# Presentation 1 - The EXCEL Progr by fjzhangweiqun

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```									           Limits

by
Andrew Winningham
UCF EXCEL Applications of Calculus
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Tips for success
Strive to understand concepts, not just memorize
equations or steps
Count on exam questions being completely different
(in details) from homework problems (concept
based)
When you study, use variations
Make connections. Don’t leave conflicts unresolved.
Ask questions. If anything is unclear, go to office
hours and use the other EXCEL resources
Don’t wait to study until the night before the exam
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 lim sin x 0
lim         x0

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
yB  yA
slope 
xB  xA

A


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
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

Nor does it say that n is ever infinite
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        2
r
We’d like to know how
kQ    k Q     the dipole’s field varies
E x   2 
x    x  a2
at distances much

 1      1 
greater than a
 kQ 2         2 
x    x  a              (x >> a)
Electric Dipole
Now, with a little factoring, we see that

2ax  a 2            2ax 1  a 2x             2ax 1  a 2x 
                           
x x  a            x  x 1  a x            x 1  a x 
2          2        2                  2         4         2
             

When x gets much larger than a, a/x goes to
zero. Therefore, when x >> a

2kQa
E x      3
x
A variation of this exercise is present in almost
every problem that you will encounter in       PHY
2048 and PHY 2049

?            F

?
F
F = mg
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.
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
the magnitude of the               Ay = A sin 
components if we         
know the magnitude
of the vector and the Ax = A cos 
angle .
A variation of this exercise is present in almost
every problem that you will encounter in       PHY
2048 and PHY 2049

?            F

?
F
F = mg
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.
choice!
Application of Limits: Treatment of
U.S. Mortality, 2004
No. of       % of all
Rank      Cause of Death                                                   deaths       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

10. Septicemia                                                        33,373         1.4
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     185.8
200                                 180.7

100
50.0            48.1
19.8
0
Heart               Cerebrovascular              Pneumonia/                   Cancer
Diseases                 Diseases                   Influenza
* 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     Women     26%   Lung & bronchus
289,550   270,100
Prostate                      9%                             15%   Breast
Colon & rectum                9%                             10%   Colon & rectum
Pancreas                      6%                             6%    Pancreas
Leukemia                      4%
6%    Ovary
Liver & intrahepatic          4%
bile duct                                                 4%    Leukemia
Esophagus                     4%                             3%    Non-Hodgkin
lymphoma
3%    Uterine corpus
Non-Hodgkin                  3%
lymphoma                                                  2%    Brain/ONS
Kidney                        3%                             2%    Liver & intrahepatic
All other sites              24%                                              bile duct
23% All other sites

ONS=Other nervous system.
Source: American Cancer Society, 2007.
2007 Estimated US Cancer Cases
Men             Women
766,860           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                    4%                                             4%       Non-Hodgkin
lymphoma                                                                              lymphoma
Melanoma of skin               4%
4%       Melanoma of skin
Kidney                         4%
4%       Thyroid
Leukemia                       3%
3%       Ovary
Oral cavity                    3%
3%       Kidney
Pancreas                       2%
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†                                             1 in 2
Prostate                                               1 in 6
Lung and bronchus                                      1 in 12
Colon and rectum                                       1 in 17
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
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†                                              1 in 3
Breast                                                  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
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
Site                                             1975-1977            1984-1986            1996-2002
All sites                                           50                   53                    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
*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.
Tobacco Use in the US, 1900 - 2003
5000                                                    100

4500                                                    90
Per Capita Cigarette Consumption

4000                                                    80

3500                                                    70

3000                                                    60

Rates*
2500                                                    50

2000                                                    40

1500                                                    30

1000                                                    20

500                                                    10

0                                                    0
1900
1905
1910
1915
1920
1925
1930
1935
1940
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
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.
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

Rectum
Hips
External contour (skin)
Patient Contours

Linear accelerator
(Linac)
Tumor Volume vs. Treatment Volume
GTV = Gross tumor              PTV
volume
CTV
CTV = Clinical tumor
volume
GTV
PTV = Planning
treatment volume

PTV must include
errors from set-up,
motion, anatomy
changes, etc.
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              DOSE and
ANATOMY
CONTOUR
OVERLAY
Respiratory Gating
Implanted Markers
Respiratory Gating
Patient Immobilization
Multiple Imaging Modalities
CT (x-rays)        MRI
Multiple Imaging Modalities

Image Fusion
Brachytherapy
Brachytherapy sources are used to
deliver radiation at a short distance.
137Cs
103Pd
Brachytherapy

MammoSite
Applications of Limits

lim Dnormal tissue  0
DCTV  DRx

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