Alma Guadalupe Luna Math IA (SL TYPE1) Circles Circles Introduction The objective of this task is to explore the relationship between the positions of points within circles that intersect. The first figure illustrates circle C1 with radius r‚ centre O‚ and any point P. r is the distance between the centre O and any point (such as A) of circle C1. Figure 1 The second diagram shows circle
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variable: Theorem 1: Suppose Y = ln X is a normal distribution with mean m and variance v‚ then X has mean exp( m + v /2 ) Proof: The density function of Y= ln X Therefore the density function of X is given by Using the change of variable x = exp(y)‚ dx = exp(y) dy‚ We have = Note that the integral inside is just the density function of a normal random variable with mean (m-v) and variance v. By definition‚ the integral evaluates to be 1. Proof of Black Scholes Formula Theorem 2: Assume
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ratio of output over input Laplace Transfer Theorem 1. L f(t)=Fs=0∞f(t)e-stdt 2. L Kf(t)=KF(s) 3. L f1t+f2t=F1s+F2s superposition theorem 4. L e-atft=Fs+a complex shifting theorem 5. L ft-a=e-as F(s) real shifting theorem 6. L fat= 1aFsa similarity theorem 7. L dfatdt=sFs-f(0) derivative theorem 8. L d2fatdt2=s2Fs-sf’0-f(0) multiple derivative theorem 9. L 0τfτdτ=F(s)s integral theorem Example Find the transfer function represented
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American Airlines – US Airways Merger Executive Summary The big news of the past few weeks in the travel field was the proposed and almost completed merger of AMR‚ the parent of American Airlines and US Airways. The merger creates the world’s biggest airline in the world. The recent injunction filed by the US Attorney General with the backing of the Government created a problem for the Airlines put a big question mark on whether the deal will go through or not. This white paper refutes the points
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Thomas Theorem William Isaac Thomas (13 August 1863 – 5 December 1947) was an American sociologist. I decided to write about the W.I Thomas after reading about the other theorist I decided that W.I Thomas theory was much more intriguing and very controversial. W. I Thomas is well known for his quote: "If men define situations as real‚ they are real in their consequences." In 1928‚ the sociologist W.I. Thomas formulated this statement which later became known as the Thomas Theorem. In other
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Young’s theorems. Euler’s theorem on homogenous functions. Taylor’s theorem for functions two variables and error estimation. Maxima and Minima‚ Lagrange’s multiplier method. SECTION-B Scalar and vector fields‚ differentiation of vectors‚ velocity and acceleration. Vector differential operators: Del‚ Gradient‚ Divergence and Curl‚ their physical interpretations. Formulae involving Del applied to point functions and their products. Line‚ surface and volume integrals‚ Greens Theorem in the Plane
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|tan t | |sin t | |sec t = |1 | |csc t = |1 | | | | |[pic] | | |[pic] | | | | |cos t | | |sin t | | | The Pythagorean formula for sines and cosines. sin2 t + cos2 t = 1 Identities expressing trig functions in terms of their complements cos t = sin([pic]/2 – t) sin t = cos([pic]/2 – t) cot t = tan([pic]/2 – t) tan t = cot([pic]/2 – t)
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Decision of Uncertainty Paper All decision-making has some level of uncertainly. “Competent researchers and astute managers alike practice thinking habits that reflect sound reasoning—finding correct premises‚ testing the connections between their facts and assumptions‚ making claims based on adequate evidence” (Cooper & Schindler‚ 2006). Data from appropriate investigations can lead to high quality decisions with a lesser amount of uncertainty. Risks in everyday life can be reduced. Our
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Module 5 Circular Functions and Trigonometry What this module is about This module is about trigonometric equations and proving fundamental identities. The lessons in this module were presented in a very simple way so it will be easy for you to understand solve problems without difficulty. Your knowledge in previous lessons would be of help in the process What you are expected to learn This module is designed for you to: 1. state the fundamental identities 2. prove trigonometric identities
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The Mean Value Theorem Russell Buehler b.r@berkeley.edu 1. Verify that f (x) = x3 − x2 − 6x + 2 satisfies the hypotheses of Rolle’s theorem for the interval [0‚ 3]‚ then find all c that satisfy the conclusion. www.xkcd.com 2. Let f (x) = tan(x). Show that f (0) = f (π)‚ but there is no number c in (0‚ π) such that f (c) = 0. Is this a counterexample to Rolle’s theorem? Why or why not? 3. Verify that f (x) = x3 − 3x + 2 satisfies the hypotheses of the mean value theorem on [−2‚ 2]‚ then
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