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Limits by Andrew Winningham UCF EXCEL Applications of Calculus i<Clickers We will be using an electronic response system called i<Clickers You are required to purchase a keypad from the book store and register your pad online Go to www.iclicker.com/registration and enter your name, NID, and clicker ID. Your grade will be derived completely from your i<Clicker responses to in-class questions No clicker No credit i<Clickers The clicker is a two-way RF transmitter/receiver Blue LED - power on Green LED = response received Red LED = response not received After each class, check WebCT Report problems immediately after class i<Clicker Test Is your clicker working? A. Yes B. No 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 xa if for every number > 0 there is a number > 0 such that f x L whenever 0 xa 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 x0 x0 x lim x 0 x0 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 xa if for every number > 0 there is a number > 0 such that f x L whenever 0 xa 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 a2 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