In: Science

Submitted By huajui

Words 6354

Pages 26

Words 6354

Pages 26

Focus on Exam 9

1

∫

3 + sin x dx = cos2 x

∫cos

3

2

x

+ sin2 x dx cos x

= ∫[3 sec2 x + (cos x)−2 sin x] dx = ∫[3 sec2 x − (cos x)−2 (−sin x)] dx (cos x)−1 = 3 tan x − +c −1 = 3 tan x + 1 + c cos x 2 ∫sin2 x cos2 x dx = ∫(sin x cos x)2 dx = = = = 1 ∫2 sin 2x

2

dx

sin 2x = 2 sin x cos x 1 sin x cos x = sin 2x 2

∫ 4 sin

1 4

1

2

2x dx

∫

1 − cos 4x dx 2

cos 4x = 1 − 2 sin2 2x sin2 2x 1 − cos 4x = 2

∫ 1 1 = x − sin 4x + c 8 4

3

1 (1 − cos 4x) dx 8

∫

du x 2 dx = 1 − u2 1 − x4

∫

= = =

1 du 1 − u2 2

∫

u = x2 du = 2x dx du x dx = 2

1 1 1+u × ln +c 2 2 1−u

1 1 + x2 ln +c 1 − x2 4

∫ a − x = − 2a dx 1

2 2

ln

a − x + c

© Oxford Fajar Sdn. Bhd. (008974-T) 2012

a+x

2

ACE AHEAD Mathematics (T) Second Term

4 Let x = sin θ dx = cos θ dθ dx = cos θ dθ

∫x

2

dx cos θ dθ = 2 2 1−x sin θ 1 − sin2 θ

∫

= =

∫ sin θ ∫ sin dθ 2

cos θ dθ 2 cos2 θ

θ q 1

x

= ∫cosec2 θ dθ = − cot θ + c 1 − x2 =− x +c 5 Let u = ln x du 1 = dx x dx = du x dx 1 dx ∫ x ln x = ∫ x ln x =

1 − x2

[Shown]

∫u

du

= ln |u| + c = ln |ln x| + c 6 d (tan3 x) = 3 tan2 x sec2 x dx = 3(sec2 x − 1) sec2 x = (3 sec2 x − 3) sec2 x = 3 sec4 x − 3 sec2 x

4 2 3

[Shown]

∫(3 sec x − 3 sec x) dx = tan x + c ∫3 sec4 x dx − ∫3 sec2 x dx = tan3 x + c ∫3 sec4 x dx = ∫3 sec2 x dx + tan3 x + c = 3 tan x + tan3 x + c 1 c ∫ sec4 x dx = tan x + tan3 x + 3 3 1 ∫sec4 x dx = tan x + tan3 x + c′ 3

© Oxford Fajar Sdn. Bhd. (008974-T) 2012

Fully Worked Solution

3

7 ∫x sin x cos x dx = x =

1 ∫ 2 sin 2x dx 1

∫ 2 x sin 2x dx 1 1 1 1 = ∫ cos 2x x − ∫ − cos 2x dx 2 2 2 2

1 1 = − x cos 2x + cos 2x dx 4 4

∫

1 1 = − x cos 2x + sin 2x + c 8 4 8

∫

x 1 dx = 2 2 0 1 + x 1 = 2 1 = 2 1 = 2 1 =…...

...on edge of rooftop. Pans back around to Close-up of Gabe. Gabe: I can’t explain it any other way, this goes down or we all do! We’ve tried to do it legally…it didn’t work…they own everyone…. Mike: I know, but ….fuck man, this is going to….I don’t know ….start a fucking war. Gabe: Exactly…a war they are not prepared to fight…..WE are! Honestly, this is our only option, unless of course you like the alternative. (Mike shakes head) They’ve been runnin’ the show for the last 2000 years. They’ve fucked up; it’s time for someone to make it right! They fall …society finally gets a chance to run themselves. No one group of people should have the kind of power they do! Mike: You know we’ll be dead inside of a week… Gabe: Bullshit. Chaos hides all, my friend! If I could die and say one thing about my life, it would be that I fought the good fight. I didn’t sit down and take it. I charged head on and made them remember why the human heart should never be underestimated. Mike: They probably already know who we are! Gabe: They know who we were. They don’t know what we are…..or what were capable of. I’m not sayin’ I don’t value our lives, but this is bigger than us. Fuck living, if it means serving them …..Fuck them… You and I know that no one plans their own lives…they do! They decide who gets to survive...who lives...who dies…who starves…You can’t walk a goddamn block without seeing their influence…their power…they own it all. Being able to kill on demand was our...

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...Chaos: A short Introduction Scientists have been coming up with theories for thousands of years now. The theories have been tried and tested have been altered and improved to better understand why certain things happen in and out of this world. Many of these theories are still in the works and our not fully comprehendible and scientists have been working hard to achieve breakthroughs in these areas. One such theory has achieved several breakthroughs within the past couple of years and that theory is called Chaos Theory. This theory first started out mainly as a sort of fictitious happening only read about in novels until lately when scientist have discovered that this theory may have more effect than previously deliberated. In Leonard Smith’s book, “Chaos: A Very Short Introduction” he gives an understandable account of what the Chaos Theory is and how it affects the environment around us. Leonard explains to the reader how the Chaos Theory works through mathematical diagrams, models, and simulations to show the possible results the theory could claim on the environment around us. The book is for the most part easy to understand and gives the reader a sufficient understanding even if the reader does not have a background in mathematics or science. Chaos theory first came about when scientist decided they wanted to be able to predict the future more accurately of certain events happening. The theory takes a look at the initial beginning of a system and tries to......

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...INTRODUCTION TO CHAOS THEORY BY T.R.RAMAMOHAN C-MMACS BANGALORE -560037 SOME INTERESTING QUOTATIONS * “PERHAPS THE NEXT GREAT ERA OF UNDERSTANDING WILL BE DETERMINING THE QUALITATIVE CONTENT OF EQUATIONS; TODAY WE DO NOT KNOW WHETHER THE EQUATIONS OF FLUID FLOW CONTAIN THE BARBER POLE STRUCTURE OF TURBULENCE; TODAY WE DO NOT KNOW WHETHER THE SCHRODINGER EQUATION CONTAINS FROGS, MUSICAL COMPOSERS OR MORALITY OR WHETHER SOMETHING BEYOND IT LIKE GOD IS NEEDED OR NOT “ PARAPHRASED FROM R.P.FEYNMAN * “ONLY A CORNER OF THE VEIL HAS BEEN LIFTED, BUT PERHAPS WE HAVE STARTED ON THE ABOVE JOURNEY ” PARAPHRASED FROM FLORIS TAKENS WHAT IS CHAOS THEORY? * CHAOS THEORY IS BASED ON THE OBSERVATION THAT SIMPLE RULES WHEN ITERATED CAN GIVE RISE TO APPARENTLY COMPLEX BEHAVIOR. * EG. LET US CONSIDER X0 RULE X1 X2 RULE | | | | X1 XN RULE X N+1 THE BASIC OBSERVATION IS ; IF IN THE IMPLEMENTATION OF THE ABOVE RULE X 0 IS CHANGED BY A SMALL AMOUNT, THE RESULTING SEQUENCE WILL BE VERY DIFFERENT. AFTER SOME ITERATIONS MATHEMATICIANS HAVE STILL NOT AGREED ON A DEFINITION OF CHAOS THAT IS ACCEPTABLE TO ALL HOW EVER AN OPERATIONAL (MEANING ONE THAT CAPTURES MOST ESSENTIAL FEATURES) DEFINITION OF A CHAOTIC SYSTEM IS ; 1) THE SOLUTION MUST BE APERIODIC 2) THE SOLUTION MUST BE BOUNDED 3) THE SOLUTION MUST BE EXPONENTIALLY SENSITIVE TO INITIAL CONDITIONS THE RULE THAT GOVERNS A CHAOTIC SYSTEM MUST SATISFY CERTAIN PROPERTIES 1) THE RULE MUST BE NONLINEAR 2) IN THE...

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...Chaos Report The Standish Group is a widely known source of independent primary research and analysis of IT project performance. In 1995, the Standish Group published the CHAOS Report, an often-quoted analysis that focuses on the failure of IT projects to better improve overall IT project performance. The CHAOS Report had documented monetary loss on software developments for unsuccessful projects. The Standish Group comprised such data using focus groups, in depth surveys and executive interviews discussing project performance of completed IT projects. The CHAOS Report provides data on the failure rate of IT projects and also details common explanations for such failure. The authors of the CHAOS Report describe the advantageous process of building bridges to accentuate the lack thereof in software development; Alfred Spector had used such a comparison originally in his article, a computer science perspective of bridge design. The concept is that an error, which resulted in the collapse in a bridge, can be so great in its severity that investigations were subsequently done, allowing those involved in the building process to learn from their mistakes. Differentially, the computer industry allegedly covers up, ignores and/or rationalizes the errors or shortcomings that led up to the failure. The Standish Group contends that project failures will decrease if the following is focused on following a project failure: the scope of software project failure, main factors that......

Words: 1523 - Pages: 7

...BORDERING ON CHAOS The book "Bordering on Chaos" tells an important stage in the history of Mexico. Everything happened just as Mexico enjoyed great success for growth and economic development in1994 when was recently signed the North American Free Trade Agreement and the president Salinas enjoyed and celebrated the economic growth of the country. The Mexicans had finally found a leader in which they could trust. Foreign investment in Mexico had grown by 98 Percent in 1993 and the streets were full of Mexican foreign brands and stores, such as Mcdonals, Dominos pizza, Pizza hut outlets,etc.. Mexico was about to be accepted as a full member of the OECD, the Organization for Economic Co-operation and Development. The whole world was surprised by the rapid growth of Mexico and its citizens were proud of it. “Polls showed that Mexican people were more optimistic about the future than they had been in years”(12).Everything seemed too good to be true, until this happiness and celebration did not last long for Mexican people. Just when everything was going perfectly well, one night in1994 when the president Carlos Salinas de Gortari was celebrating the New year with his family and friends, it was reported that there had just been a guerrilla uprising in the southern state of Chiapas remote, “More than two thousand Mayan Indians, carrying everything from machetes to AK47 riffles, had taken the city of St Cristobal de las Casas shortly after midnight and were reported to have also......

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...Divorce Chaos Rebecca H. Fischer, Esq. Howard Chusid, Ed.D, NCC, LMHC The scenario is one that is known to many. Your spouse and you are continually arguing. You can’t agree on anything. Everything, be it little or large, becomes an argument. The children hear it, your friends hear it in your voice, and your parents hear it in your interactions with them. You decide to visit a counselor, together and individually, but it doesn’t help. You still are not getting along and are left with few options. So, your friends tell you of a great attorney who has done great for many. You visit with the attorney and are told that you need a $5000 retainer but you have no idea where that is going to come from. The attorney also tells you that the cost for the divorce will be in the $50,000 – $80,000 range by the time you get through. You don’t have that kind of money to throw away on attorneys’ fees. If you are feeling this way, there is a high probability your spouse is feeling the exact same way. The exclusion of happiness is always felt by both partners. Problems and irritations are not just a unitary function of a marriage; both partners experience it, usually at the same time. Moreover, we should not forget the kids because they always know what is going on. There are three models of way to get through a divorce: the litigated model, collaborative model and the mediation option. The last two are similar in ideology but are different in approach. The litigated model is the one......

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...Film Analysis 1 Fine Arts Film- Spring 2014 Order and Chaos in Jurassic Park First time watching Jurassic Park the theme that stands out most is Man vs. Nature but this is not the best fit. The true theme is almost shown straight forward during the helicopter ride to the island when Dr. Malcolm states, “John doesn't subscribe to Chaos, particularly what it has to say about his little science project!” (Koepp 19) pitting him against Hammond. These two are the most obvious players in the chaos game, but Dr. Sattler against Dr. Grant and the Tyrannosaurus Rex (T. rex) against the Velociraptors (Raptors) are also players. The other characters do not have as defined roles but still play into the game. They all have a common theme: Chaos vs. Order. Most of the main characters arrive on the island on the same helicopter. The first view inside shows Hammond, Dr. Sattler, and Dr. Grant on the right with Dr. Malcolm and Gennaro on the left. This scene shows who is on which side of the line, but Hammond’s and Dr. Malcolm’s conflict with each other is most prevalent. Hammond is wearing all white while Dr. Malcolm is, “dressed all in black” (Koepp 18) adding visual representation of the polar difference between chaos and order. Chaos and order is the root topic during the Velociraptor birth and lunch scene. Dr. Sattler and Dr. Grant debate which will win in the end but Dr. Malcolm and Hammond have already made up their minds. Chaos is thrown its first blow when Dr. Malcolm is hurt......

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...Chao Feng Chinese Orchestra’s 31st Anniversary Concert was held on Sunday, 25th August 2013 from 4-6pm at Ivanhoe Girls’ Grammar School Performing Arts Centre. Traditional Chinese instruments - Gaohu, Erhu, Zonghu, Jinghu, Guqin, Dizi (flute), Sheng (mouth organ), Suona, Yangqin, Liuqin, Pipa, Zhongruan, Sanxian, Daruan and Guzheng, joined together to perform several Chinese pieces as in ensemble conducted by Mr. Yat Ping Chan. The performance included pieces played by the whole ensemble as well as solo and duet performances. Instruments that had the chance to perform in solo performances without accompaniment included Guzheng and Guqin. Guzheng and Guqin is both plucked zither, but Guzheng has more strings and movable bridges. These instruments produce a very clear and soothing sound with a hint of sadness. However, when they are played in a faster tempo, they can sound quite cheerful and energetic. The 2 songs performed by Guzheng and Guqin were “Jackdaws frolicking in the Water” and “Phoenix Tree Leaves dancing in the Autumn Breeze” respectively. Both songs depicts a scene of birds dancing and enjoying themselves in the nature. Due to the nature of the zithers to produce slightly melancholic sound, it felt as though the happiness won’t last long. The most interesting duet of the performance was the duet by Sheng. Sheng, the Chinese mouthorgan, is always used to harmonise the song in ensembles. Thus, it is quite different to hear the instruments playing a melody......

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...Chaos and Fractals Chaos is a fairly new science. The definition of chaos is complete confusion and disorder, but more generalized to everyday speech it means a random pattern that can occur. (Mirriam - Webster) For scientist this can be a good thing. Henry Adams is quoted as saying "Chaos often breeds life as order breeds habit." (STSCI) Fractals simply put, is a repeat design that as you zoom in is repeated indefinitely. Above is the Sierpenski Triangle, as you zoom in you see three more of the triangle, and in each of those is three more triangle, and as you zoom in you see three more triangles. I imagine that on a big enough scale, or with a strong enough computer there is no end to the number of triangles you can put in the Sierpenski Triangle. (STSCI) If you break a straight line into an equal number N than the equation to describe those parts would be r=1/N(1/d) In the case of the Sierpenski Triangle, a self repeating object, the equation is D = log (N) / log (1/r) (STSCI) The Koch Snowflake is designed by: * divide a line segment into three equal parts * remove the middle segment (= 1/3 of the original line segment) * replace the middle segment with two segments of the same length (= 1/3 the original line segment) such that they all connect (i.e. 3 connecting segments of length 1/3 become 4 connecting segments of length 1/3.) The top row of shapes is the pattern to make the snowflake with a triangle, the bottom shape is a blowup of the......

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...A Constitutional chaos. On September 17th, 1787, The Constitution of the United states of America was signed by the delegates on the conclusion of the Constitutional Convention in Philadelphia. Weeks later after many arguments between the same delegates about other rights of American citizens, the Bill of Rights was written to solidify all of the basic individual freedoms of the people. It was on this document that several laws were written and many great freedoms given. But because these amendments didn't give a specific definition of these freedoms and rights problems ranging from wars being fought because of conflicting ideals in people to poor moral choices being unjustified because there is no line drawn to decide how far things will go; and the vague rules that allow and cause these things to happen need to rewritten . To learn where it all this mess starts is easy; it starts almost immediately after the signing of the Constitution. The document states that there will be freedom of speech, assembly, the right to bear arms, and other freedoms. The cause of one of the first issues that plagued America was not whether the rules are fair and good or not, but who the rules and freedoms are meant for. This was cause of the continuation of slavery in America even after men were supposed to be free because the Bill of rights did not state that a man, either black or white or red or green, has these freedoms. It just simply says that the citizens of America have these......

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...Compromising toward Chaos: Failure to Lead results in Civil War David J. Chasmer Fairleigh Dickinson University Author Note David J. Chasmer holds a bachelor’s degree in Political Science from Rutgers University, and is a candidate for a master’s degree in Public Administration through Fairleigh Dickinson University. Correspondence concerning this article should be addressed to David Chasmer, 1310 Morris Street, Union City NJ 07087. Abstract This paper seeks to examine the presidency of Millard Fillmore, in the context of leadership. It is my objective to prove Fillmore to be a ‘bad leader.’ Fillmore’s actions and leadership skill will be analyzed against the five aspects of leadership, specifically shared vision, modeling the way, challenging the process, encouraging the heart, and enabling others to act. Compromising toward Chaos: Failure to Lead results in Civil War Highly divisive times in world history have provided the opportunities for some of our nation’s best leaders to take charge and guide our great nation on a path that leads to success. The particular brand of success sought after could be any, from security from attack, to economic success. The great leaders of history can be measured in any number of quantitative or qualitative standards, but the ultimate issue is whether their actions benefited the nation in the long run. With that being said, I believe that the opposite of a path toward success would be a path toward division and......

Words: 1658 - Pages: 7

...Chaos and Fractals What is a Fractal A fractal is a never ending pattern. Fractals are never ending complex patterns that are self-similar across different scales. They are created by by being repeated over and over in a feedback loop. Fractals are everywhere. They can be found in nature, algebra, and math. Nature Algebra Geometry Mandelbrot Set This was created in 1980 shortly after the first personal computer in order to calculate numbers thousands and sometimes millions of times. Equation (old)Z=(new)Z2+C We start by plugging a value for the variable C into the simple equation below. Each complex number is actually a point in a 2-dimensional plane. The equation gives an answer, Znew . We plug this back into the equation, as Zold and calculate it again. We are interested in what happens for different starting values of C. When you square a number it gets bigger. Then you square the answer and it get even bigger.. Eventually, it goes to infinity. This is the fate of most starting values of C. Some values of C do not get bigger, but get smaller, or alternate between a set of fixed values. These are the points inside the Mandelbrot Set, which we color black. Outside the Set, all the values of C cause the equation to go to infinity, and the colors are proportional to the speed at which they expand. Ju Fractals in Nature Natural fractals can be found anywhere in nature. Even in our body. It can be the blood vessels in our arms, or...

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...A Solution: Urban Chaos in Cairo What is known today as the Greater Cairo Region is in fact a combination of a number of neighbouring cities and governorates (including Cairo, Giza, Helwan, Qalyubia and 6th of October. Since the first establishment in the region, Al- Fustat in 641 AD, Cairo was in continuous growth. The numerous and different reigns over Egypt since the establishment of Cairo, different Islamic periods, Ottoman, French, English until the revolution in 1952, have led to the inconsistency in development and urban planning which gave rise to cumulative problems. During the course of the last fifty years, the population of Cairo has increased dramatically as a result of high birth rates and large rural-urban migration from other parts of the country (Yousry and Aboul Atta). Currently Cairo is considered one of the biggest cities in the world with a population averaging at 14-15 million (Sims, 2003). The fast and continued increase in the population has been a major urban challenge that the city encountered. Today, the urban landscape of the Greater Cairo Region (GCR) is somewhat unclear: main built up, dense areas are surrounded by a number of satellite settlements at the periphery, but most importantly occupied by infinite unplanned and random constructions in the core and surroundings of the city. Informal settlements, that are named after their informal legal condition, gradually, but rapidly, took over a large area of Cairo’s landscape and crawled over......

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...Mayhem- Danielle Hamilton Chaos- Melissa Mayhem’s Victim- Crystal Chaos’ Victim- Whitney K. Hass & May Hemm, Attys. At Law Commercial Mayhem- I’m the face of your ex-boyfriend’s new girlfriend that you just can’t stop thinking about and the more you drink that tequila the harder it is to forget me. But go ahead, try one more. Maybe I’ll go away then. (Crystal is sitting at a table obviously intoxicated looking sad and mad crying and so on.) Chaos- O my god! Don’t you just love me! I’m the new ipod and wow I love this song too! Let’s get jiggy with it! Oh no! What’s happening? I’m stuck. I’m stuck. I’m stuck. (Whitney is pretending to drive jamming on her ipod when something goes wrong causing her to take her eyes off the road and focus on her device) At the same time Crystal gets up to leave and drive home. Swaying as she walks she says: Crystal- “I’m fine…I’mmmm fine I shay….” (While driving Whitney knocks over a street sign for a one way road but she is too focused on her ipod to even notice) Chaos- Weeeeeeee!!!! Allright! Let’s get the jams going again!!!! Now in her car Crystal is driving and attempting to watch the road. Mayhem- I’m still here. And the longer you focus on me instead of the road with your double vision glasses on the madder you become… and forget where to turn. Yup, this looks good. (Crystal turns onto the one way road and runs into the pole)...

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...Chaos in the Caribbean Strayer University Forensic Accounting and Fraud Examination ACC 571 Dr. Timothy Brown ------------------------------------------------- 1. Evaluate Avey’s role as an expert witness for the Jamaican government. Avey and his firm were hired by the Jamaican government starting in the early in the 90s to investigate accusations of fraud and mismanagement and prepare reports outlining his findings starting with the Blaise Merchant Bank and Trust Co which spread to two similar but larger cases involving Century National Bank and its related financial entities and Eagle Merchant Bank. As an expert witness, Avey was hired by the Jamaican government to provide forensic investigation and audit support. He utilized specialized investigative skills in carrying out an inquiry conducted in such a manner that the outcome would be applicable to a court of law. In addition he examined evidence regarding assertions to determine its correspondence to established criteria carried out in a manner suitable to the court. Avey conducted his investigation grounded in sound forensic accounting principles where he quickly discovered self dealing in the Blaise Case where money was lent from one Blaise financial entity to companies controlled by its principal shareholders. In the Century Case, dishonesty was the main cause of the problems where the use of depositors’ funds to acquire such assets as real estate (which had also been the case with Blaise) for the...

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