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Theorem 1 Let ˘be an equivalence relation on X and let g X !Y be a function with the property that g(x) = g(x0) whenever x˘x0 Then there exists a (wellde ned) function f X=˘!Y given by the formula f(x) = g(x) Before we prove this theorem, let us return to Example 3 We started with the function r 5 Z !f0;1;2;3;4g Here, r% " % ( % ) ) / & , 0 % 0 % 1 2 3 4 5 4 6 7 8 9 7 ;V W \ v \ Z c r v b >4E g C T I 8 ò r K 8 U } ?

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Title Qualtrics Survey Software Author rmarriott Created Date AM94 7 Metric Spaces Then d is a metric on R Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolutevalue metric Example 74 Define d R2 ×R2 → R by d(x,y) = √ (x1 −y1)2 (x2 −y2)2 x = (x1,x2), y = (y1,y2)Then d is a metric on R2, called the Euclidean, or ℓ2, metricIt corresponds toSCRABBLE® is a registered trademark All intellectual property rights in and to the game are owned in the USA and Canada by Hasbro Inc, and throughout the rest of the world by JW Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc Mattel and Spear are not affiliated with Hasbro

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The mean or expected value of g(X) is E(g(X)) = Z g(x)dF(x) = Z g(x)dP(x) = (R 1 1 g(x)p(x)dx if Xis continuous P j g(x j)p(x j) if Xis discrete Recall that 1 Linearity of Expectations E P k j=1 c jg j(X) = P k j=1 c jE(g j(X)) 2 If X 1;;X n are independent then E Yn i=1 X i!The Fundamental Theorem of Calculus, Part 1 If f is continuous on a,b, then the function g defined by g(x) = Z x a f(t)dt a ≤ x ≤ b is continuous on a,b and differentiable onThe CDC AZ Index is a navigational and informational tool that makes the CDCgov website easier to use It helps you quickly find and retrieve specific information Find links to key CDC topic areas in this alphabetical index Skip directly to site content Skip directly to AZ link Español

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Counterclockwise Then for any z 0 inside C f(z 0) = 1 2ˇi Z C f(z) z z 0 dz (1) Re(z) Im(z) z0 C A Cauchy's integral formula simple closed curve C, f(z) analytic on and inside C This is remarkable it says that knowing the values of fon the boundary curve Cmeans we knoweverythingabout f inside C!!Purple Line Transit Neighborhood Plan Land Use & Zoning Concept MixedUse Corridors & Character Residential Areas S Y O S T ER D R W CARUSO PL W 1ST ST(10) Z x a2 x2 dx= 1 2 lnja2 x2j (11) Z x2 a 2 x dx= x atan 1 x a (12) Z x3 a 2 x dx= 1 2 x2 1 2 a2 lnja2 x2j (13) Z 1 ax2 bx c dx= 2 p 4ac b2 tan 1 2ax b p 4ac b2 (14) Z 1 (x a)(x b) dx= 1 b a ln a x b x;

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Homework1 Solutions 2 Compute the distances d1(f,g) and d∞(f,g) when f,g ∈ C0,1 are the functions defined by f(x)=x2 and g(x)=x3 Since x2 ≥ x3 for all x∈ 0,1, the first distance is given by d1(f,g)= Z 1 0 (x2−x3)dx= x3 3 − x4 4 1 = 1 3 − 1 4 = 1 12 To compute the second distance, we need to find the maximum of a–c, Spatial pattern of trends in growing season integrated LAI derived from three remote sensing data setsa, GIMMS LAI3gb, GLOBMAP LAIc, GLASS LAIAll data sets cover the period 19 to 090 ' ( 2 3 4 5 4 6 7 8 9 & ' !

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Let sets R and S be defined as follows R = {x ∈ Z x is divisible by 2} S = {y ∈ Z y is divisible by 3} a Characterize the set R ∩ S b Characterize the set R ∪ S c Characterize the set (R ∪ S) c More Concepts • Disjoint sets ∩ • Mutually disjoint sets Definition Sets A 1, A 2, A= C B = H H l 7 1 } 5 8 b b _ m 7 8 j 0 2 b Z 9 0 ~ 3 8 0 d 9 7 8 7 6 9 0 c< Z d < a 7 Z j 0 a 0 9 b a 1 d 7 a 7 4 i ;C A Bouman Digital Image Processing 2 Properties of Chromaticity Coordinates x = X X Y Z y = Y X Y Z z = Z X Y Z • xyz =1

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(a) Z ∞ 1 e−x2 dx, (b) Z ∞ 1 sin2(x) x2 dx Solution Both integrals converge (a) Note that 0 < e−x2 ≤ e−x for all x≥ 1, and from example 1 we see R∞ 1

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