1 sqrt x 2 a

• [PDF File]Square Roots via Newton’s Method - Massachusetts Institute of Technology

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  x n+1 = 1 2 x n + a x n : The intuition is very simple: if x n is too big (> p a), then a=x n will be too small (< p a), and so their arithmetic mean x n+1 will be closer to p a. It turns out that this algorithm is very old, dating at least to the ancient Babylonians circa 1000 BCE.1 In modern times, this was seen to

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• [PDF File]RESIDUE CALCULUS, PART II

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  ln(x2 +1) x2 +1 Take principal branch of log. Branch cut ΓR dz ln ( z + i ) ( z + 1 ) 2 Im z Re z i −i R Γ R Consider •By residue theorem I ΓR ln(z +i) z2 +1 dz = 2πi Resz=if = 2πi ln2i 2i = π ln2 + iπ 2 . •By Jordan lemma Z SR →0 for R → ∞ . • Z R −R ln(x +i) x2 + 1 dx = Z R 0 ln(−x+ i) x2 +1 dx+ Z R 0 ln(x+ i) x2 +1 ...

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• [PDF File]Lecture1.TransformationofRandomVariables - University of Illinois ...

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  7 2.3ATypicalApplication Let Xand Ybe independent,positive random variables with densitiesf X and f Y,and let Z= XY.We find the density of Zby introducing a new random variable W,as follows: Z= XY, W= Y (W= Xwould be equally good).The transformation is one-to-one because we can solve for X,Yin terms of Z,Wby X= Z/W,Y= W.In a problem of this type,we must always

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• Evaluation of the Integral f0 t2a-Jv (X +t 2) dt - JSTOR

  The special cases Iv /2 1 (x) and ]a 12 + 1`(x), which, as was mentioned in Section I, are often found in tables of integrals of Bessel functions, also follow from (3). Thus, with the G-function representation [10] of the product Jv(x)Yj(x), the

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• [PDF File]Solution: i - Department of Mathematics and Statistics

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  Math 132 Spring 2013 Final Exam 1. Let F( )x d = == = 0 x 2 ( )ln 1 ++ + + + t ++ + 2 t 3 ln ( )1 + ++ + t t.Calculate F ' (4), the derivative of F(x) at x = 4. a) 0 b) 1 c) 2 d) 3 e) 4

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• [PDF File]Solutions 1.2-Page 17 Problem 5 - University of Florida

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  Problem 25 A diesel car gradually speeds up so that for the first 10s its acceleration is given by t t dt dv =(0.12) 2 +(0.6) (ft/s2) (Note: the form it is written in the book is wrong.It should not be dx dy.) If the car starts from rest (x0 =0 , v0 =0), find the distance it has traveled at the end of the first 10s and its velocity at that time.

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• [PDF File]11 The normal distribution and the central limit theorem 11.1 The ...

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  standard deviation σ f(x; μ,σ) = 1 2πσ e-(x-μ)2/2σ2. f[x_,μ_,σ_] = (1 / (Sqrt[2 Pi] *σ)) E^(-((x -μ)^2) / (2σ^2)) ⅇ-(x-μ)2 2σ2 2πσ It turns out that again about 95% of the time an observation from a normal distribution will be within two standard deviations from the mean. We showed this above for the standard normal distribution.

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• [PDF File]Lab #1 - University of California, Berkeley

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  EE227A 8/25/10 Lab #1 1. Install and play with CVX1.Be sure to learn how to solve a least-squares problem. 2. Reformulating constaints in cvx. Each of the following cvx code fragments describes

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• [PDF File]Finding Square Roots Using Newton’s Method - University of Pennsylvania

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  x1 ≥ x2 ≥ x3 ≥ ..., is monotone decreasing and non-negative. By the monotone conver-gence property, it thus converges to some limit x. I claim that x2 = A. Rewrite (2) as A − x2 k = 2xk(xk+1 − xk) and let k → ∞. Since xk+1 −xk → 0 and xk is bounded, this is obvious. We now know that √ A exists as a real number. then it is ...

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• [PDF File]1 Evaluating an integral with a branch cut - University of Arizona

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  approaches zero the real part converges to 1 −1/x < 0 and the imaginary part approaches zero from the half-plane in which y is taken. So the boundary value is i p 1/x−1 when we approach from the upper half plane and is −i p 1/x−1 when we approach from the lower half plane. Thus the boundary values of f(z) are f(x+i0) = 1 ±ix q 1− 1 x ...

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• [PDF File]CS 194-10, Fall 2011 Assignment 2 Solutions - University of California ...

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  1x 2 = − 1 and the negative examples have x 1x 2 = + 1. The maximum margin separator is the line x 1x 2 =0, with a margin of 1. The separator corresponds to the x 1 =0 and x 2 =0 axes in the original space—this can be thought of as the limit of a hyperbolic separator with two branches. (b) Recall that the equation of the circle in the 2 ...

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• [PDF File]MA 104 Graded Homework 2 Solutions - Brigham Young University

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  Plot[2 + Sqrt[9 - x^2], {x, -3, 0}] Integrate[2 + Sqrt[9 - x^2], {x, -3, 0}] (2) (5 Points) Evaluate the following definite integral Z 4 0 e √ x √ x dx. This problem is done with Mathematica using the following command. Integrate[Exp[Sqrt[x]]/Sqrt[x], {x, 0, 4}] Manually, we set u = √ x and get du = dx 2 √ x or 2du = dx √ x. We have ...

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• [PDF File]Trigonometric Substitutions Math 121 Calculus II

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  x2 for 0 x 1. 2. Since the derivative of f(x) = 1 2 x2 is x, the length is L= Z 1 0 p 1 + x2 dx: We’ll use the trig sub of the second kind with x= tan , dx= sec2 d , and p 1 + x2 = sec . Then the integral becomes L= Z ˇ=4 0 sec3 d : It takes an application of integration by parts to nd that an antiderivative of sec3 is 1 2 sec tan + 1 2

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• [PDF File]Building Java Programs - University of Washington

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  // A Point object represents an (x, y) location. public class Point {private int x; private int y; public Point(int initialX, int initialY) {x = initialX; y = initialY;} public double distanceFromOrigin() {return Math.sqrt(x * x + y * y);} public int getX() {return x;} public int getY() {return y;} public void setLocation(int newX, int newY) {x ...

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• [PDF File]Linear Regression (continued)

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  Linear regression Setup Input: x2RD (covariates, predictors, features, etc) Output: y2R (responses, targets, outcomes, outputs, etc) Model: f: x!y, with f(x) = w 0 + P d w dx d = w 0 +wTx I w= [w1 w2 w D]T: weights, parameters, or parameter vector I w0 is called bias I We also sometimes call w~ = [w0 w1 w2 w D]T parameters too Training data: D= f(x

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• 1. THE SQRT ONE-STEP DIFFERENCE SCHEME - JSTOR

  1. THE SQRT ONE-STEP DIFFERENCE SCHEME A standard approximation for the differential equation in the linear boundary value problem (BVP) u'(x) = A(x) u(x)+ g (x) X E [0 1], U(X) ER, ... which has the practical consequence that for a O 0 the matrix X (cf. (1.5)) may be computed by (2.3) instead of (2.1), saving half the computational effort ...

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• [PDF File]Table of Integrals - UMD

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  ©2005 BE Shapiro Page 3 This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or

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• [PDF File]Multiplying by the Conjugate - University of Washington

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  x+x2 x−2. Suppose we want to eliminate the square root from the numerator (this is sometimes called rationalizing the numerator). What we can do it multiply the entire expression by √ x−x2 √ x−x2. Since this is essentially equal to 1 (that is, it is 1 unless x = 1or x = 0, in which case it is

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• [PDF File]Real Variables: Solutions to Homework 2 - Mathematics

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  Exercise 0.1. Chapter 2, # 1: Let f(x) = xsin(1=x) for x2(0;1] and f(0) = 0. Show that fis bounded and continuous on [0;1] but V[f;0;1] = +1. Proof. To see that fis bounded it is enough to realize that jsin(x)j 1 for x2[0;1], so jf(x)j= jxsin(1=x)j 1: To see that fis continuous, because it is a product of continuous functions on the interval

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• 1. Comparison Tests

  2 2.5 3 3.5 sqrt(x) ln(x) Figure 1: Graph of f(x) = ln(x) (red-solid graph) and g(x) = √ x (blue-dashed graph) Therefore, since ln(n) < √ n then squaring both sides yields (lnn)2 < n. Therefore, a n = (lnn)2 n 3 < n n = 1 n2. The series P 1 n2 converges, and therefore, by the Comparison Test Part (i), the given series P (lnn)2 n3 also ...

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