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Limits
by
Andrew Winningham
UCF EXCEL Applications of Calculus
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equations or steps
Count on exam questions being completely different
(in details) from homework problems (concept
based)
When you study, use variations
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Limits
So, what is a limit exactly?
The formal definition:
Let f be a function defined on some open
interval that contains the number a, except
possibly a itself. Then we say that the limit of
f(x) as x approaches a is L, and we write
lim f x   L
xa
if for every number  > 0 there is a number
 > 0 such that
f x   L  
whenever
0  x a 
Is this formal definition necessary?
Consider the following expression:
sin x
x
This expression is central to the description
of the intensity of light after it has passed
through a diffraction grating.
sin x 0
sin x limx0
lim


x0
x
lim x
0
x0
History
In the beginning, Newton’s and Leibniz’s
work was based on the idea of ratios and
products of arbitrarily small quantities or
numbers.
Newton  fluxions, Leibniz  differentials
These ideas arose from the work of Wallis,
Fermat, and Decartes.
The root of these issues is the following
question:
How close can two numbers be without being
the same number?
History
Equivalently, since we can also consider the
difference of two such numbers:
How small can a number be without being zero?
The effective answer provided by Newton and
others is the concept of infinitessimals, positive
quanitites that are smaller than any non-zero real
number. (What?)
This concept was needed because differential
calculus relies crucially on the consideration of
ratios in which both numerator and denominator go
to zero simultaneously.
History
Critics complained that it is impossible to
imagine in a concrete way something that is
infinitely small.
More importantly, without a theoretical basis
for infinitessimals, mathematicians couldn’t
be sure that there methods were correct.
D’Alembert introduced a new way of thinking
about ratios of vanishing quantities, the
method of limits.
History
D’Alembert saw the tangent to a curve as a
limit of secant lines.
As the end point of the secant converges on
the point of tangency, it becomes identical to
the tangent “in the limit.”
B
A

yB  yA
slope 
xB  xA
History
This is precisely how a derivative is
motivated in your calculus course, but it is
still only a geometrical argument.
It is subject to objections like those seen in
Zeno’s paradoxes.
Cauchy finally provided the rigorous
formulation of the limit concept that we just
discussed.
Archimedean Principle
Let a and b be real numbers. Then we may
find some natural number n such that a < nb.
Dividing by n, we see that a/n < b.
So, for any real number r (e.g. b/a), we may
find an n such that 1/n < r.
Notice that no matter how large n is, 1/n is
never zero.
Also note that no matter what positive
number r that you pick, an n can be found so
that 1/n is closer to zero than r.
Archimedean Principle
Since we are getting closer and closer to
zero as n gets larger, we say that the limit of
1/n as n goes to infinity is zero.
1
lim  0
n n
Note: This does not say that 1/n is ever equal
to zero
 it say that n is ever infinite
Nor does
Archimedean Principle
Note: This does not say that 1/n is ever equal
to zero
Nor does it say that n is ever infinite
Instead, it means that - by choosing n
sufficiently large - the quantity 1/n can be
made as close to zero as desired.
Limits
The formal definition:
Let f be a function defined on some open
interval that contains the number a, except
possibly a itself. Then we say that the limit of
f(x) as x approaches a is L, and we write
lim f x   L
xa
if for every number  > 0 there is a number
 > 0 such that
f x   L  
whenever
0  x a 
The Real World
It’s rare that one ever calculates a limit using
the formal definition. The limit rules usually
work.
Moreover, limits are usually calculated to
understand the behavior of an expression at
some extreme.
We’re going to take a look at example of how
limits are used in “the real world.”
Electric Dipoles
In general, the
magnitude of an
electric field due to a
point charge is given
by
kQ
E r 
kQ k Q
E x   2 
2
x
x  a
 1
1 
 kQ 2 
2 
x  a 
x
We’d like to know how
the dipole’s field varies
at distances
much

greater than a
(x >> a)
r
2
Electric Dipole
Now, with a little factoring, we see that
2ax  a
2
x x  a 
2
2

2ax 1  a 2x 
x  x 1  a x 
2
2

2ax 1  a 2x 
x 1  a x 
4
2
When x gets much larger than a, a/x goes to
zero. Therefore, when x >> a
2kQa
E x  
3
x
Limits can reduce your mistakes
A variation of this exercise is present in almost
every problem that you will encounter in
PHY
2048 and PHY 2049
F
?

?
F = mg
F
Vectors and Scalars
A scalar quantity is completely specified by a
single value with an appropriate unit and has no
direction (e.g., temperature, pressure, mass,
speed, etc.)
A vector quantity is completely described by a
number and appropriate units plus a direction
(e.g., force, displacement, velocity,
acceleration)
Vectors
We represent vectors graphically using
arrows.
The magnitude of the quantity is represented
by the length of the vector (arrow).
The direction of the quantity is represented
by the direction of the vector (arrow).
The distance traveled
is a scalar.
The displacement is a
vector.
Adding vectors graphically
Vector components
A component is a part
The components of A
shown here, Ax and Ay,
are just the projections of
A along the x- and y-axes.
Any vector can be
written as the sum of
two components that
lie in orthogonal
directions.

Vector components
Because the vector
and the components
form a right triangle,
we can use trig
functions to calculate
A
Ay = A sin 
the magnitude of the
components if we

know the magnitude
of the vector and the Ax = A cos 
angle .
Limits can reduce your mistakes
A variation of this exercise is present in almost
every problem that you will encounter in
PHY
2048 and PHY 2049
F
?

?
F = mg
F
Limits can reduce your mistakes
Suppose we choose to call
the bottom angle 
F
F = mg

F
F = F cos 
We can use a limit to quickly
determine if we made the right
lim F||  lim F cos  F
 0
 0
choice.
So we made the wrong
choice!
Application of Limits: Treatment of
cancer with radiation
U.S. Mortality, 2004
Rank
Cause of Death
No. of
deaths
% of all
deaths
1.
Heart Diseases
652,486 27.2
2.
Cancer
553,888 23.1
3.
Cerebrovascular diseases
150,074
6.3
4.
Chronic lower respiratory diseases
121,987
5.1
5.
Accidents (Unintentional injuries)
112,012
4.7
6.
Diabetes mellitus
73,138
3.1
7.
Alzheimer disease
65,965
2.8
8.
Influenza & pneumonia
59,664
2.5
9.
Nephritis
42,480
1.8
33,373
1.4
10. Septicemia
Source: US Mortality Public Use Data Tape 2004, National Center for Health Statistics, Centers for Disease Control
and Prevention, 2006.
Change in the US Death Rates by Cause,
1950 & 2004
Rate Per 100,000
600
586.8
1950
500
2004
400
300
217.0
193.9
180.7
200
185.8
100
50.0
48.1
19.8
0
Heart
Diseases
Cerebrovascular
Diseases
Pneumonia/
Influenza
Cancer
* Age-adjusted to 2000 US standard population.
Sources: 1950 Mortality Data - CDC/NCHS, NVSS, Mortality Revised.
2004 Mortality Data: US Mortality Public Use Data Tape, 2004, NCHS, Centers for Disease Control and
Prevention, 2006
2007 Estimated US Cancer Deaths
Lung & bronchus
31%
Men
289,550
Women
270,100
26%
Lung & bronchus
15%
Breast
Colon & rectum
Prostate
9%
Colon & rectum
9%
10%
Pancreas
6%
6%
Pancreas
Leukemia
4%
Liver & intrahepatic
bile duct
4%
6%
Ovary
4%
Leukemia
Esophagus
4%
3%
Urinary bladder
3%
Non-Hodgkin
lymphoma
Non-Hodgkin
lymphoma
3%
3%
Uterine corpus
2%
Brain/ONS
Kidney
3%
2%
All other sites
24%
Liver & intrahepatic
bile duct
23% All other sites
ONS=Other nervous system.
Source: American Cancer Society, 2007.
2007 Estimated US Cancer Cases
Men
766,860
Women
678,060
Prostate
29%
26%
Breast
Lung & bronchus
15%
15%
Lung & bronchus
Colon & rectum
10%
11%
Colon & rectum
Urinary bladder
7%
6%
Uterine corpus
Non-Hodgkin
lymphoma
4%
4%
Non-Hodgkin
lymphoma
Melanoma of skin
4%
Kidney
4%
4%
Melanoma of skin
Leukemia
3%
4%
Thyroid
Oral cavity
3%
3%
Ovary
Pancreas
2%
3%
Kidney
3%
Leukemia
All Other Sites
19%
21%
All Other Sites
*Excludes basal and squamous cell skin cancers and in situ carcinomas except urinary bladder.
Source: American Cancer Society, 2007.
Lifetime Probability of Developing Cancer, by
Site, Men, 2001 - 2003*
Site
Risk
All sites†
Prostate
1 in 2
1 in 6
Lung and bronchus
1 in 12
Colon and rectum
1 in 17
Urinary bladder‡
1 in 28
Non-Hodgkin lymphoma
1 in 47
Melanoma
1 in 49
Kidney
1 in 61
Leukemia
1 in 67
Oral Cavity
1 in 72
Stomach
1 in 89
* For those free of cancer at beginning of age interval. Based on cancer cases diagnosed during 2001 to 2003.
† All Sites exclude basal and squamous cell skin cancers and in situ cancers except urinary
bladder.
Source: DevCan: Probability of Developing or Dying of Cancer Software, Version 6.1.1 Statistical Research and
Applications Branch, NCI, 2006. http://srab.cancer.gov/devcan
‡ Includes invasive and in situ cancer cases
Lifetime Probability of Developing Cancer, by
Site, Women, 2001 - 2003*
Site
Risk
All sites†
Breast
1 in 3
1 in 8
Lung & bronchus
1 in 16
Colon & rectum
1 in 19
Uterine corpus
1 in 40
Non-Hodgkin lymphoma
1 in 55
Ovary
1 in 69
Melanoma
1 in 73
Pancreas
1 in 79
Urinary bladder‡
1 in 87
Uterine cervix
1 in 138
* For those free of cancer at beginning of age interval. Based on cancer cases diagnosed during 2001 to 2003.
† All Sites exclude basal and squamous cell skin cancers and in situ cancers except urinary bladder.
‡ Includes invasive and in situ cancer cases
Source: DevCan: Probability of Developing or Dying of Cancer Software, Version 6.1.1 Statistical Research and
Applications Branch, NCI, 2006. http://srab.cancer.gov/devcan
Five-year Relative Survival (%)* During Three
Time Periods by Cancer Site
1975-1977
50
1984-1986
53
1996-2002
66
Breast (female)
75
79
89
Colon
51
59
65
Leukemia
35
42
49
Lung and bronchus
13
13
16
Melanoma
82
86
92
Non-Hodgkin lymphoma
48
53
63
Ovary
37
40
45
Pancreas
2
3
5
Prostate
69
76
100
Rectum
49
57
66
Urinary bladder
73
78
82
Site
All sites
*5-year relative survival rates based on follow up of patients through 2003.
†Recent changes in classification of ovarian cancer have affected 1996-2002 survival rates.
Source: Surveillance, Epidemiology, and End Results Program, 1975-2003, Division of Cancer Control and
Population Sciences, National Cancer Institute, 2006.
5000
100
4500
90
4000
80
3500
70
3000
60
2500
50
2000
40
1500
30
1000
20
500
10
0
1900
1905
1910
1915
1920
1925
1930
1935
1940
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
0
Year
*Age-adjusted to 2000 US standard population.
Source: Death rates: US Mortality Public Use Tapes, 1960-2003, US Mortality Volumes, 1930-1959, National
Center for Health Statistics, Centers for Disease Control and Prevention, 2005. Cigarette consumption: US
Department of Agriculture, 1900-2003.
Age-Adjusted Lung Cancer Death
Rates*
Per Capita Cigarette Consumption
Tobacco Use in the US, 1900 - 2003
So how does the concept of a limit
apply to cancer treatment?
The effectiveness
of a treatment
improves as the
limit of the volume
treated
approaches the
volume
encompassed by
the disease.
CT Simulation
CT scan (3-5 mm cuts)
Volume determination
Treatment Planning
Volume determination
Outline structures:
Target:
Prostate only
Prostate + Seminal
vesicles
Bladder
Rectum
Hips
External contour (skin)
Patient Contours
Radiation Delivery
Linear accelerator
(Linac)
Radiation Delivery
Tumor Volume vs. Treatment Volume
GTV = Gross tumor
volume
PTV
CTV
CTV = Clinical tumor
volume
PTV = Planning
treatment volume
PTV must include
errors from set-up,
motion, anatomy
changes, etc.
GTV
Tumor Volume vs. Treatment Volume
We try to develop
plans so that the CTV
gets at least 100% of
the prescribed dose
while minimizing the
dose to normal
tissue.
lim Dnormal tissue  0
DCTV  DRx
Multiple Angles for Conformity
Tomotherapy
Imaging and Alignment
Helical Tomotherapy
Imaging and Alignment
MVCT
IMAGE
KVCT
IMAGE
Imaging and Alignment
MVCT
IMAGE
ANATOMY
OVERLAY
DOSE and
CONTOUR
Respiratory Gating
Implanted Markers
Respiratory Gating
Patient Immobilization
Multiple Imaging Modalities
CT (x-rays)
MRI
Multiple Imaging Modalities
Image Fusion
Brachytherapy
Brachytherapy
Sealed radioactive sources are used to
deliver radiation at a short distance.
137Cs
103Pd
Brachytherapy
MammoSite
Applications of Limits
lim Dnormal tissue  0
DCTV  DRx