text:
User:BrittanyW

=Analytical Writing=

Statement of Intent
Although this project is still confusing to me, I would like to make a booklet (possibly online) for people called "Living Green." I find this to be very interesting because I care about our earth and would like to find out more about what we can do to help in our everyday lives. For my project I will research ways to use more products that are better for the environment. Today people are trying to find ways to use more green products. Recycling and saving energy is becoming more important. I want to make this booklet for myself and others for an easy reference for green products and tips. I want to find the best cleaning products, water bottles, household items and clothing materials that would best suit the environment. I would also like to list the different stores that offer the greenest products and give discounts for using reusable grocery bags. In addition, I will also list the different things we can all already do at home to use less energy as well as writing a personal essay on what I decided to include in the booklet.

Overview
I want to accomplish a booklet of at least 5-10 pages, a 2-5 page personal essay and to use the research I find at home.

Rationale
I chose this for my project because I believe in using 'green' products and I hope to get a lot of helpful information that everyone will be able to use.

Publication
I have no idea how I would publish the booklet but I know I want it to be available online. I could post it on a blog so it’s accessible to everyone and FREE.

Schedule

 * 1) Write Statement of Intent
 * 2) Research at library, Statement DUE
 * 3) Research
 * 4) Research
 * 5) Start putting together booklet
 * 6) 1st meeting with Steve
 * 7) WORKSHOP/Design art for booklet
 * 8) Finish personal essay
 * 9) Finish Booklet
 * 10) Final meeting with Steve/Make final revisions to booklet
 * 11) Living Green Booklet and Personal Essay DUE

Anticipated Problems
I don't see how I would not be able to finish this project. However, I could get sick due to the weather or become overwhelmed by my workload in my other classes. I plan on sticking to my schedule to get the booklet done and balancing all my various projects.

Week 6 Project Review
Where I am now for my project is continuing to research. I’m finding different products to list and to describe for the booklet as well as listing other ways we can “live green.” For example, turning off the water when brushing your teeth and using kitchen scraps for compost. In addition, I’ve been listing products such as the stainless steel water bottles, recycled aluminum foil and household cleaning products that are friendly to the environment. I’ve been looking online for research mostly and I will be going to the school library more to look for some books that I can use in addition to online articles. After I finish my research I will start to write my personal essay on what I decided to include in the booklet and my thought process through the whole project. Some of the challenges I have faced while working on this project is scheduling around other class projects. I found myself juggling a lot of my time between this class and my speech communications class. I tend to focus my time more on my video classes and had to remind myself of my general education classes. Despite all that, I know I am on the right track with my project and that I will be ready for my workshop on week seven.

Before Or During Week 6

 * Research project from a former quarter in Analytical Writing.
 * Products/consumption paper: 1,000wds
 * Personal essay: 1,000wds.
 * NuBe Green store on Cap Hill.

 * Reducing consumption
 * Green products - affordability

Project
I found the process of what to include and exclude in my booklet to be very difficult. While doing research I found new products and issues that I could write entire papers on. So for my booklet I decided to keep it to some local stores in Seattle listing a handful of products. I listed the stores websites and addresses so students could explore further. In addition, I wrote a persuasive paper instead to convince others to follow the green movement with facts, statistics and easy tips.

Many of the global warming stories are scary and we see them on the news and on educational channels almost every day about melting ice caps and climate changes. Some people may think there is nothing we can personally do about the issue, especially poor college students. Most of the people who can afford greener products are well off. Thankfully, there are others ways we can live green with cheap products and free tips. We all need to think globally and act locally. Some people say that consumption is the cause of global warming but we do not have to stop consuming products; we just have to consume the right kinds of products. When we were young most of us were taught the three R’s to reduce, reuse and recycle. We should reduce but consume the right products. We need to reuse and donate items which lead into recycling. Most of us have forgotten to use the three R’s which results in global warming. Global warming is the result of greenhouse gases, such as methane and carbon dioxide, being released into the O-Zone which causes the worlds climates to change. Humans seem to be the major factor in global warming. According to Andrew Revkin, New York Times, “since 1950, the world's climate has been warming, primarily as a result of emissions from unfettered burning of fossil fuels and the razing of tropical forests.” Methane from landfills, cattle and oil facilities comes second to carbon dioxide for causes of global warming (New York Times). Al Gore alerted the world to warming’s risks through a 2006 documentary and book called An Inconvenient Truth. Such as, some fluctuations in the Earth's temperature are inevitable because of decades-long ocean cycles (New York Times). The Intergovernmental Panel on Climate Change shared the 2007 Nobel Peace Prize with former Vice President Al Gore for alerting the world to warming's risks. Everyone needs to be reminded by the three R’s to start bringing our planet back to health. The first R is to reduce. The U.S. Environmental Protection Agency says, that “between 1960 and 2008 the amount of waste each person creates has almost doubled from 2.7 to 4.5 pounds per day.” In addition more than “7,000 communities have instituted ‘pay-as-you-throw’ programs where citizens pay for each can or bag of trash they set out for disposal rather than through the tax base or a flat fee (EPA).” Washington State has over 200 communities with this program but unfortunately King County is not one of them listed on the EPA’s website. What we can do about making a pay-as-you-throw program in Seattle is to go to City Hall and raise a citizens’ initiative. The EPA also states that, “containers and packaging represented approximately 28 percent of the materials source reduced in 2000, in addition to nondurable goods (newspapers, clothing) at 17 percent, durable goods like appliances, furniture and tires at 10 percent and other solid waste like yard trimmings and food scraps at 45 percent.” Instead of throwing out your banana peels and apple cores you should start your own compost if you have the yard to do so. If you do not have the room for a compost heap or cannot dispose of food scrapings in your yard waste bin I suggest investing in a garbage disposal for your kitchen sink.

The best thing to do is to reduce our consumption but that doesn’t mean we have to stop buying all together. We should buy “green” products. According to Conservation Value, what makes a product ‘green’ is they must be “energy efficient and saving, durable, and have low maintenance requirements.” In addition green products are readily recycled, biodegradable, made using natural and/or renewable resources, and that they are obtained from local resources and manufacturers (Conservation Value). There are many local stores that sell organic foods or products made out of entirely raw materials which tend to be pricey, such as NuBe Green located in Seattle on 921 East Pine Street, which sells bowls made out of lemons and Bok Choy leaves for $40, chipboard binders, and soy candles. Another important item to purchase that is sold in most stores is stainless steel water bottles that cost around $10. According to the Earth Policy Institute, “Americans used 50 billion plastic water bottles in 2006 and sent 38 billion water bottles to landfills, the equivalent of 912 million gallons of oil.” Those bottles could take up to 1,000 years to biodegrade in a landfill (Earth Policy Institute). Reducing doesn’t always mean stop shopping, just shop for better items. The second R is to reuse. Shopping at the Salvation Army or Goodwill is a great way to reuse items. If you have clothes, toys or any other things you don’t want; you should donate them to your nearest thrift store. Not only are you helping others you are also getting a deduction on your taxes. If you have old clothes that are torn instead of throwing them out you could repair them buy sewing. There are many other items throughout your house that can be turned into crafts or used for others purposes. Using old silverware and metals to make crafts for your yard or garden such as wind chimes. Old milk jugs can be used to water pumpkins by burying them in the ground with holes in the bottom, which I personally do. Giving away unwanted items and making new things out of the old can benefits others and reduce waste. The final R is to recycle. Recycling benefits everyone by preventing pollution caused by the manufacturing of  products from virgin materials. In addition, it reduces greenhouse gases and the need for landfills and incinerators as well as preserving natural resources. The city of Seattle offers free recycling service for Seattle residents. The first step of recycling is collection and processing. We can recycle either by the curb-side or drop-off centers. Recyclables are bought and sold and prices for the materials change with the market (EPA). Things like phone books, magazines, glass, plastic and aluminum products can be recycled. But batteries, ceramics and Styrofoam packaging cannot be recycled. The second step is manufacturing. The recycled goods are first sorted and cleaned. Then the goods are used for partially recycled and totally recycled products such as newspaper, pop cans, plastic containers and glass bottles. The final step to recycling is purchasing recycled products. If consumers start to demand products that are environmentally friendly then manufactures will produce high-quality recycled goods to meet their needs (EPA). Most companies are already using recycled products such as, Sun Chips has the first 100% biodegradable chip bag that is said to release on Earth Day of 2010. Recycling may seem like it takes a lot out of your day, but in the end it helps you and the rest of the world. Global warming is happening and we need to do whatever we can to bring our planet back to health. There are cheap ways that we can all do to live green and we need to remember to reduce, reuse and recycle. If we start to demand recycling and buy green products it will help our Earth and put more money in our wallets. The next time you go to the store start with buying a stainless steel water bottle to do your part. We may be the cause of global warming but we can also try to stop it.

LIVING GREEN

I wanted to create a “Living Green” booklet for students because I strongly believe in preserving our Earth for generations of people and wildlife to come as well as continuing the green movement that started in America during the 1970s’. Humans are the main cause of global warming and I wanted to see what college students could personally do to help with stopping the problem. After my parents went to a local hardware store to change all of our housing light bulbs to CFL’s for free I wanted to learn more about green products and free tips. My parents went to get their free CFL bulbs and there was no line because no one seemed to think CFL bulbs were important or worth their time.

According to the CLF bulbs’ website, “If every household in the U.S. replaced just ONE incandescent light bulb with an energy efficient compact fluorescent light bulb (CFL), it would eliminate the equivalent of the emissions created by one million cars (CFL).”

So I want to let people know benefits of living green and that there are cheap and many free things we can be doing to live green.

-Brittany Weible

What We Can Do for FREE, REDUCE REUSE AND RECYCLE!

Heating and Energy We can reduce our energy bills by turning off lights and using natural light from outside. Bundle up during winter and use thick blankets for bed instead of turning up your heat. In the summer open your windows instead of using a cooling system.

Water Usage Turning off the water when soaping up in the shower and while brushing our teeth

Transportation Carpooling with co-workers, buses, trains, biking or skateboarding. Recycling Recycling services for the residents of Seattle is FREE. Use old silverware to make wind chimes or old milk jugs for watering pumpkins (which I personally do).

Clothing Sew old clothing and give clothes to relatives instead of throwing them out. Donate clothing to thrift store (Goodwill or Salvation Army) to help others and receive a deduction on taxes.

Yard Waste Use yard waste and food scraps for a compost heap if you have room. If you do not have room for a compost heap dispose of yard waste and food scraps in Yard Waste bin that is collected like garbage and recycling on the curb-side.

ORGANIC vs. NATURAL

In 1990, Congress passed the Organic Foods Production Act (OFPA). “OFPA required the U.S. Department of Agriculture (USDA) to develop national standards for organically produced agricultural products to assure consumers that agricultural products marketed as organic meet consistent, uniform standards (Gold).”

Natural does not mean organic. Free-ranged, natural and hormone-free can be labeled on food but it does not mean it is organic.

The USDA organic seal means the product is at least 95% organic.

The USDA National Organic Standards Board (NOSB), in April 1995, defined organic to be:

“Organic food handlers, processors and retailers adhere to standards that maintain the integrity of organic agricultural products. The primary goal of organic agriculture is to optimize the health and productivity of interdependent communities of soil life, plants, animals and people (Gold).”

Local Stores

NuBe Green at 912 East Pine Street, Seattle, WA 98122 http://www.nubegreen.com/

Goods for the Planet at 525 Dexter Ave North, Seattle, WA 98109 http://www.goodsfortheplanet.com/

PCC Natural Markets located in Fremont and Seward Park http://www.pccnaturalmarkets.com/

Fred Meyer stores http://www.fredmeyer.com/healthy_living/Pages/default.aspx

Safeway stores http://www.safeway.com/IFL/Grocery/Shop

Fred Meyer and Safeway Lifetime Guarantee - If your reusable bag ever wears out, just return it and Fred Meyer will replace it free of charge. Fred Meyer: Reg. Size (89 cents) Large size (99 cents) each

Fred Meyer gives you a 5 cent discount for each reusable bag you use at the checkout. These reusable bags can be used for everyday use like a purse. If you have plastic shopping bags they can be recycled at most stores, such as Fred Meyer which uses recycled plastic for their indoor benches. Furthermore, Fred Meyer has a “Green Living” link that give tips for conserving energy.

Safeway sells reusable bags like many stores and has been a major recycler for almost 50 years. Both Fred Meyer and Safeway carry many similar products such as Stainless steel water bottles, Sun Chips in 100% compostable bags and Clorox ‘Green Works’ goods that are made out of 99.9% natural ingredients.

NUBE GREEN PRODUCTS Cedar Herb Planters - Size: 3 Long ($25) Big Dipper Beeswax Pillar Candle ($17) Lemon and Bok Choy Bowls ($40) Fruit/Veggie Wash - Spray 16oz. ($7.50) Recycled Cardboard Coasters ($17) 100% Organic Cotton “Circle Trees” shirts ($41)

GOODS FOR THE PLANET

Bags, Clothing and Bedding Backpacks made from recycled pop bottles Backpacks, purses, wallets & belts made from hemp & recycled tires Organic Cotton/Hemp t-shirts, robes, baby diapers and socks 100% organic cotton sheets, pillowcases, blankets & mattress pads

Bath Organic cotton towels and bathrobes Local handmade and organic soaps Stainless steel & recycled glass soap dishes

Toys for Baby Lush toys made from organic cotton Wood toys made from sustainably harvested organic rubber tree wood & certified non-toxic stains.

Home Décor Bowls and vases made from recycled wood Recycled glass bowls and vases

Energy Conserving Products Fire logs made from recycled coffee grounds energy efficient light bulbs (full-spectrum, dimmable, 3-way, candelabra base) Wind to charge rechargeable LED flashlights

Kitchen Hemp placemats, napkins, aprons & potholders Long-lasting cast iron pots and pans Bamboo cutting boards

Cleaning supplies Reusable viscose cleaning towels & sponges Unscented, natural, flushable wipes Biodegradable sandwich bags

Garden Worm bins Hemp garden gloves Natural and organic fertilizers Watering cans made from recycled plastic

Office Supplies Recycled cardboard binders and folders Recycled paper and file folders

Solar Solar attic fans; address lights; security lights; battery chargers Lighted stepping stones Backpacks with built-in solar panels

Yard Outdoor furniture made from recycled plastic lumber Garden sculptures made from recycled and reclaimed metal Cast iron wind bells and birdbaths

PCC Natural Markets The PCC Natural Markets have a wide range of foods for Seattle residents from natural sweeteners, soy foods, tofu, and seafood to fresh local produce. According to the PCC Natural Markets website, they support local farmers such as “Farmer Nash Huber of Nash’s Organic Produce (Sequim, Wash.); farmers JoanE McIntyre and Mike Shriver of Rent’s Due Ranch (Stanwood, Wash.); and Farmers Andrew Stout and Wendy Munroe of Full Circle Farm in Carnation, Wash” as well as local ranchers “including Draper Valley Farms (Mount Vernon, Wash.), Thundering Hooves (Walla Walla, Wash.), and Umpqua Valley Lamb(Oregon).”

Some of the products listed on the PCC Natural Markets website are:

Date Sugar Brown Rice Syrup Sugar Cane Juice Brand name ZERO, a certified organic calorie free erythritol Edamame Soybeans, soy nuts and soy butter Soy flour Soy milk and cheese Firm, soft and silken tofu Kale Chard Collards and collard greens Root vegetable tops (beets, turnips and carrots) Organic Beef, poultry, lamb, pork and bison

title: User:BrittanyW


text:
User:Hollypetrie

=Humanities I/The Commons=

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Hollypetrie


text:
User:Cassandras

=Humanities I/The Commons=

title: User:Cassandras


text:
User:DevonnRonning

=Analytical Writing=

Statement of Intent

Overview
During the course of this class, I would like to do research on the origins of some common typographic characters, and how they have changed and evolved over time. At the moment it is my plan that I would do this as a set of smaller individual pieces, each relating to a single character. I would start by identifying the character and its current usage. Then I would go back and explain when it was first used and how it has changed and evolved since then, I would also include if an older one inspired it, or if it’s a ligature of sorts. I would like to look into things like the ampersand, the dagger, currency symbols, asterisk, at, and quote. Because I’m not sure of the dept of these subjects, I may have to expand the range of the project; I would probably choose a pictorial symbol that has a longer history and cover that.

Rationale
I want to peruse this subject because I have a personal interest in it, and I think that a better understanding of how symbols evolve can help me with ideas on how to evolve them further.

Publication
The first form of publication I will use for this will be on wikiversity. If I am satisfied with what I come up with, I will also probably also post the final version on my personal website so that others who are interested in the subject can read it too.

Schedule

 * 1) Week 1: Finish statement of intent.
 * 2) Week 2: Beginning research gathering sources and decide what characters are the best candidates.
 * 3) Week 3: Continued research, write outlines for first draft.
 * 4) Week 4: Write first drafts and edit.
 * 5) Week 5: Pick one character to bring to final draft status. Midterm project assessment.
 * 6) Week 6: Continue finalizing all drafts.
 * 7) Week 7: Review project. Do I need to spend the rest of my time finalizing and polishing the current pieces (or) do I need to move on to research another symbol?
 * 8) Week 8: Continue finalizing papers, (or) outline and first draft on new symbol.
 * 9) Week 9: Final edits on papers, (or) finish next draft on new symbol.
 * 10) Week 10: Individual meeting, and final touches getting ready for the reading.

Anticipated Problems
I'm concerned that this subject may be difficult to get information on, or if these characters even have much of a history. I am very limited when it comes to time because I am a full time student, as well as a having a full time job. Most of my "down time" happens in the middle of the night so my research needs to be limited mostly to books and online sources, and its not very likely I can interview someone about this. Because I’m not sure how much information will be available on each of the characters, my schedule will most likely change quite a few times when I’m in the beginning research stage. When setting up the schedule I left the last four weeks open to two different paths, depending on how things look on week 7. I also expect to have a lot more work in my other classes during this time.

Week 6 Project Review

Before Or During Week 6

 * See Helvetica.
 * at sign
 * look at comments to decide on second symbol
 * design--large versions of symbols.
 * textism
 * each symbol 1,000 words? 4.5 pages double spaced.

Project
This is a result of both verbal and written interviews that I did, both formal and informal.

As a river runs on through a landscape it constantly changes. Its banks slowly erode in some places and build themselves back in others. Water runoff steadily joins the river adding its own influence. People build dams and levees to try and contain its flow, but sometimes-natural influences like too much rain at once, can cause it to spill over these structures and take a natural course. In ways, the English language can be like a river. It is always gradually evolving and constantly moving forward. We have set rules and restrictions to force it into the path that we want it to be. Even though we put so much effort into creating a proper structure, there are still outside influences that can be more powerful than the rules we set and can change the way we perceive it. Sometimes our languages natural evolution is unavoidable. The way that people react to these changes can vary greatly from person to person. Some people view these changes as a devolving, a bastardization of the proper English. Others see these changes as an improvement, preferring a language built on logic and phonetics over one ruled by conventional ideas. No matter what side of the river you’re standing on, one thing is for certain; there is no stopping the current. There are two interesting ideas that have emerged in punctuation. In an attempt to create a better way to communicate an idea or make their intentions clearer, some people have started enlisting the help of the interrobang and the irony mark. Both of these are meant to help the reader understand the emotional context of the preceding sentence. The interrobang glyph itself is a superimposition of the exclamation point and the question mark. It is meant to combine the emotions of both these. The interrobang, also sometimes referred to as an interabang or quesclamation mark, indicates that the sentence asks a question in an excited manner, expresses excitement or disbelief in the form of a question, or asks a rhetorical question. “American Martin K. Speckter invented the interrobang in 1962. As the head of an advertising agency, Speckter believed that advertisements would look better if copywriters conveyed surprised rhetorical questions using a single mark. He proposed the concept of a single punctuation mark in an article in the magazine TYPEtalks. Speckter solicited possible names for the new character from readers. Contenders included rhet, exclarotive, and exclamaquest, but he settled on interrobang. He chose the name to reference the punctuation marks that inspired it: interrogatio is Latin for "a rhetorical question" or "cross-examination";bang is printers' slang for the exclamation point. Graphic treatments for the new mark were also submitted in response to the article.” Some writers feel that its unnecessary to create a combined single glyph to represent this idea, because the same idea can be conveyed by simply ending the sentence with a question mark followed by an exclamation mark. Others feel that it would be beneficial to use it, because it can be quickly read and has a more unified visual impact. It could also be argued that it helps keep a consistent flow to the content, instead of breaking it with multiple marks. The irony mark is meant to be used to indicate that a sentence should be understood at a second level. Some writers reject it, with the opinion that if a special mark is needed to point out irony or sarcasm, then it’s simply a matter of rewriting it to be understood with contextual clues only. Using multiple punctuation marks at the end of a sentence is grammatically incorrect. If the interrobang became a standard part of the English language, people who were concerned with that, could get their idea across better without breaking the rule. It could also be argued that its unnecessary to properly include it in the English language because if people writing informally were going to use it, they would be less concerned with its correctness, and the average person will be able to infer the meaning of the mark without having to be taught what it is. To look forward into whether or not marks like these will catch on, it helps to take a step back and try and understand what other parts of English are changing. These are both good examples of a gradual relaxation of English. It is also very evident in other aspects of our language such as grammar and spelling. Our current standard of living, the use of technology and the amount of interactions we have all influence what we regard as important or necessary to communicate an idea. With text messages and emails, people often pay less time and attention to their spelling and grammar. Juliet B. Schor a Harvard professor commented on the subject, stating: “The rise of worktime was unexpected. For nearly a hundred years, hours had been declining […] In the last twenty years the amount of time Americans have spent at their jobs has risen steadily. Each year the change is small, amounting to about nine hours, or slightly more than one additional day of work. In any given year, such a small increment has probably been imperceptible. But the accumulated increase over two decades is substantial. When surveyed, Americans re port that they have only sixteen and a half hours of leisure a week, after the obligations of job and household are taken care of. Working hours are already longer than they were forty years ago. If present trends continue, by the end of the century Americans will be spending as much time at their jobs as they did back in the nineteen twenties.” I believe these are all indications that our language is on its way to becoming more informal and relaxed. Personally I don’t think that this is a bad thing. Legibility isn’t a major concern, because the human minds are always going to be able to pronounce things based on logic, and will usually be able to understand the meaning of a sentence whether we use special defining marks or not. English seems to be on its natural course.

title: User:DevonnRonning


text:
User:SabrinaPhillips

=Analytical Writing=

Statement of Intent

 * I'd like to mention that these ideas and opinions are in no way berating or belittling the medium of 3D animation, or CG cinematic compositions. It is simply to establish that this form would not be possible, nor would successful executions of these animations be possible without a full understanding of the foundations in 2D animation.*

Overview
Honestly, I'm not entirely sure what I hope to gain from this class in particular. I do know, however, that I have one year exactly until I graduate from the Art Institute of Seattle. My area of expertise is animation; and I have every intention of spending this next coming year devoted to my craft. I currently work in the industry. I'm interning at a video game studio under Warner Brothers. I worked hard to get this job on my own out side of school. Aside from being paid less than most of my other full time co workers; I am treated well; just like any other full time animator.I've realized now with industry experience, that the grad illusion of animation and the industry is not all that I thought. And my goals seem far more attainable and I not as talentless as I often felt. I know now what is required by me. Not only by the schools standards, but also those of the industry and myself. I want to make great strides in my field by bringing unique style and technique learned from a thorough understanding of those who came before me. My intent this quarter in this class is to research and write a paper about the recent claims of 2D animation being a dying art form. I disagree. Perhaps in the box office, 2D has taken a sharp decline since the established awe of Pixar. However, I do not believe that any of that could have been possible without the 2D aspect. I think that this gross misconception is resulting in a decline in the quality of animation being produced. As well as a decline in the skill and drive of the aspiring next generation of animators. While technology is developing into some fantastic capabilities, in my opinion, it is causing a dependence that is deteriorating the skill of the craft. These opinions are my own and are not entirely shared with many of my peers at school, or even those in the industry. I intend to research and back my opinions with these into a paper I would hope is actually publishable. I hope to inspire those to change their thoughts about 2D animation, and perhaps work harder to strengthen their skill and knowledge by understanding the foundations that lead to the awe inspiring visuals they feel are the current and future of animation.

Rationale
I really hope that I will be able to record the personal opinions of a great many of persons, both in and out of the industry. I don't actually think it will be too difficult, since I'm quite sure that it is a subject of great concern for those in my field and are passionate about the subject. My desire is to, sort of, think of this as my thesis paper. I have a hope that this will be something that garners the attention of those in my industry and awaken some to the undeniable value of this art form. On a more personal note. I hope that writing this with the intent of publishing, will give me the courage to make myself noticed. On a grand scale, if I am to be successful in my field, not only do I need to be the best in my school, but I need to be one of the best on the west coast. It is difficult to accept, however, it is highly competitive and you need everything you can to get a leg up. If you want to compete you need to make yourself and your passion known. I hope that with this paper I will accomplish that and help me to gain the confidence to stand behind it.

Publication
I have published in the school paper, Fuel, before and I think I would like to start there. I also have the intention of starting an animation blog for myself to post not only my work; but also my thoughts and hopefully, this paper. My most principal aspiration would be to have it published, or posted, on/in a frequented magazine or website read by all walks of animators world wide. As I said I really hope to write and share my thoughts and opinions, hoping to awaken the seemingly dulling sense of significance regarding the foundations of animation. Reestablishing the kind of passion and quality that made the art great and gave it meaning.

Schedule

 * 1) week 2
 * 2) week 3
 * 3) week 4
 * 4) etc.

Anticipated Problems

Week 6 Project Review

Project

title: User:SabrinaPhillips


text:
User:Egm6341.s10.Team4.andy

I, Anandan Rangasamy, work with Team4 for EGM 6341 [Spring 2010] homeworks or assigments assigned by Prof. Vu-Quoc.

Please find my work on

 * Homework1 due by 27 January 2010 5pm EST.
 * Homework2 due by 10 February 2010 5pm EST.
 * Homework3 due by 17 February 2010 5pm EST.
 * Homework4 due by 3 March 2010 5pm EST.
 * Homework5 due by 24 March 2010 5pm EST.
 * Homework6 due by 7 April 2010 5pm EST.
 * Homework7 due by 23 April 2010 5pm EST.

The probable latest archived versions of our homework submissions can be found at:

 * Team4

title: User:Egm6341.s10.Team4.andy


text:
User:Egm6341.s10.Team4

Homework 1

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

Homework 7

title: User:Egm6341.s10.Team4


text:
User:Egm6341.s10.Team4/HW1

= Problem 1: Limits=

Given
Refer Lecture slide 2-1 for problem statement

 * {| style="width:70%" border="0" align="center"

f(x) = \frac{e^x - 1}{x} $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)

 * }.
 * }.

Find
1. Find

 * {| style="width:70%" border="0" align="center"

\lim_{x \rightarrow 0}f(x) $$
 * $$\displaystyle

 * }.
 * }.

2. Plot
 * {| style="width:70%" border="0" align="center"

f(x),\ x\in[0,1] $$
 * $$\displaystyle

 * }.
 * }.

Solution
1. Limit

The Taylor series expansion for $$\displaystyle e^x $$ is given by,

 * {| style="width:70%" border="0" align="center"

\begin{align} e^x &= \sum_{i=0}^n \frac{x^i}{i!} \\ &= 1 + \sum_{i=1}^n \frac{x^i}{i!} \end{align} $$
 * $$\displaystyle

 * }.
 * }.

Dividing throughout by x,

 * {| style="width:70%" border="0" align="center"

 * $$\displaystyle

\Rightarrow \frac{e^x - 1}{x} = \sum_{i=1}^n \frac{x^{i-1}}{i!}

$$

 * }.
 * }.

which is equal to $$\displaystyle f(x)$$ in Equation(1).

 * {| style="width:70%" border="0" align="center"

\begin{align} \Rightarrow f(x) &= \frac{e^x - 1}{x} \\ &= \sum_{i=1}^n \frac{x^{i-1}}{i!} &= \frac{x^0}{1!} + \frac{x^1}{2!} + \frac{x^2}{3!} + \cdots \end{align} $$
 * $$\displaystyle

 * }.
 * }.

Applying limits on both sides,

 * {| style="width:70%" border="0" align="center"

 * $$\displaystyle

\Rightarrow \lim_{x \rightarrow 0}f(x) = \frac{0^0}{1!} + \cancelto{0}{\frac{0^1}{2!}} + \cancelto{0}{\frac{0^2}{3!}} + \cdots

$$

 * }.
 * }.

Hence,

 * {| style="width:70%" border="0" align="center"

$$\displaystyle \Rightarrow \lim_{x \rightarrow 0}f(x) = 1 $$ $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$\displaystyle \longrightarrow (2)

 * style= |

 * }.
 * }.

2. Plot convergence of $$\displaystyle f(x) $$ .

To visualize the limit as in Equation (2), $$\displaystyle f(x) \to 1 \ as\ x \to 0 $$, two plots are provided below one with $$\displaystyle n = 10 $$ in the Taylor series, & with 10 points on x-axis scale, and another with $$\displaystyle n = 100 $$, & with 100 points on x-axis scale.

Matlab Code:

Plots:

.

Author
Solved by - --Egm6341.s10.Team4.roni 04:31, 18 February 2010 (UTC) Typed and Reviewed by - Egm6341.s10.Team4.andy 21:59, 17 February 2010 (UTC) .

= Problem 2: Taylor Series Terms =

Given
Refer Lecture slide 2-3 for problem statement.

From lecture slide 2-2, for a continuous function $$\displaystyle f(x) $$, the Taylor series is given by

 * {| style="width:100%" border="0" align="left"

f(x)= p_n(x) + R_{n+1}(x) $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

such that $$\displaystyle f^{n+1}(x) $$ exists and continuous,

 * {| style="width:100%" border="0" align="left"

f^{n+1}(x) := \frac{d^{n+1}}{dx^{n+1}} f(x) $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

where,

 * {| style="width:100%" border="0" align="left"

p_n(x):= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \cdots + \frac{(x-x_0)^n}{n!} f^n(x_0) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

and,

 * {| style="width:100%" border="0" align="left"

R_{n+1}(x):= \frac{1}{n!} \int_{x_0}^{x}(x-t)^nf^{n+1}(t)dt $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

Find
To find $$\displaystyle p_n(x) $$ and $$\displaystyle R_{n+1}(x) $$ for $$\displaystyle e^x $$.

Solution
Taking $$\displaystyle x_0 = 0 $$, Equation (1) for $$\displaystyle f(x) = e^x $$ becomes,

 * {| style="width:100%" border="0" align="left"

p_n(x)= e^0 + \frac{(x-0)}{1!} e^0 + \frac{(x-0)^2}{2!} e^0 + \cdots + \frac{(x-0)^n}{n!} e^0 $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

which is,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle p_n(x)= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} $$ $$
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * $$\displaystyle \longrightarrow (3)

 * }.
 * }.

From Lecture slide 2-3, Equation(2) is given by,

 * {| style="width:100%" border="0" align="left"

R_{n+1}(x)= \frac{(x-x_0)^{n+1}}{(n+1)!} f^{n+1}(\xi) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \xi \in [x_0,x] $$
 * $$\displaystyle \longrightarrow (4)
 * }.
 * }.

For $$\displaystyle x_0 = 0 $$ and $$\displaystyle f(x) = e^x $$ Equation (4) becomes,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle R_{n+1}(x)= \frac{x^{n+1}}{(n+1)!} e^\xi $$ $$
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style = |
 * $$\displaystyle \longrightarrow (5)
 * }.
 * }.

Author
Solved by - --Egm6341.s10.Team4.roni 04:41, 18 February 2010 (UTC) Typed and Reviewed by - Egm6341.s10.Team4.andy 22:00, 17 February 2010 (UTC) .

= Problem 3: Infinity Norms =

Given
Refer Lecture slide 3-3 for problem statement.

 * {| style="width:100%" border="0" align="left"

f(x)= sin x $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

g(x)= sin (x-\frac{\pi}{2}) = -cos x $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

Find
1. Plot $$\displaystyle f(x), g(x) $$ given $$\displaystyle x \in [0,\pi] $$.

2. Find $$\displaystyle \parallel f \parallel_{\infty}, \parallel g \parallel_{\infty}, \parallel f - g\parallel_{\infty} $$

Solution
1.

Matlab Code:

Plots:

.

2.

The infinity norms are given as,

 * {| style="width:100%" border="0" align="left"

\parallel f \parallel_{\infty} = \underset{x}{max} |f| $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

or

 * {| style="width:100%" border="0" align="left"

\parallel f - g\parallel_{\infty} = \underset{x}{max} |f - g| $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

Min and Max Values from Matlab:

So, the infinity norms are given by,

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} \parallel f \parallel_{\infty} &= 1 \\
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style="width:35%; padding:10px; border:2px solid #8888aa" |

\parallel g \parallel_{\infty} &= 1 \\ \parallel f - g\parallel_{\infty} &= 1.414214 \end{align} $$
 * style = |

 * }.

Author
Solved by - --Egm6341.s10.Team4.roni 04:43, 18 February 2010 (UTC) Typed and Reviewed by - Egm6341.s10.Team4.andy 22:00, 17 February 2010 (UTC) .

= Problem 4: Integral Mean Value Theorem (IMVT)=

Given
From lecture slide [[media:Egm6341.s10.mtg2.pdf|2-4]], any function $$\displaystyle f(.) $$ continuous on $$\displaystyle [a,b] $$ with $$\displaystyle m = min(f(x)) $$ and $$\displaystyle M =  max(f(x)) $$,

 * {| style="width:70%" border="0" align="center"

m \le f(x) \le M $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)

 * }
 * }

Find
Lecture Slide [[media:Egm6341.s10.mtg5.pdf|5-1]]:

1. Prove IMVT for non-negative $$\displaystyle W(.) \Rightarrow W \ge 0$$, given by:

 * {| style="width:70%" border="0" align="center"

\int_{a}^{b}W(x)f(x)dx = f(\xi)\int_{a}^{b}W(x)dx $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)

 * }
 * }

where $$\displaystyle \xi \in [a,b] $$

2. Prove IMVT for non-zero $$\displaystyle W(.) \Rightarrow W \ne 0$$.

Solution
1. $$\displaystyle W(x) \ge 0$$ .

Equation (1),

 * {| style="width:70%" border="0" align="center"

m \le f(x) \le M $$
 * $$\displaystyle

 * }.
 * }.

Multiplying by non-negative $$\displaystyle W(x) $$,

 * {| style="width:70%" border="0" align="center"

\Rightarrow m W(x) \le W(x)f(x) \le M W(x) $$
 * $$\displaystyle

 * }.
 * }.

Integrating from a to b with respect to x,

 * {| style="width:70%" border="0" align="center"

\Rightarrow m \int_{a}^{b} W(x)dx \le \int_{a}^{b} W(x)f(x)dx \le M \int_{a}^{b} W(x)dx $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (3)

 * }.
 * }.

Let the integral $$\displaystyle I $$ be,

 * {| style="width:70%" border="0" align="center"

I:=\int_{a}^{b} W(x)dx $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (4)
 * }.
 * }.

For $$\displaystyle I = 0 $$, the non-strict inequality Equation (3) holds true. So, we will take the case when $$\displaystyle I > 0$$.

Dividing (4) throughout (3),

 * {| style="width:70%" border="0" align="center"

\Rightarrow m \le \frac{1}{I} \int_{a}^{b} W(x)f(x)dx \le M $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (5)

 * }.
 * }.

From Lecture slide [[media:Egm6341.s10.mtg3.pdf|3-4]], we know that for every value in the closed interval $$\displaystyle [a,b] $$ there exists at least a value [need not be unique] $$\displaystyle \xi \in [a,b] $$ such that $$\displaystyle \zeta = f(\xi) = \frac{1}{b-a} \int_{a}^{b} f(x)dx$$, specifically for a continuous function $$\displaystyle f(x) $$.

With the same argument, there exists at least a value $$\displaystyle \xi_1 \in [a,b] $$ such that for a continuous function $$\displaystyle f(x) $$,

 * {| style="width:70%" border="0" align="center"

\Rightarrow \zeta_1 = f(\xi_1) = \frac{1}{I} \int_{a}^{b} W(x)f(x)dx $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (6)

 * }.
 * }.

Hence,

 * {| style="width:70%" border="0" align="center"

\Rightarrow f(\xi_1) I = \int_{a}^{b} W(x)f(x)dx $$
 * $$\displaystyle

 * }.
 * }.

proving Equation (2) for a non-negative $$\displaystyle W(x) $$,

 * {| style="width:70%" border="0" align="center"

$$\displaystyle \Rightarrow f(\xi_1) \int_{a}^{b} W(x)dx = \int_{a}^{b} W(x)f(x)dx $$ $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$\displaystyle \longrightarrow (7)
 * style= |

 * }.
 * }.

2. $$\displaystyle W(x) \ne 0$$ .

For the case when $$\displaystyle W(x) > 0$$ the above proof holds.

Next we deal the case when $$\displaystyle W(x) < 0$$.

Multiplying Equation(1) by $$\displaystyle -W(x)$$ (this makes $$\displaystyle W(x) > 0$$ in the below equations),

 * {| style="width:70%" border="0" align="center"

\Rightarrow -m W(x) \ge -W(x)f(x) \ge -M W(x) $$
 * $$\displaystyle

 * }.
 * }.

Integrating from a to b with respect to x gives,

 * {| style="width:70%" border="0" align="center"

\Rightarrow -m \int_{a}^{b} W(x)dx \ge -\int_{a}^{b} W(x)f(x)dx \ge -M \int_{a}^{b} W(x)dx $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (8)

 * }.
 * }.

Multiplying Equation(8) throughout by -1, we get Equation (3). Right from there, we can prove IMVT following the same procedure as explained above.

Author
Solved by - Egm6341.s10.Team4.andy 22:00, 17 February 2010 (UTC), Egm6341.s10.team4.anandankala 13:43, 19 February 2010 (UTC)

Typed by - Egm6341.s10.Team4.andy 22:00, 17 February 2010 (UTC).

= Problem 5: IMVT Appication Example =

Given
The Integral Mean Value Theorem (IMVT) is given by,

 * {| style="width:70%" border="0" align="center"

\int_{a}^{b}W(t)f(t)dt = f(\xi)\int_{a}^{b}W(t)dt $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (1)

 * }
 * }

where $$\displaystyle \xi \in [a,b] $$

Find
Refer Lecture slide [[media:Egm6341.s10.mtg5.pdf|5-1]] for problem statement.

From Lecture slide [[media:Egm6341.s10.mtg2.pdf|2-2]] the Remainder $$\displaystyle R_{n+1}(x)$$ of the Taylor series is given by,

 * {| style="width:70%" border="0" align="center"

R_{n+1}(x) := \frac{1}{n!} \int_{x_0}^{x} (x-t)^n f^{n+1}(t)dt $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (2)

 * }.
 * }.

where $$\displaystyle f^{n+1}(t) $$ is the $$\displaystyle (n+1)^{th} $$ derivative of $$\displaystyle f(t) $$.

Using IMVT show that,

 * {| style="width:70%" border="0" align="center"

R_{n+1}(x) = \frac{(x-x_0)^{n+1}}{(n+1)!} f^{n+1}(\xi) $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (3)

 * }.
 * }.

where $$\displaystyle \xi \in [x_0,x] $$.

Solution
Equation (2) is given by,

 * {| style="width:70%" border="0" align="center"

R_{n+1}(x) := \frac{1}{n!} \int_{x_0}^{x} \underbrace{(x-t)^n}_{W(t)} \underbrace{f^{n+1}(t)}_{g(t)}dt $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (3)

 * }.
 * }.

Now, using IMVT we can pull out $$\displaystyle g(t) $$ valued at $$\displaystyle \xi $$ resulting in,

 * {| style="width:70%" border="0" align="center"

\Rightarrow R_{n+1}(x) = g(\xi)\frac{1}{n!} \int_{x_0}^{x} \underbrace{(x-t)^n}_{W(t)} dt $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (4)

 * }.
 * }.

The integrand $$\displaystyle W(t) = (x-t)^n $$ is integrated from $$\displaystyle x_0 $$ to $$\displaystyle x $$. So, $$\displaystyle t \in [x_0,x]$$, which means $$\displaystyle (x-t) \ge 0$$. $$\displaystyle \Rightarrow W(t) \ge 0 $$.

 * {| style="width:70%" border="0" align="center"

\begin{align}
 * $$\displaystyle

\Rightarrow \int_{x_0}^{x} W(t)dt & = \int_{x_0}^{x} {(x-t)^n} dt \\ & = -\frac{1}{n+1} \left [ (x-t)^{n+1}\right]_{x0}^{x} \\ & = -\frac{1}{n+1} \left [ \cancelto{0}{(x-x)^{n+1}} - (x-x_0)^{n+1} \right ] \\ & = -\frac{1}{n+1} \left [- (x-x_0)^{n+1} \right ] \end{align} $$

 * }.
 * }.

So,

 * {| style="width:70%" border="0" align="center"

 * $$\displaystyle

\Rightarrow \int_{x_0}^{x} W(t)dt = \frac{(x-x_0)^{n+1}}{n+1}

$$ $$
 *
$$\displaystyle \longrightarrow (5)

 * }.
 * }.

Substituting (5) in (4),

 * {| style="width:70%" border="0" align="center"

\begin{align} \Rightarrow R_{n+1}(x) &= g(\xi)\frac{1}{n!} \frac{(x-x_0)^{n+1}}{n+1} \\ &= g(\xi)\frac{(x-x_0)^{n+1}}{(n+1)!} \\ &= f^{n+1}(\xi)\frac{(x-x_0)^{n+1}}{(n+1)!} \end{align} $$
 * $$\displaystyle

 * }.
 * }.

Hence Equation(3) is proved,

 * {| style="width:70%" border="0" align="center"

$$\displaystyle
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

\Rightarrow R_{n+1}(x):= \frac{1}{n!} \int_{x_0}^{x} (x-t)^n f^{n+1}(t)dt = \frac{(x-x_0)^{n+1}}{(n+1)!} f^{n+1}(\xi)

$$ $$
 *
$$\displaystyle \longrightarrow (6)
 * style= |
 * }.
 * }.

Author
Solved and Typed by - Egm6341.s10.Team4.andy 22:01, 17 February 2010 (UTC) .

= Problem 6: Taylor Series: Integration by Parts !!=

Given
Use following Integration

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle f(x)=f(x_0)+\dfrac{x-x_0}{1!}f^1(x_0)+\int\limits_{x_0}^{x} (x-t)f^2(t)\, dt

 * }
 * }

Find
Repeat integration by parts to reveal:

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \dfrac{(x-x_0)^2}{2!}f^2(x_0)+\dfrac{(x-x_0)^3}{3!}f^3(x_0)

 * }
 * }

Solution
For solving this problem we should show that:

I=\int\limits_{x_0}^{x} (x-t)f^2(t)\, dt = \dfrac{(x-x_0)^2}{2!}f^2(x_0)+\dfrac{(x-x_0)^3}{3!}f^3(x_0)+.... $$

From integration by parts; we can define functions u and dv as:

f^2(t)=u; (x-t)dt=dv \rightarrow\ v=\int\limits_{x_0}^{x}(x-t)\, dt = xt-\dfrac{(t^2)}{2} $$

\Rightarrow\ \int\limits_{x_0}^{x} (x-t)f^2(t)\, dt=[(xt-\dfrac{t^2}{2})f^2(t)]-\int\limits_{x_0}^{x}(xt-\dfrac{t^2}{2})f^3(t)\, dt= $$

=\dfrac{x^2}{2}f^2(x)-xx_0f^2(x_0)+\dfrac{x_0^2}{2}f^2(x_0)-\int\limits_{x_0}^{x} (xt-\dfrac{t^2}{2})f^3(t)\, dt $$

Then, by adding and subtracting following term to the integral,

\dfrac{x^2}{2}f^2(x_0) $$

We obtain:

=\dfrac{x^2}{2}f^2(x)-\dfrac{x^2}{2}f^2(x_0)+\dfrac{x^2}{2}f^2(x_0)-\dfrac{2xx_0}{2}f^2(x_0)+\dfrac{x_0^2}{2}f^2(x_0)-\int\limits_{x_0}^{x} (xt-\dfrac{t^2}{2})f^3(t)\, dt= $$

=\dfrac{(x-x_0)^2}{2!}f^2(x_0)+\dfrac{x^2}{2}\int\limits_{x_0}^{x} f^3(t)\, dt-\int\limits_{x_0}^{x} (xt-\dfrac{t^2}{2})f^3(t)\, dt= $$

=\dfrac{(x-x_0)^2}{2!}f^2(x_0)-\int\limits_{x_0}^{x} (\dfrac{x^2}{2}-xt+\dfrac{t^2}{2})f^3(t)\, dt $$

The first term of the expression has already been attained. Now, we should reveal that:

\int\limits_{x_0}^{x} (\dfrac{x^2}{2}-xt+\dfrac{t^2}{2})f^3(t)\, dt=\dfrac{(x-x_0)^3}{3!}f^3(x_0)+.... $$

Again by integration by parts we have:

f^3(t)=u; (\dfrac{x^2}{2}-xt-\dfrac{t^2}{2})dt=dv \rightarrow\ v=\int\limits_{x_0}^{x}(\dfrac{x^2}{2}-xt+\dfrac{t^2}{2})\, dt = (\dfrac{x^2}{2}t-\dfrac{x}{2}t^2+\dfrac{1}{6}t^3) $$

\Rightarrow\ \int\limits_{x_0}^{x} (\dfrac{x^2}{2}-xt+\dfrac{t^2}{2})f^3(t)\, dt=[(\dfrac{x^2}{2}t-\dfrac{x}{2}t^2+\dfrac{t^3}{6})f^3(t)]-\int\limits_{x_0}^{x} (\dfrac{x^2}{2}t-\dfrac{x}{2}t^2+\dfrac{1}{6}t^3)f^4(t)\, dt $$

=\dfrac{x^3}{6}f^3(x)-\dfrac{x^2}{2}x_0f^3(x_0)+\dfrac{xx_0^2}{2}f^3(x_0)-\dfrac{x_0^3}{6}f^3(x_0)-\int\limits_{x_0}^{x} (\dfrac{x^2}{2}t-\dfrac{x}{2}t^2+\dfrac{1}{6}t^3)f^4(t)\, dt $$

=\dfrac{x^3}{6}f^3(x)+\dfrac{(-3x_0x^2+3x_0^2x-x_0^3)}{6=3!}f^3(x_0)-\dfrac{x^3}{6}f^3(x_0)+\dfrac{x^3}{6}f^3(x_0)+....=\dfrac{(x-x_0)^3}{3!}f^3(x_0)+\int\limits_{x_0}^{x} (\dfrac{x^3}{6}-\dfrac{x^2}{2}t+\dfrac{x}{2}t^2+\dfrac{1}{6}t^3)f^4(t)\, dt $$

 * {| style="width:60%" border="0"

$$ \Rightarrow\ f(x)=f(x_0)+\dfrac{x-x_0}{1!}f^1(x_0)+\int\limits_{x_0}^{x} (x-t)f^2(t)\, dt+\dfrac{(x-x_0)^2}{2!}f^2(x_0)+\dfrac{(x-x_0)^3}{3!}f^3(x_0)+.... $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }.

Author
Solved and Typed by - Egm6341.s10.Team4.nimaa&m 02:33, 18 February 2010 (UTC) .

= Problem 7: Taylor series extension =

Given

 * {| style="width:70%" border="0" align="center"

f(x)= \sin(x),  x \in [0,\pi] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (1)

 * }
 * }

Find
1. Construct Taylor series of $$\displaystyle f(x)$$ around $$x_o=\dfrac{\pi}{4}$$  for n = 0, 1,..., 10

2. Plot these series (for each n)

3. Find (estimate of max)of R(x) at $$x=\dfrac{\pi}{2}$$

Solution
1. Construct Taylor series of $$\displaystyle f(x)$$ around $$x_o=\dfrac{\pi}{4}$$  for n = 0, 1,..., 10

when n=0,

 * {| style="width:70%" border="0" align="center"

P_0(x)= \sin(\dfrac{\pi}{4}) = \dfrac{1}{\sqrt{2}} $$
 * $$\displaystyle

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

R_0(x) = \sin(x)-\sin(\dfrac{\pi}{4}) $$
 * $$\displaystyle

 * }.
 * }.

when n=1,

 * {| style="width:70%" border="0" align="center"

P_1(x)= \sin(\dfrac{\pi}{4}) + (x-\dfrac{\pi}{4})\cos(\dfrac{\pi}{4}) = \dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2}}(x-\dfrac{\pi}{4}) $$
 * $$\displaystyle

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

R_1(x) = -\sin(\xi)\dfrac{(x-\dfrac{\pi}{4})^2}{2!}, \xi \in [0,\pi] $$
 * $$\displaystyle

 * }.
 * }.

when n=2,
 * {| style="width:70%" border="0" align="center"

P_2(x)= \sin(\dfrac{\pi}{4}) + (x-\dfrac{\pi}{4})\cos(\dfrac{\pi}{4}) - \dfrac{(x-\dfrac{\pi}{4})^2}{2!}\sin(\dfrac{\pi}{4}) = \dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2}}(x-\dfrac{\pi}{4})- \dfrac{1}{\sqrt{2}}\dfrac{(x-\dfrac{\pi}{4})^2}{2} $$
 * $$\displaystyle

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

R_2(x) = -\cos(\xi)\dfrac{(x-\dfrac{\pi}{4})^3}{3!}, \xi \in [0,\pi] $$
 * $$\displaystyle

 * }.
 * }.

when n=3,

 * {| style="width:70%" border="0" align="center"

P_3(x)= \sin(\dfrac{\pi}{4}) + (x-\dfrac{\pi}{4})\cos(\dfrac{\pi}{4}) - \dfrac{(x-\dfrac{\pi}{4})^2}{2!}\sin(\dfrac{\pi}{4}) - \dfrac{(x-\dfrac{\pi}{4})^3}{3!}\cos(\dfrac{\pi}{4}) = \dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2}}(x-\dfrac{\pi}{4})- \dfrac{1}{2\sqrt{2}}(x-\dfrac{\pi}{4})^2 - \dfrac{1}{6\sqrt{2}}(x-\dfrac{\pi}{4})^3 $$
 * $$\displaystyle

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

R_3(x) = \sin(\xi)\dfrac{(x-\dfrac{\pi}{4})^4}{4!}, \xi \in [0,\pi] $$
 * $$\displaystyle

 * }.
 * }.

when n=4,

 * {| style="width:70%" border="0" align="center"

P_4(x)= P_3(x) + \dfrac{(x-\dfrac{\pi}{4})^4}{4!}\sin(\dfrac{\pi}{4}) $$
 * $$\displaystyle

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

R_4(x) = \cos(\xi)\dfrac{(x-\dfrac{\pi}{4})^5}{5!}, \xi \in [0,\pi] $$
 * $$\displaystyle

 * }.
 * }.

when n=5 ~ 10,

 * {| style="width:70%" border="0" align="center"

P_n(x)= P_{n-1}(x) + \dfrac{(x-\dfrac{\pi}{4})^n}{n!}f^{(n)}(\dfrac{\pi}{4}) $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (2)

 * }.
 * {| style="width:70%" border="0" align="center"
 * {| style="width:70%" border="0" align="center"

R_n(x) = f^{(n+1)}(\xi)\dfrac{(x-\dfrac{\pi}{4})^{n+1}}{(n+1)!} $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (3)

 * }.
 * }.

where,
 * {| style="width:70%" border="0" align="center"

 * $$\displaystyle f^{(n)}(x) = \sin(x)$$ n=4k,   (k=0,1,2,.......)

$$\displaystyle f^{(n)}(x) = \cos(x)$$ n=4k+1,   (k=0,1,2,.......)

$$\displaystyle f^{(n)}(x) = -\sin(x)$$ n=4k+2,   (k=0,1,2,.......)

$$\displaystyle f^{(n)}(x) = -\cos(x)$$ n=4k+3,   (k=0,1,2,.......)

 * }.
 * }.

2. plot $$\displaystyle P_n(x)$$ for n=0, 1, 2, ... 10

 Matlab Code:

 Plots:  The plots for $$\displaystyle P_n(x)$$ with $$\displaystyle n= 0, 1, 2,. . . 10 $$ are shown below.

3. Find (estimate of max)of R(x) at $$x=\dfrac{\pi}{2}$$

from the equation $$\displaystyle (3)$$
 * {| style="width:70%" border="0" align="center"

R_n(x) = f^{(n+1)}(\xi)\dfrac{(x-\dfrac{\pi}{4})^{n+1}}{(n+1)!} $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (4)

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

\left | R_n(\dfrac{\pi}{2}) \right |\leqslant \dfrac{(\dfrac{\pi}{2}-\dfrac{\pi}{4})^{n+1}}{(n+1)!} \underbrace{\max \left | f^{(n+1)}(\xi)\right |, \xi \in [0,\dfrac{\pi}{2}]}_{\leqslant 1} = \dfrac{(\dfrac{\pi}{4})^{n+1}}{(n+1)!} $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (5)

 * }.
 * }.

$$\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|c|}\hline n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline R_n(\dfrac{\pi}{2})&\dfrac{(\pi/4)^{1}}{1!}&\dfrac{(\pi/4)^{2}}{2!}&\dfrac{(\pi/4)^{3}}{3!}&\dfrac{(\pi/4)^{4}}{4!}&\dfrac{(\pi/4)^{5}}{5!}&\dfrac{(\pi/4)^{6}}{6!}&\dfrac{(\pi/4)^{7}}{7!}&\dfrac{(\pi/4)^{8}}{8!}&\dfrac{(\pi/4)^{9}}{9!}&\dfrac{(\pi/4)^{10}}{10!}&\dfrac{(\pi/4)^{11}}{11!}\\ \hline &7.854E-01&3.084E-01&8.075E-02&1.585E-02&2.490E-03&3.260E-04&3.658E-05&3.591E-06&3.134E-07&2.461E-08&1.757E-09\\ \hline \end{array} $$.

Author
Solved and Typed by - Egm6341.s10.Team4.yunseok 02:07, 18 February 2010 (UTC) .

= Problem 8: Integration Methods - A Comparison=

Given

 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle f(x)= \frac{e^x-1}{x} $$
 * $$\displaystyle I= \int\limits_{0}^{1}\frac{e^x-1}{x}\, dx $$
 * }
 * }
 * }

Find
Use 3 methods to find $$I_n$$

1. Taylor series expansion of $$f_n$$.

2. Composite Trapezoidal Rule.

3. Composite Simpson Rule.

n=2, 4, 8,. . . until error of order $$10^{-6}$$

Solution
'''1. Taylor series expansion of $$f_n$$. '''

The function $$e^x$$ can be approximated using Taylor series to find that $$e^x=\displaystyle\sum_{j=0}^{\infty}\frac{x^j}{j!}.$$ Inserting this into our function, $$f(x)$$, and simplifying the expression, we find that $$f(x)=\displaystyle\sum_{j=1}^{\infty}\frac{x^{(j-1)}}{j!}.$$ While this form of our function, $$f(x)$$, presents an exact solution, we need to truncate this in order to solve it numerically. Therefore, let $$f_n(x)$$ be our approximation of $$f(x)$$ and let $$f_n(x)=\displaystyle\sum_{j=1}^{n}\frac{x^{(j-1)}}{j!}.$$ While $$f_n(x)$$ can be integrated numerically, it still has some degree of error because it is only an approximation of $$f(x)$$. The error in this approximation can be analyzed by integrating the remainder of $$f_n(x)$$, where $$R \displaystyle\ _{n}(x)=f(x)-f_n(x)=\frac{x^{n}}{(n+1)!}exp(\xi),$$ for $$\displaystyle\ \xi   \epsilon   [0,x] $$.

$$\displaystyle f_n(x)= \sum_{j=1}^n \frac{x^{j-1}}{j!}$$

$$\displaystyle I_n= \int\limits_{0}^{1}f_n(x)\, dx = \int\limits_{0}^{1} \sum_{j=1}^n \frac{x^{j-1}}{j!}\, dx = \sum_{j=1}^n \frac{1}{j j!}$$

The erroer$$\displaystyle (E_n) $$ is

$$\displaystyle E_n= I-I_n = \int\limits_{0}^{1}f(x)-f_n(x)\, dx = \int\limits_{0}^{1} \frac{(x-0)^{n}}{(n+1)!} \frac{e^x-1}{x}\, dx \underbrace{=}_{IMVT} \max \left | \frac{e^\xi-1}{\xi} \right |\int\limits_{0}^{1} \frac{x^{n}}{(n+1)!} dx, \xi \in [0,1] \leqslant \frac{e-1}{(n+1)(n+1)!}$$

$$\displaystyle E_8 \leqslant 5.327\times10^{-6}$$

So, $$\displaystyle I_8 = \sum_{j=1}^7 \frac{1}{j j!} = 1.3179018$$

'''2. Composite Trapezoidal Rule. '''

The trapezoidal rule is a method of numerical integration that uses linear interpolation between two points in order to calculate the area under a given curve, $$f(x)$$.

For two points on a curve, where x=a,b. The trapezoidal rule can be expressed as $$\displaystyle\ I_1=\int_a^b f_n(x)dx =\frac{b-a}{2}(f(a)+f(b)).$$

For n+1 number of points, the domain of $$ \displaystyle\ x \epsilon [a,b] $$ can be broken up into n segments of equal length, h, and the trapezoidal rule works out such that $$ \displaystyle\ I_n=\frac{h}{2}(f_0+2f_1+2f_2+...+2f_{n-1}+f_n)$$, where $$\displaystyle\ f_i=f(x_i)$$.

For the case that $$f(x)=\frac{e^x-1}{x}$$, results of this method are shown below.

The error of this case, based on the result of using the Taylor series approximation of f(x) for a large value of n, is also available below.

'''3. Composite Simpson's Rule. '''

Whereas the trapezoidal rule uses a linear method of interpolation, Simpson's Rule uses a 2nd order polynomial to determine the area under a given curve, $$f(x)$$.

For three points on a curve, for x=a, (a+b)/2,b. $$\displaystyle\ I_2=\int_a^b f_n(x)dx =\frac{b-a}{6}(f(a)+4f(\frac{a+b}{2})+f(b))$$.

For n+1 number of points, the domain of $$ \displaystyle\ x \epsilon [a,b]$$ can be broken into n segments of equal length, h, and Simpson's Rule works out such that $$ \displaystyle\ I_n=\frac{h}{6}(f_0+4f_1+2f_2+4f_3+...2f_{n-2}+4f_{n-1}+f_n)$$, where $$ \displaystyle\ f_i=f(x_i)$$.

For the case that $$f(x)=\frac{e^x-1}{x}$$, results of this method are shown below.

The error of this case, based on the result of using the Taylor series approximation of f(x) for a large value of n, is also available below.

 Matlab Code: For $$ E_{n,Taylor} $$

.

 Matlab Code: For $$ E_{n,trap} $$

.

 Matlab Code: For $$ E_{n,Simpson} $$

.

Author
Solved by - --Egm6341.s10.Team4.roni 04:47, 18 February 2010 (UTC)

Egm6341.s10.Team4.riherd 14:53, 18 February 2010 (UTC)

Egm6341.s10.team4.anandankala 13:43, 19 February 2010 (UTC)

Typed by - Egm6341.s10.Team4.riherd 14:53, 18 February 2010 (UTC) .

= Problem: 9 Taylor Series Terms=

Given

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \dfrac{e^x}{x}=\dfrac{1}{x}[e^x-1]=:f(x)

 * }
 * }

Find
1) Expand
 * {| style="width:100%" border="0" align="left"

e^x $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

in Taylor series with remainder.

2) Find Taylor series expansion and remainder of f(x).

Solution
Generally, f(x) can be expressed in Taylor series as follows:

 * $$\displaystyle

f(x)=P_n(x)+R_{n+1}(x) $$

\Rightarrow\sum_{i=0}^n \dfrac{(x-x_0)^n}{n!}f^n(x_0)+\dfrac{(x-x_0)^{n+1}}{(n+1)!}f^{n+1}(\gamma) $$

\gamma\in [0,x] $$

\Rightarrow\ e^x=\sum_{i=0}^n \dfrac{x^n}{n!}f^n(0)+\dfrac{x^{n+1}}{(n+1)!}f^{n+1}(\gamma)=\sum_{i=0}^n \dfrac{x^n}{n!}+\dfrac{x^{n+1}}{(n+1)!}e^\gamma $$

Using the first part will lead us to solve second part:

\Rightarrow\ e^x=1+\sum_{i=1}^n \dfrac{x^n}{n!}+\dfrac{x^{n+1}}{(n+1)!}e^\gamma $$

\Rightarrow\ e^x-1=\sum_{i=1}^n \dfrac{x^n}{n!}+\dfrac{x^{n+1}}{(n+1)!}e^\gamma $$

\Rightarrow\ \dfrac{e^x-1}{x}=\sum_{i=1}^n \dfrac{x^{n-1}}{n!}+\dfrac{x^n}{n+1}e^\gamma $$

\gamma\in[0,x] $$

Author
Solved and Typed by - Egm6341.s10.Team4.nimaa&m 02:34, 18 February 2010 (UTC) .

= Problem: 10 Trapezoidal Simple Rule: Lagrange Basis!!=

Given
Use following Integration

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \int\limits_{a}^{b} \sum_{i=0}^{n=1} l_i(x)f(x_i)\, dx

 * }
 * }

Find
To obtain

 * {| style="width:100%" border="0" align="left"

I=\dfrac{b-a}{2}(f(a)+f(b)). $$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle

 * }
 * }

Solution
First we expand the summation inside the integral:

I=\int\limits_{a}^{b} \sum_{i=0}^{n=1} l_i(x)f(x_i)\, dx = \int\limits_{a}^{b}[l_0(x)f(x_0)+l_1(x)f(x_1)]\, dx $$

From Lagrange basis equation we have:

l_i(x)=\prod_{j=0,j \ne i}^{n=1} \dfrac{x-x_j}{x_i-x_j} $$

\Rightarrow\ l_0(x)= \dfrac{x-x_1}{x_0-x_1}; l_1=\dfrac{x-x_0}{x_1-x_0} $$

Trivially,

x_0=a;   x_1=b $$

\Rightarrow\ I= \int\limits_{a}^{b}[\dfrac{x-b}{a-b}f(a)+\dfrac{x-a}{b-a}f(b)]\, dx $$

\dfrac{1}{b-a} \int\limits_{a}^{b}[(b-x)f(a)+(x-a)f(b)]\, dx $$

=\dfrac{1}{b-a} [\int\limits_{a}^{b}(b-x)f(a)\, dx]+[\int\limits_{a}^{b}(x-a)f(b)\, dx] $$

Now, integration can be easily solved in its interval:

=\dfrac{1}{b-a}[f(a)(bx-\dfrac{x^2}{2})+f(b)(\dfrac{x^2}{2}-ax)]=\dfrac{1}{b-a}[f(a)(\dfrac{b^2}{2}-ab+\dfrac{a^2}{2})+f(b)(\dfrac{b^2}{2}-ab+\dfrac{a^2}{2})] $$

=\dfrac{1}{2(b-a)}[f(a)(b-a)^2+f(b)(b-a)^2]=\dfrac{b-a}{2}[f(a)+f(b)] $$

So,

 * {| style="width:20%" border="0"

$$\displaystyle \Rightarrow I = \dfrac{b-a}{2}[f(a)+f(b)] $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

 * style= |

 * }.
 * }.

Author
Solved and Typed by - Egm6341.s10.Team4.nimaa&m 02:35, 18 February 2010 (UTC) .

=Problem 11: Simpson's Rule - Simple=

Given
From Lagrange basis equation we have:
 * {| style="width:70%" border="0" align="center"

 * $$l_i= \displaystyle\prod_{\underset{j\neq i}{j=0}}^{2}\frac{x-x_j}{x_i-x_j}$$
 * }.
 * }.

Find
Show that,

 * {| style="width:70%" border="0" align="center"

 * $$p_2(x_j)= \displaystyle\sum_{i=0}^{2} \underbrace{l_i(x_j)}_{\delta_{ij}}f(x_i)= f(x_j)$$
 * }.
 * }.

Solution

 * {| style="width:70%" border="0" align="center"

 * $$\underset{i=0}{l_0}= \displaystyle\prod_{\underset{j\neq i}{j=0}}^{2}\frac{x-x_j}{x_i-x_j}= \frac{(x-x_1)(x-x_2)}{(x_0-x_1)(x_0-x_2)}$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$\underset{i=1}{l_1}= \displaystyle\prod_{\underset{j\neq i}{j=0}}^{2}\frac{x-x_j}{x_i-x_j}= \frac{(x-x_0)(x-x_2)}{(x_1-x_0)(x_1-x_2)}$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$\underset{i=2}{l_2}= \displaystyle\prod_{\underset{j\neq i}{j=0}}^{2}\frac{x-x_j}{x_i-x_j}= \frac{(x-x_0)(x-x_1)}{(x_2-x_0)(x_2-x_1)}$$
 * }.
 * }.

@x=x0
 * {| style="width:70%" border="0" align="center"

 * $$l_0= \frac{(x_0-x_1)(x_0-x_2)}{(x_0-x_1)(x_0-x_2)}= 1$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$l_1= \frac{ \cancelto{0}{(x_0-x_0)}(x_0-x_2)}{(x_1-x_0)(x_1-x_2)}= 0$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$l_2= \frac{ \cancelto{0}{(x_0-x_0)}(x_0-x_1)}{(x_2-x_0)(x_2-x_1)}= 0$$
 * }.
 * }.

Similarly @x=x1

 * {| style="width:70%" border="0" align="center"

 * $$l_0= \frac{ \cancelto{0}{(x_1-x_1)}(x_1-x_2)}{(x_0-x_1)(x_0-x_2)}= 0$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$l_1= \frac{(x_1-x_0)(x_1-x_2)}{(x_1-x_0)(x_1-x_2)}= 1$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$l_2= \frac{(x_1-x_0) \cancelto{0}{(x_1-x_1)}}{(x_2-x_0)(x_2-x_1)}= 0$$
 * }.
 * }.

Similarly @x=x2

 * {| style="width:70%" border="0" align="center"

 * $$l_0= \frac{(x_2-x_1) \cancelto{0}{(x_2-x_2)}}{(x_0-x_1)(x_0-x_2)}= 0$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$l_1= \frac{(x_2-x_0) \cancelto{0}{(x_2-x_2)}}{(x_1-x_0)(x_1-x_2)}= 0$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$l_2= \frac{(x_2-x_0)(x_2-x_1)}{(x_2-x_0)(x_2-x_1)}= 1$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$ p_2 (j=0)_{x=x_0}= l_0.f(x_0)+\cancelto{0}{l_1}.f(x_1)+\cancelto{0}{l_2}.f(x_2)= 1.f(x_0)= f(x_j)$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$ p_2 (j=1)_{x=x_1}= \cancelto{0}{l_0}.f(x_0)+l_1.f(x_1)+\cancelto{0}{l_2}.f(x_2)= 1.f(x_1)= f(x_j)$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$ p_2 (j=2)_{x=x_2}= \cancelto{0}{l_0}.f(x_0)+\cancelto{0}{l_1}.f(x_1)+l_2.f(x_2)= 1.f(x_2)= f(x_j)$$
 * }.
 * }.

Hence proved.

Author
Solved by - --Egm6341.s10.Team4.roni 04:54, 18 February 2010 (UTC)

Typed by - Egm6341.s10.Team4.riherd 14:53, 18 February 2010 (UTC)

Egm6341.s10.team4.anandankala 13:43, 19 February 2010 (UTC) .

= Contributing Team Members =

Egm6341.s10.Team4.andy 22:03, 17 February 2010 (UTC) - Author of Problems:4, 5 ; Proof Read: 1,2,3,9,10,11

Egm6341.s10.Team4.riherd 1:48, 25 January 2010 (UTC) - Author of Problems 8, 11

Egm6341.s10.Team4.nimaa&m 13:54, 19 February 2010 (UTC) 10:50, 26 January 2010 (UTC) - Author of Problems: 6, 9, 10

Egm6341.s10.Team4.yunseok 08:43, 25 January 2010 (UTC) Author of Problems:7

Egm6341.s10.Team4.roni 05:10, 26 January 2010 (UTC) - Author of Problems:1,2,3,8,11 ; Proof Read: 1,2,3,4,5,7,8,9,10,11

Egm6341.s10.team4.anandankala 12:05, 27 January 2010 (UTC)- Author of problems:4,8,11; Proof Read: 2,3,5,6.

title: User:Egm6341.s10.Team4/HW1


text:
User:Bonny~enwikiversity

1. http://en.wikiversity.org/wiki/Category:The_web_design_project

2. http://en.wikiversity.org/wiki/Web_design

3. http://en.wikiversity.org/wiki/Introduction_to_HTML

title: User:Bonny~enwikiversity


text:
User talk:JackBailey

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title: User talk:JackBailey


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User talk:Ebouchut

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title: User talk:Ebouchut


text:
User talk:Pam~enwikiversity

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Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

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Renamed
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title: User talk:Pam~enwikiversity


text:
User talk:Htetaung

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User talk:FionaNicholson

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title: User talk:FionaNicholson


text:
Votian/Potential

== Potential mood  ==

Negative potential:

 * The very same endings of the first and second person are used in all  moods  and  tenses.
 * The negation verb will be conjugated in various moods. See the present tense negation chapter about dialectal and morphophonological alternatives of it.

At Ariste's time potential could be found almost nowhere but in runic songs. Even though Ahlqvist described it in his grammar as an ordinary feature and gave us a full inflectional paradigm.

'' The inflectional paradigm above is constructed following the example of modern Finnish. ''

''' Please fix it if it does not match the one of Ahlqvist. '''

Morphophonology

 * The morpheme -ne takes a consonant stem if possible e.g. tuлne̮n  ('Perhaps I shall come.') vs. tuлe̮  ('Come!').
 * Therefore it should be in the same grade as all the suffixal imperative forms except the singular of the second person.
 * It's first n goes through the same assimilation like the one of -nut participle  e.g. Ahlqvist's tulenen ~ tulnen ~ tullen  ('Perhaps I shall come.').

Ahlqvist does not mark the alternation of  -ne-, -ne̮-   ( depending in vowel harmony ) e.g. sȫ-nen ('Perhaps I shall eat.'),  jōnen  ('Perhaps I shall drink.'),  wõttanen  ('Perhaps I shall take.'),  wajeltanen  ('Perhaps I shall change.')

In Votic runic songs  the potential marker may be even doubled -nene-.

=== Phonological history ===

The potential marker -ne- is represented in other Finno-Ugric languages, but in other branches of this language group it may be used for  conditional mood. It's out of active use in most of Baltic Finnic  languages - except  Finnish,  South Estonian  and some  Karelian  dialects.

Usage
Potential may express
 * (future or present)  expectations  - e.g.  jōnen  ('Perhaps I shall drink.')

Votic potential is already substituted by constructions of ehtši / Kukkuzi  (and  Luuditsa ):  ehki  ("perhaps"): e.g.  johzamma ehki sinne  ("Perhaps we should run there."  Kukkuzi ),   miä arvān,  ehtši tämä tue̮b ōme̮n̄    ("I suppose she will probably come tomorrow."  Luuditsa ),   ehtši jumal lähetep läsivelle tervüs̄    ("May be God will send health to the sick one."   Jõgõperä ).

=== Other Finnic dialects ===

The first person of singular
 * Standard Finnish : luke-ne-n ('Perhaps I shall read.')
 * Vesjegonski Karelian dialects : luge-nne-n ('Perhaps I shall read.')
 * Vepsian dialects :  anda-ne̮-k  ('Perhaps I'll give?')
 * Ingrian dialects : lukkē-ne-n ('Perhaps I shall read.')

The first person of plural
 * Standard Finnish : luke-ne-mme ('Perhaps we shall read.')
 * Vesjegonski Karelian dialects : luge-nne-mma ('Perhaps we shall read.')
 * Ingrian dialects : luGe-ne-(m)man ('Perhaps we shall read.')

The second person of singular
 * Standard Finnish : luke-ne-t ('Perhaps you will read.')
 * Vesjegonski Karelian dialects : luge-nne-t ('Perhaps you will read.')
 * Ingrian dialects : lukkē-ne-D ('Perhaps you will read.')

The second person of plural
 * Standard Finnish : luke-ne-tte ('Perhaps you will read.')
 * Vesjegonski Karelian dialects : luge-nne-tta ('Perhaps you will read.')
 * Ingrian dialects : luGe-ne-tta ('Perhaps you will read.')

The third person of singular
 * Standard Finnish : luke-ne-e ('Perhaps she will read.')
 * Vesjegonski Karelian dialects : luge-nn-ou ('Perhaps she will read.')
 * Ingrian dialects : luG-en-ōn ('Perhaps she will read.')
 * South Estonian Võro dialects :  sā-ne̮-s̀  ('Perhaps he will get.')

The third person of plural
 * Standard Finnish : luke-ne-vat ('Perhaps they will read.')
 * Karelian dialects use indefinite voice instead : luge-nne-tta-h ('Perhaps they will read.').
 * Ingrian dialects : luG-en-osseG ('Perhaps they will read.')
 * South Estonian Võro dialects :  läǹ-ne-Zeˀ ('Perhaps they will go.')

Finnish potential may be substituted by constructions of ehkä ~ kai ("perhaps"): e.g. '' Tietänet, että hän on jo kotona. ~ Tiedät kai, että hän on jo kotona.  ("You probably know, that he is at home already."),  Muistanette, mikä hänen nimensä on. ~ Ehkä muistatte, mikä hänen nimensä on. '' ("You probably remember, what's her name.").

== Indefinite potential ==

At Ariste's time indefinite potential could be found nowhere but in runic songs. Even though Ahlqvist described it in his grammar as an ordinary feature e.g. (Ahlqvist's)  kōrittanes  ('Perhaps somebody will peel.'),  ihottanes  ('Perhaps somebody will grind.'),  ujuttanes  ('Perhaps somebody will swim.').

 * See the indefinite conditional chapter  about dialectal and morphophonological alternations of indefinite voice markers.

== Future tense  ==

Negative future:

The category of grammatical tense  in  Baltic Finnic  languages is rather disputable as the whole grammar of those languages is created under influence of Indo-European  linguists and then taught at every school, which has unrecoverably ruined the original linguistic intincts of the native speakers.

Actually the so called present tense  is just an unmarked verbal paradigm, which may represent an action happening in present, future or even past time as well as a  condition,  proposal or order.

Thus processes taking place in future are usually expressed by present tense  e.g.  kui tämä tuлe̮p  ̮ kotōse̮, eittäp  ̮ ke̮haллā makāmāsē̮   ('When he'll come home, he will go to sleep immediately.'),  nätilpǟn tuлe̮mma teilē  ('We shall come to visit you on Sunday.').

Nevertheless Dmitri Tsvetkov, the first native ( Jõgõperä ) Votic speaker to write a Votic grammar, denying Russian and German influences claims to a  sovereign right  of future tense for both Votic and Estonian languages opposing Estonian writer Ernst Peterson , who calls barbarisms  constructions like  mina saan küpsma  ("I shall mature." ~  German   Ich werde reifen. ), while an Estonian would say "Ma küpsen", "küll ma küpsen" or "eks ma küpse".

Yet the  simple future  of Tsvetkov's grammar coincides with present tense  again, but he describes also a  composed future , composed of future forms of  e̮лe̮ma   auxiliary verb  +  an infinitive  of a main verb.
 * 1) I did not find any examples of such a construction in his book. Instead of that he uses another  auxiliary verb   ne̮ise̮ma  +  ma infinitive  of a main verb e.g.  mǖ ne̮ise̮mm' лauлomā  ('We shall sing.' composed future),  emm' ne̮ise̮iz лauлomā  ('We should not sing.' conditional future).
 * 2) Still there exist future forms of  e̮лe̮ma   auxiliary verb  +  past participle  of a main verb e.g.  mǖ lēmm tšüzüttü  ('We shall be asked.' passive future).
 * 3) And there exist future forms of  e̮лe̮ma   auxiliary verb  +  ta infinitive  of a main verb in Votic grammar of Ariste e.g.  sinū tavākā liep  ̮ tauge̮ta  ('One of your habits should die.' future obligation),  lieb  ̮ mennä mettsǟ obahkā  ('We should go to forest to pick mushrooms.' future proposal  Lempola ).

 * 1) Ariste in his grammar describes those  future forms of  e̮лe̮ma   auxiliary verb   as  forms of a future  auxiliary verb   lid́d́ä   e.g.  i siл̄   lēväd  ̮ лahze̮D  ('You too will have children.' future expectation),  m a t́ ō   ku kazvab, lēB  ̮ mokoma niku emä  ('When Matyo will grow up, she will be like her mother.' future expectation).
 * 2) Ariste interpretes the future forms of  e̮лe̮ma (~ lid́d́ä )   auxiliary verb  +  past participle  of a main verb as  futurum exactum  e.g.  kui miä lēn tšüzünnü, sis siä anna  ("If I shall ask, then you give me!"),  siллē̮  ke̮ik̄   lēp   ̮ prostittu  ("Everything will be forgiven you!"  indefinite voice ),  leiväd  ̮ lēvät  ̮ senelē лavvaлē̮  pantu  ("Bread will be layed on this table."  indefinite voice ).
 * 3) He also notes constructions of  auxiliary verb   ne̮isa  +  ma infinitive  of a main verb e.g.  miä nē̮n patoi tetšemǟ  ('I shall make pottery.' future expectation).  kui mēt  ̮ kotōsē̮  , sinua isä tunte̮mā eb ne̮ize̮  ('When you shall go home, your father will not recognize you.' future expectation).
 * 4) Moreover he describes constructions of  auxiliary verb   ve̮ttā  +  ta infinitive  of a main verb e.g.  miä ävitin se̮rmuhsē̮  ; tšen ve̮tab  ̮ leütǟ ?  ('I lost my ring; who could find it?' future expectation).

 * And future may be expressed by total object  with  present tense  e.g.  лe̮ikkān sigā  ('I shall slaughter a pig.' genitive object, future confirmation) vs.  лe̮ikkān sikā  ('I am slaughtering a pig.' partitive object, present continuous);  tšünnättä pe̮ллō   ('You will plough the field.' genitive object, future confirmation) vs.  tšünnättä pe̮лtua   ('You are ploughing.' partitive object, present continuous);  e̮samma ope̮zē̮    ('We shall buy a horse.' genitive object, future confirmation) vs.  e̮samma ove̮ssa   ('We are buying a horse.' partitive object, present continuous).

=== Usage ===

 lee-tak  may express also
 * 1)  obligation  - e.g. Lempola dialect  lieb minū manaχann  ̮ e̮ллa ke̮ik̄   itšä  ("I must be a monk all my lifetime.")
 * 2)   indefinite relative pronouns   - e.g.  siä ku b lie tahod mennä tšäümǟ enellēz nuorikke̮ata  ("You probably want to go wooing."),  Savvokkala dialect  tšel   ̮ lēp tuli   n a r v a s̄    ("Somebody came from Narva.")

=== Dialectal alternatives ===

Both of these auxiliary verbs  may be contracted.
 * 1) ne̮ise̮ma / ne̮isa (nòis-tak) : ne̮ise̮vaD / nē̮vaD, ne̮ize̮B / nē̮B etc.
 * 2) e̮лe̮ma / lid́d́ä (lee-tak) : lēneväD / lēväD, lēneD / lēD etc.

 * Votic of Ingria
 * Eastern Votic dialect
 *  lee-tak  auxiliary verb  is pronounced  lēäG  e.g.  eb lēk kuhe̮p pannaG  ("There will be no place to put it to."),  miä tänä vōna läz̆zī, a tuлavōna tah̆hō e̮ллaG tervēnä, lēäG tervēnä  ("This year I'm sick, but the next year I want to be well.")
 * A metanalytic morpheme -k has been added to the potential-future  negation   ettak lee-ne  >  ettak lee-k  e.g.  ettäg  ̮ lēG  ('You won't be.'). The suffix -k is probably borrowed from the present tense negation here, where it was a present tense marker.

 * Western Votic dialects
 * Hill dialect:
 * Kattila dialect
 *  lēd läsivä terve vōsi  ("You will be sick the whole year long.")
 * Lempola dialect
 *  mā lieneb raute̮ne̮ , a taivaz lieb vahtšine̮    ("The earth becomes iron and the sky becomes copper.")

 * Valley dialect:
 * Mati dialect
 *  lee-tak  auxiliary verb  is pronounced  lid́d́ä  e.g.  mill lehmä nellättä vazikkā lēB  ("My cow will have a calf for the fourth time.")
 * Savvokkala dialect
 *  tšel ̮ lēp tuli   n a r v a s̄    ("Somebody came from Narva.")

 * Vaipooli
 * Luuditsa dialect
 *  lee-tak  auxiliary verb  is pronounced  lēvvä  e.g.  lēd läsivä terve vōsi  ("You will be sick the whole year long."),  kassin piäB lēvvä üvä sarka, üvä koto  ("Here should be a good field, a good household.")
 * Jõgõperä dialect
 *  kase pāri jo on tšihle̮Z, tšīre lēvvä pulme̮D  ("This pair is engaged already, there will be weddings soon.")
 * Liivtšülä dialect
 *  elä märnä, lēb ohto märnǟ  ("Don't shriek, you have shouted enough.")

 * Kukkuzi dialect
 *  kai tahot t́śītǟ, ni kīrē vana lēD  ("If you want to know everything, then you'll olden quickly.")

=== Other Finnic dialects ===

The first person of singular
 * Standard Finnish : lie-ne-n ('Perhaps I am.')

The first person of plural
 * Standard Finnish : lie-ne-mme ('Perhaps we are.')

The second person of singular
 * Standard Finnish : lie-ne-t ('Perhaps you are.')

The second person of plural
 * Standard Finnish : lie-ne-tte ('Perhaps you are.')

The third person of singular
 * Standard Finnish : lie-ne-e ('Perhaps he is.')

The third person of plural
 * Standard Finnish : lie-ne-vät ('Perhaps they are.')

In Finnish  grammar such constructions are interpreted as potential of  auxiliary verb   olla ~ Votic: e̮ллa  even though they may express future expectations as well e.g. '' Hän palannee huomenna.   ("He'll probably come tomorrow." ),  Suutuneeko isäsi tästä? ''  ("Will your father get angry of this?" ~  Tuleeko isäsi ehkä suuttumaan tästä?).

Finnish potential marker -ne- may be omitted exactly like the Votic one:
 * lienevät / lievät, lienet / liet etc.

Finnish does have even more regular potential of auxiliary verb   olla , but it's seldom used:
 * lienevät / ollevat, lienet / ollet etc.

title: Votian/Potential


text:
User:Harkey Lodger/Activity unit template 2

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Header 2
Below are sample quiz questions. Edit the question and responses so that they pertain to your lesson plan.

{Sample true statement. + True - False
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{Sample false statement. - True + False
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{Type a question here... - Wrong answer - Wrong answer + CORRECT answer - Wrong answer
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 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Module::
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 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Techniques::
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 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Food knowledge::
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 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Equipment & appliances:
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Objectives
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Essential reading
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Assignment
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Vocabulary
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Preparation skills
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Methods and techniques

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Experience with:

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Experience with:

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Purchasing and budgeting
Insert text here

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Self assessment
Below are sample quiz questions. Edit the question and responses so that they pertain to your lesson plan.

{Sample true statement. + True - False
 * type=""}

{Sample false statement. - True + False
 * type=""}

{Type a question here... - Wrong answer - Wrong answer + CORRECT answer - Wrong answer
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{Type a question here... + CORRECT answer - Wrong answer - Wrong answer - Wrong answer
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{Type a question here... - Wrong answer + CORRECT answer - Wrong answer - Wrong answer
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See also:

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