The Mathematics of CT-Scans

Transcription

1 The Mathematics of CT-Scans Tomography has become one of the most important applications of mathematics to the problems of keeping us alive. Modern medicine relies heavily on imaging methods, beginning at the start of the 20th Century with the early use of X-Rays. The basic mathematics behind tomography was worked out by the mathematician Johann Radon in In the 1960 s Allan McLeod Cormack, working in collaboration with Electric and Musical Industries Ltd and Godfrey Hounsfield, developed the first practical scanning device, the celebrated EMI scanner. For this work, Cormack won the Noble Prize. More info, and some more.

2 Example: A tiny CT-Scan

3 Example: A tiny CT-Scan A B C D

4 Example: A tiny CT-Scan A B C D A + B = 2 A + C = 3 A + D = 4 B + C = 3 C + D = 5

5 Example: A tiny CT-Scan A B C D A + B = 2 A + C = 3 A + D = 4 B + C = 3 C + D = 5 Unique solution! A = 1, B = 1, C = 2, D = 3.

6 Testing your clickers How do you feel about Mathematics? (A) I LOVE math (B) I like math, but I am not in love with it. (C) It s complicated. Math and I have a love/hate relationship. (D) I do not like math, and yet, here I am. (E) Math makes me physically ill! Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 4 / 37

7 Testing your clickers How do you feel about Mathematics? (A) I LOVE math (B) I like math, but I am not in love with it. (C) It s complicated. Math and I have a love/hate relationship. (D) I do not like math, and yet, here I am. (E) Math makes me physically ill! Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 4 / 37

8 Our FINAL EXAM will be on SATURDAY 12/13, 1-3pm Two locations (check the registrar s website) Sections 81, 83, 84, 85, 86, 88, 89, 90 and 92 at ITE C80. Sections 91, 93, 94, 95 at OAK 101. There is a practice exam and solutions in the outline section of the website: Covers all Sections Emphasis is on There will be (some) review during class on Thursday 12/4. My office hours this week are as usual: Tuesdays 10:15-11:15 and Thursdays 11-12, at MSB 312. TA s office hours also as usual (see website). My office hours during finals: Wednesday and Friday, 11-12, at MSB 312.

9 Our FINAL EXAM will be on SATURDAY 12/13, 1-3pm Register your iclicker!! Select NO for the question about Learning Management Systems. Use your Net ID as your Student ID Remember to double check you Remote ID is correct.

10 Our FINAL EXAM will be on SATURDAY 12/13, 1-3pm Register your iclicker!! Select NO for the question about Learning Management Systems. Use your Net ID as your Student ID Remember to double check you Remote ID is correct. There will be two Q Center review sessions: December 8th, 9:30-10:30am, LH 205. December 9th, 7-8pm, LH 201. I will be running a review session during the week of finals on Thursday 12/11, 10:15am-12pm, at TLS 154. Prof. Savkar will be running a review session during the week of finals on Friday 12/12, 1pm-4pm, at LH 101.

11 MATH 1131Q - Calculus 1. Álvaro Lozano-Robledo Department of Mathematics University of Connecticut Day 28 - Last Lecture! Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 7 / 37

12 The Fundamental Theorem of Calculus

13 The Fundamental Theorem of Calculus, Part I If f is continuous on [a, b], then the function g defined by x g(x) = f (t) dt is continuous on [a, b] and differentiable on (a, b), and In other words, g (x) = f (x). a dg dx = d x f (t) dt = f (x). dx a

14 The Fundamental Theorem of Calculus Part I d dx x a f (t) dt = f (x).

15 The Fundamental Theorem of Calculus, Part II If f is continuous on [a, b], then b a f (x) dx = F(b) F(a) where F is any antiderivative of f, that is, a function such that F = f.

16 The Fundamental Theorem of Calculus Part II b a f (x) dx = F (b) F (a) where F = f. Alternative notation: b b f (x) dx = F (x) a a

17 The Fundamental Theorem of Calculus Part II (The Net Change Theorem) b a F (x) dx = F(b) F(a) where F = f.

18 Definite Integrals versus Indefinite Integrals A definite integral b a f (x) dx is a number, that represents the area (with sign) below the graph of y = f (x). The indefinite integral f (x) dx is the collection of all antiderivative functions of the function y = f (x), i.e., f (x) dx = F(x) + C where F(x) is an antiderivative of f (x) (i.e., F (x) = f (x)), and C is an arbitrary constant. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 14 / 37

19 Function f (x) f (x) = 0 f (x) = k, a constant f (x) = x Family of Antiderivatives 0 dx = C k dx = kx + C x dx = x C f (x) = x 2 x 2 dx = x C f (x) = x n, a real number n = 0, 1 x n dx = x n+1 n+1 + C f (x) = 1 x for any constant C. 1 x dx = ln( x ) + C Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 15 / 37

20 Function f (x) f (x) = e x f (x) = a x, with a > 0 f (x) = sin x Family of Antiderivatives e x dx = e x + C a x dx = ax ln a + C sin x dx = cos x + C f (x) = cos x cos x dx = sin x + C f (x) = 1/ 1 x 2 1/ 1 x 2 dx = arcsin x + C f (x) = 1/ 1 x 2 1/ 1 x 2 dx = arccos x + C f (x) = 1/(1 + x 2 ) 1/ 1 + x 2 dx = arctan x + C for any constant C. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 16 / 37

21 Rules of Antidifferentiation Suppose the functions f (x) and g(x) have antiderivatives F(x) and G(x), respectively. Function k f (x), with k constant f (x) + g(x) f (x) g(x) f (x) g(x) f (g(x)) g (x) Family of Antiderivatives kf (x) dx = k f (x) dx f (x) + g(x) dx = f (x) dx + g(x) dx f (x) g(x) dx =??? f (x) dx =??? g(x) f (g(x)) g (x) dx = f (g(x)) + C for any constant C. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 17 / 37

22 Indefinite Integrals: Method of u-substitution The u-substitution rule is based on the fact that if u = g(x), then f (g(x))g (x) dx = f (u) du.

23 Indefinite Integrals: Method of u-substitution The u-substitution rule is based on the fact that if u = g(x), then f (g(x))g (x) dx = f (u) du. When to use it: We use the method of substitution for indefinite integrals which look like the result of a chain rule. In particular, try to use this method when you see a composition of two functions.

24 Indefinite Integrals: Method of u-substitution The u-substitution rule is based on the fact that if u = g(x), then f (g(x))g (x) dx = f (u) du. When to use it: We use the method of substitution for indefinite integrals which look like the result of a chain rule. In particular, try to use this method when you see a composition of two functions. How to use it: In this method, we go from integrating with respect to x to integrating with respect to a new variable, u, which makes the integral much easier. 1 Find inside the integral the composition of two functions and set u = the inner function. Also, calculate u (x) and write du = u (x)dx. 2 Substitute everything in the integral that depends on x in terms of u. 3 Integrate with respect to u. 4 Once we have the result of integration in terms of u (+C), substitute back in terms of x.

25 Indefinite Integrals: Method of u-substitution Example 1 Find (3x + 5) dx. 3 Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 19 / 37

26 Indefinite Integrals: Method of u-substitution Example cos(ln x) Find dx. x Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 37

27 Indefinite Integrals: Method of u-substitution Example Find tan x dx. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 21 / 37

28 Indefinite Integrals: Method of u-substitution Example Evaluate 2 0 x x dx. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 22 / 37

29 Indefinite Integrals: Method of u-substitution Example Evaluate e 1 ln x x dx. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 23 / 37

30 Review

31 Is there a curve in this class? The Normal Distribution: In probability theory, the normal (or Gaussian) distribution is a continuous probability distribution that has a bell-shaped probability density function, known as the Gaussian function or informally the bell curve. The normal distribution is considered the most prominent probability distribution in statistics. The normal distribution arises as the outcome of the central limit theorem, which states that under mild conditions the sum of a large number of random variables is distributed approximately normally. The parameter μ is the average value and σ is the standard deviation. μ = 1 n x i, σ = 1 N (x i μ) 2. N i=1 N i=1 f μ,σ(x) = 1 (x μ) 2 σ 2π e 2σ 2 Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set, while two standard deviations from the mean (medium and dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue) account for about 99.7%.

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33 An Example: Area Under the Bell Curve x Let g(x) = e t2 dt. Analyze the properties of g(x), and g (x) Graph of g(x). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 27 / 37

34 An Example: Area Under the Bell Curve x Let g(x) = e t2 dt. Analyze the properties of g(x), and g (x) Graph of g (x). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 28 / 37

35 Let g(x) = x 0 e t2 dt. Analyze the properties of g (x), including: domain, possible horizontal asymptotes, increasing and decreasing intervals, intervals of concavity, and local max/min, global max/min.

36 x Let g(x) = e t2 dt. Analyze the properties of g (x),... 0

37 Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 35 / 37