Carbon Dating
• Log, Semi-Log and Log-Log Graphs
• Graphical discrimination of Exponential Power
  Functions
• Arc Length
• Circular Motion
• Trigonometric Functions
• Trigonometric Functions in the Right Triangle
• Inverse Trigonometric Functions
• Graphs of Trigonometric Functions. Periodicity
  and Parity
• Simple Harmonic Motion

2
Special Functions
Log, Exponential and Trigonometric functions are
essential to understand and describe many
phenomena in science. They are based in two
numbers e and p, which are fundamental in
mathematics.
The Exponential Function
There is an old legend about a scholar who could
choose his own reward for some job. He asked to
place grains of corn on a chess board, starting
with one, on each field the double amount of the
former...
Some applications of the exponential functions

• In the phenomenon of growth
• Populations grow
• WORLD POPULATION
•              

• Money deposited at compound interest
• Decay processes (radioactivity, capacitor
  discharge, etc.)

3
The Exponential Function
Lets consider a process where certain types of
bacteria increase in number by the process of
subdivision. Assuming that a bacterium subdivides
once every hour we have that the number of
bacteria N at the end of any number of hours, t ,
is
The table shows that when the values of t form
an arithmetic progression, the corresponding
values of N form a geometric progression. The
equation representing this kind of correspondence
is called an exponential equation.
Then, starting with 1 bacteria at the end of t
hours there are clearly 2t bacteria. Thus, the
relation between N and t is expressed by the
equation
See that the factor 2 is obtained from the fact
that the quantity present at the end of any
interval is proportional to the quantity present
at the beginning of the interval Nn1 k Nn. The
proportionality constant in our case is k 2.
4
The Exponential and Linear Trends
If we compare the linear increase of a variable
With the exponential increase
We see that while the ratio ?N/ ?t is constant in
the first case, in the second is increasing
forming a geometric progression.
5
The Number e
In the above example if we had started with 5
bacteria there would have been just five times as
many at the end of t hours
And if we start with N0 bacteria, the equation is
Now suppose that half of the bacteria reproduce
at the half-hour. After ½ hour we would have
And after a whole hour
If 1/n of the bacteria are allowed to replicate n
times an hour, then after an hour there will be
When n becomes very large then
Then for the growth of continuously reproducing
bacteria, after t hours we have
6
The Slope of the Exponential Function
Lets find the slope ?N/?t of the function N N0
et
To interpret e ?t we can use the interpretation
of e introduced by Euler
Then
Hence
and
Thus, we conclude that the slope of the
exponential function is the exponential function.
This means, that as the function grows, the slope
grows. The more there is, the faster the rate of
change.
7
The Graph of the exponential function
An exponential function applies whenever values
of the independent variable forming an arithmetic
progression are accompanied by values of the
dependent variable that form a geometric
progression. Thus the function y 3x, which fits
the values in the table is an exponential
function.
To recognize an exponential trend in a data set,
we make use of the key algebraic property of
exponential functions  f(x) a bx . Namely f
(x ?x) ab(x ?x) f(x) b?x We can read this
equation this way If the input  x  is increased
by a constant interval (?x), then the output
 f(x)  will increase by a constant multiple (b
?x) . In this example b ?x 3 .
8
Growth and Decay Graphs
The growth graph has the following basic shape
This shape occurs whenever the base of the
exponential function is greater than 1.  Using
function notation, we have a growth graph for
The decay graph has the following basic shape
This shape occurs whenever the base of the
exponential function is between 0 and 1.  Using
function notation, we have a decay graph for
9
Negative Growth and Decay Graphs
We can see that the negative growth and decay
graphs are the mirror reflections about the
X-axis of the corresponding positive ones.
10
Exponential Function Graphic Representation

• Exponential graphs are asymptotic to the x-axis
  In the long term, in one direction or the other,
  they become arbitrarily close to, but never
  touch, the x-axis.
• The graph of yax always intercepts the Y-axis at
  y 1 (because a0 1 independent of the value
  of a).
• They live entirely in a half plane on one side or
  the other of the x-axis.
• The curves may either pull away from or approach
  the X-axis. These changes may occur at increasing
  rates (more positive or less negative) or
  decreasing rates (less positive or more
  negative).
• The above graphs show just a few of the members
  of the family of exponential functions.

11
Rate of Increase
y
r .6
r .4
r .2
x
This is the general form of the exponential
functions in applications, where the parameter r
can be a positive or negative constant. No
matter what the value of r is, when t 0, N
N0. Thus N0 is the quantity present at the
instant from which we begin to measure time. In
the graph it is represented how increasing r the
shorter it will take for the function to increase
(more sharply the graph).
12
Questions
Would the graph of show
exponential growth or exponential decay?

• Growth
• Decay

?
Would the graph of show
exponential growth or exponential decay?

• Growth
• Decay

?
13
The Argument of the Exponential Function should
be Dimensionless
Lets consider the growth process
The process we were analyzing originally
Can be expressed by the function
Note that if t has units of time, then T has
units of time so that the unit cancel. You cannot
take the exponential of a unit, be it a second or
a meter!
14
Exponential Decay
An exponential decay in general is represented by
the function
Where is called the decay constant.
In what of these graphs is the decay constant
larger?
See that a larger decay constant causes the graph
to decay more slowly.
15
Terminal Speed
As an object falls through the air, it
accelerates, gaining speed however the speed
does not increase indefinitely. If the object
falls for a long enough time, the air resistance
becomes large enough to completely negate the
effect of gravity. When this point is reached,
the object has attained a terminal speed and no
longer accelerates.
Velocity of a falling object in vacuum
The speed reached from a high of 2 km is 100 m/s.
On the other hand, it is known that raindrops
falling from clouds reach terminal speed
approximately of 5 20 m/s.
Terminal velocity ( v0 ) of falling object in a
medium
16
The velocity of a certain falling object (which
is being affected by air resistance) is given
by v 95(1 - e-0.1t ) where v is in km/h and t
is the time of fall in hours. Sketch the graph.
Answer A table of values gives us
The sketch is
We see that the terminal velocity of the object
is 95 km/h.
17
How to get variables out of the exponent The
Logarithms
Given the function
The inverse function is defined as
The logarithm is the inverse of exponentiation.
We call the function ln the natural logarithm,
and it corresponds to the base of e. For other
bases
In general, the log undoes the base raised to a
power, giving the value of the power.
18
Here are some places to look for logarithms

• Musical staff for notes (distance vs frequency)

• Richter scale for earthquakes (magnitude vs
  energy)

• Measurement of sound. (decibels vs. intensity)

• Star magnitudes. (magnitude vs. brightness)

   

• Cram studying. (amount forgotten vs. time)

19
The Properties of Logarithms and Exponents
From this table we can see that the properties of
the logs directly follow from the function
definition and the exponent laws.
20
Base Change
Base other that 10 and exponents other than
natural numbers 1, 2, 3, are needed for
applications. For example, the world population t
years from now is predicted to grow by a factor
close to N (1.02)t . Certainly t does not need
to be a whole number of years and the base 1.02
should not be 10 (or we are going to be in real
trouble). If we want to rewrite N to base 10 we
can do
Since

1
In general, the change between base a and b is
given by
21
Half-Life t1/2
Suppose that the elements in a sample are
decreasing in time through an exponential decay
function
Where is the decay constant. Find the
time when only half the sample is left.
We need to find the time when
Since the variable is in the exponent, and
because the log undoes exponentiation, we have
Thus, the half-life is a simple multiple of the
decay constant.
22
Carbon Dating
Cosmic ray protons blast nuclei in the upper
atmosphere, producing neutrons which in turn
bombard nitrogen, the major constituent of the
atmosphere. This neutron bombardment produces the
radioactive isotope carbon-14. The radioactive
carbon-14 combines with oxygen to form carbon
dioxide and is incorporated into the cycle of
living things.
23
Carbon Dating
Carbon-14 is a beta emitter and decays back into
nitrogen. All living things contain some
carbon-14 obtained from the amount existing in
our atmosphere.
The carbon-14 in living things changes into
stable nitrogen. But because living things
breath, this decay is compensated keeping a fixed
ratio of carbon-14 to carbon-12. When a plant or
animal dies, however, replenishment stops. Then,
the percentage of carbon-14 steadily decreases-
at a known rate. The half-life of carbon-14 is
about 5730 years.
24
Ex. 3-3. While laying a new road, a work crew
uncovers the remains of an ancient boat. The
archeologist employed to investigate brings a
wood sample to you for dating. It weighs 50 grams
and shows a C14 activity of 200 disintegrations
per minute. If the C14 activity of living plants
is 12 disintegrations per minute and per gram,
estimate the age of the wood.
We know that the activity obeys an exponential law
The present activity A is given, and we also
know
Then, the age of the wood can be found from
Since the initial activity A0 is that of living
plants, and the present activity A is
and the decay const. can be found from the
half-life
Finally, the age of the dead wood is
25
Log Graph
y
x
The log is a very slowly function of x. Because
ln (x) is the inverse function of e, we have that
y ln (x) is the mirror reflection about the
line y x of y ex. Because ex grows at such a
large rate, the ln grows slowly. Note from the
graph that the negative numbers are not part of
the domain of the logarithm. It is a consequence
of the fact that there is not value of x that can
produce a negative value of y in y ex.
26
The Electromagnetic Spectrum
Radio Waves
Micro Waves
Infrared
Visible Light
Ultraviolet
X-Rays
Gamma-Rays
27
Semi-Log Graph
Log (y)
1000
800
600
400
200
x
2
4
6
8
10
Often, when plotting a function to see its
behavior over a large range of values, or if f(x)
changes over many orders of magnitude for a
relatively small change in x, then the semi-log
graph can be used.
On the other hand, if we plot log (y) vs x, we
can make the plot quite easily, as shown in the
Fig. Of course, log (y) log(100)x log(2), so
we obtain a straight line.
28
Semi-log Paper
A special kind of graph paper makes the analysis
of exponential growth and decay problems much
simpler. Consider that knowing a set of values
(x, y) corresponding to the function
we want to find the value of b. From the plot of
this exponential function for both positive and
negative values of x is not trivial how to get b.
Taking the log in both sides of the equation we
have
Which is the equation of a straight line with
slope
intercept
and dependent variable
Special graph paper is available on which the
vertical axis is marked in a logarithmic fashion.
When we use this semi-log paper we dont have to
compute the values of the logarithms before
plotting but can label the graph with the values
of the function directly.
29
Semi-logarithmic Plot
30
The slope of a line in a semi-log graph
To find b in we can consider
the values of x corresponding to a jump of 10 in
the y-axis
The values of x2 and x1 can be found in the graph.
31
Ex. 3-4. To determine the acceleration of gravity
in the Lab you can use a pendulum. Suppose that
changing the length L of the pendulum and
measuring the corresponding period T its
obtained
How can we obtain the acceleration of gravity g
from those data?
If we plot those values we obtain the following
graph
T(s)
L (m)
From this graph is difficult to realize how to
get g. Nevertheless, if we know that it is a
power-law relation we can use the log-log graph
to find easily g.
log T
log L
32
Exponential (y aex) Power Functions (y axm )

• To determine if a function is exponential or a
  power law we can follow the rule
•     On a semi-log plane, an exponential function
  is linear.
•     On a log-log plane, a power law is linear.

Exponential Function
Slope
Power Law
Slope
33
(No Transcript)
34
Three representations for the function
35
Three representations for the function
36
Three representations for the function
37
(No Transcript)
38
Arc Length
If we divide the perimeter of the circumference L
by its radius r we obtain
Independent of the value of r. In this expression
p is an irrational number that can be linked to
the integers numbers through the series
We can introduce an angular measure ? defined by
Then, ? is given in radians which is a fraction
of 2p . Note that radians are dimensionless. We
see that 2p corresponds to the whole 360 angle.
39
Circular Motion
Suppose that an object is traveling with a
constant speed v around a circle of radius r.
Then, we can write
Defining the angular velocity as
We have the following relation between the linear
and the angular velocities
Because the angular velocity in this case is also
constant we will have for ? 2p the
corresponding time t T, which is the period of
rotation.
40
What is the relation between the linear
velocities of the objects A and B, if they rotate
with equal constant angular velocity as shown in
the Fig.?
A
rA
B
rB
41
A person travels with a constant linear speed of
5 m/s around a circle of radius 4 m. Find how
long it takes to complete one cycle.
Data v 5 m/s, r 4 m
Question T ?
Then, we have that in 5 s the person completes
one circle.
42
Linear Frequency
Since the period is the number of seconds per
cycle, then its inverse is the number of cycles
per second. The inverse of the period T is
defined as the linear frequency f .
The unit of f is 1/s which is defined as Hertz
(Hz).
From the formulas for the angular and linear
velocities we have
As can be seen from these formulas, since 2p is
in radians the dimension of ? is that of f.
That is why ? is some times called an angular
frequency.
43
A turbine in an electric power plant rotates at
3600 rpm. Find its angular speed in rad/s, and
determine the linear speed of a point 72 cm from
the rotation axis.
Data f 3600 rpm, r 72 cm
Question ? ?, v (r 72 cm) ?
How will behave the angular and linear speeds of
a point at 36 cm from the rotation axis?

• The angular velocity will be the same

• The linear velocity will decrease to one half

44
Trigonometric Functions
If the position of an object is changing on a
circle we see that the angle ? is also changing.
We can also note that if the angle ? changes, x
and y also change. It seems plausible that these
two coordinates must be connected to ?.
The connection between ?, the slop of r, and the
coordinates x and y, are given by the
trigonometric functions
Since x and y can never be larger that r , so sin
? and co ? must never be larger than 1.
45
Inverse Transformations
The Cartesian coordinates x and y are given as
functions of the polar coordinates ? and r by
the equations
The inverse transformations that connect ? and r
with the coordinates x and y, are given by
The inverse trigonometric functions are referred
by prefixing the name with arc. Note that the
inverse trigonometric functions return an angle.
46
Signs of the Trigonometric Functions
y


-
-
y-

-
x
x-
-

-

-

47
Trigonometric Functions in the Right Triangle
Inverse Trigonometric Functions
48
Graphs
49
(No Transcript)
50
Parity

• The cosine is an even function

• The sine is an odd function

51
Rotation of Trigonometric Functions
at different angles
52
Simple Harmonic Motion
?
A
?
x
shadow
Suppose that a ball on the circle of radius A
starts on the x-axis at x A and moves through
the angle ? in a time t with constant angular
velocity ?. Therefore, the angle ? changes with
time as
The displacement x of the shadow is just the
projection of the radius A onto the x axis
As time passes, the shadow of the ball oscillates
with period T 2p/? between the values of x A
and x -A. This motion is called simple harmonic
motion, a term that means that the periodic
motion is a sinusoidal function of time. There
are many physical systems, as the simple
pendulum, or an object attached to an ideal
spring, etc., that oscillate in a harmonic way.
53
Suppose that the ball on the circle of radius A
10 m starts on the x-axis at x A and moves
through the angle ? with constant angular
velocity ? 3 rad/s. Find the position of the
shadow at t p/4 s .
We know that the shadow displacement will change
in time as
Then
54
Give the displacement as a function of time for
the shadow of a ball moving at a constant angular
velocity on the circle of radius A 10 m, if it
starts on the x-axis at x 0.
We know that the shadow displacement will change
in time as a sinusoidal function of time, but now
the initial condition is
Then, we should introduce an initial phase f
to be determined by the initial condition
Thus
55
Simple Harmonic Motion Velocity
vT
?
?
V
A
?
shadow
The drawing indicates that the velocity V of the
shadow is just the x component of the tangential
velocity VT , that is
Taking into account that
Its obtained
This velocity is also a harmonic function of
time. We can see that when the shadow changes
direction at either end of the oscillatory
motion, the velocity is zero. When the shadow
passes through x 0 position the velocity has a
maximum amplitude A? .
56
A satellite circles the Earth once every 5300 s
in an orbit that passes over the north and south
poles. The radius of the orbit is 6.5 ? 106 m.
The Sun is shining and the satellite casts a
shadow on the Earth . What is the speed of the
shadow as the satellite passes over the equator,
where the Suns rays strike the Earth
perpendicularly?
T 5300 s
A 6.5 ?106 m
The shadow will have the maximum velocity when is
passing through the equator, then