Presentation 1 - The EXCEL Progr

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Presentation 1 - The EXCEL Progr Powered By Docstoc
					           Limits


              by
      Andrew Winningham
UCF EXCEL Applications of Calculus
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your i<Clicker responses to in-class questions
 No clicker  No credit
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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
 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
         
 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
 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
                                      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.
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
 the magnitude of the               Ay = A sin 
 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
                                      F = mg
  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
                                                                             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
Urinary bladder               3%
                                                             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
     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†                                              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
         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
     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
     Urinary bladder                                        73                    78                     82
*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




                                                                                                            Age-Adjusted Lung Cancer Death
                                        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

                     Bladder
                     Rectum
                     Hips
                     External contour (skin)
Patient Contours
Radiation Delivery




    Linear accelerator
         (Linac)
Radiation Delivery
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
Sealed radioactive
 deliver radiation at a short distance.
                                          137Cs
                      103Pd
Brachytherapy



     MammoSite
Applications of Limits




                         lim Dnormal tissue  0
                    DCTV  DRx