Free Essay

Submitted By garganmr

Words 893

Pages 4

Words 893

Pages 4

The limx→afx=L means that as x becomes closer and closer to a specific value a, the function f(x) will approach L. L is the number that the value approaches. The function never actually reaches this “a” value, so instead we find what it approaches. Let’s take the function fx=x2-1x-1, at x=1, because it equals 00 so we can take the limit to find what y value, x is approaching. I can either take the limit from the right, or the left. For this example I took the limit as it approached from the right. We get the equation: limx→1+x2-1x-1 = L. Now we can create a table of values where it is approaching x from the right. x | 2 | 1.5 | 1.1 | 1.01 | 1.001 | 1.0001 | f(x) | 3 | 2.5 | 2.1 | 2.01 | 2.001 | 2.0001 |

As you can see from the table, as x approaches 1, f(x) approaches 2 but is never actually 2. Since we can’t actually write that f (1)=2 you express it as, limx→1+x2-1x-1=2. The limit does not exist if limx→a-fx≠limx→a+f (x) because in order for the limit to exist, the function must be approaching the “L” value from both the left and the right towards the same number. In the same example above, yes it is possible for this statement f(a)≠L to be true because in in the above example the limit value above is 2 but the function does not exist.

Asymptote

An asymptote is a value that f(x) approaches when x is approaching an “a” value. In the equation limx→a+fx=∞, this is saying that as x becomes sufficiently close to an “a” value, the function f(x) will approach infinity. This means that there is a vertical asymptote at x=a. For example: looking at the graph pictured on the right, as x approaches 0, f(x) is approaching infinity. This limit does not exist because limx→0+1x=∞ and the limx→0-1x=-∞. Technically, the limit doesn’t exist because it is approaching infinity, which isn’t a specific number. These two limits do not head towards the same number or in this case infinity; therefore the limit does not exist. The limx→∞f(x)=L means that there is a horizontal asymptote. For example, in the same graph pictured above, as x approaches infinity, the function f(x)=1x is becoming closer and closer to 0. So we write the limx→∞1x=0. In this case the limit does exist because it is a one-sided limit that is settling on a specific number.

Continuity

For a function f to be continuous at a the limit of f(x) as x approaches a has to equal f(a). There are three conditions that f must satisfy. First, if there is a specific f(a) value for “a”. Meaning that there can’t be a hole at f(a) for a value “a”. Second, the limx→af(x) has to exist. Meaning that as you approach “a” from the left and the right, it has to settle on the same number. Lastly, the limx→afx=f(a). F is discontinuous at x=a if it doesn’t satisfy at least one of the three conditions mentioned above.

The first picture on the left exemplifies the first condition. The second picture exemplifies the second condition because as x approaches 3 from the right its 4 but as x approaches 3 from the left it is approaching 1. Since it does not settle on a single number, this is a point of discontinuity. The third example exemplifies the third condition. This is a picture where the limx→afxdoes not equal f (a). This is because even though the limit at x=2 exists, it is not equal to the value of the function, which is 1.5.

Derivative

The derivative of a function represents the tangent line. When you compute the slope of a tangent line, the instantaneous rate of change of a function, and the instantaneous velocity of an object at x=a all require us to compute the limit . This quantity represents the slope of the tangent line. On a graph it would look like this graph on the right. You can see that the black line represents the tangent line, which is the slope of the derivative of the original function.

There are three situations where a function is not differentiable. First, if the graph has a kink or a cusp then f is not differentiable there. This fails the definition of differentiability because they break the rule that right and left hand limits have to settle on the same number. On the left side of this example, it is approaching infinity, but from the right it is approaching negative infinity. Second, if there is any discontinuity, then f is not differentiable. For example if there is a hole or jump discontinuity. This is not differentiable because in order for it to be differentiable it has to be continuous on an interval [a,b]. Third, if there is a vertical tangent line. Meaning that if the line is completely vertical, it is not differentiable. This is not differentiable because it has an infinite slope

Free Essay

...A Guide for Writing a Technical Research Paper Libby Shoop Macalester College, Mathematics and Computer Science Department 1 Introduction This document provides you with some tips and some resources to help you write a technical research paper, such as you might write for your required capstone project paper. First, congratulations are in order– you are embarking on an activity that is going to change the way you think and add to the overall body of human knowledge. The skill of gathering information, deciding what is important, and writing about it for someone else is extremely valuable and will stay with you for the rest of your life. Because we humans have been doing this for quite some time, we have some reasonably standard forms for technical research papers, which you should use for your capstone. You should do this because your paper will better understood by readers who are familiar with this form. Before you can begin writing your paper, you need to have a sense for what research entails, so I’ll start there. Then I will give you some tips about writing, including connecting with your readers, deﬁning your topic, the format of your paper, and how to include references from the literature. I am a computer scientist, so be aware that parts of this paper are biased toward my discipline. 2 What is Research? A short deﬁnition of research, as given by Booth, Colomb, and Williams (Booth et al., 1995) is “gathering the information you need to......

Words: 3479 - Pages: 14

Free Essay

...are often used interchangeably to describe work which previously was done with paper, but which now has been adapted to information & communication technology (ICT) devices and software. The Information Technology Association of America (ITAA) has defined information technology (IT) in the electronic era as "the study, design, development, implementation, support or management of computer-based information systems, particularly software applications and computer hardware." IT entails processes involving the use of computers and software to create, convert, store, process, transmit, and retrieve information securely. The term has recently been broadened to ICT (Information and Communications Technology), so as to include the idea of electronic communication. To be paperless means essentially that the traditional paper-based practices-such as writing, note taking, reading, editing, communicating, and even drawing-are instead performed electronically with ICT devices and software. Much has been said and written about the paperless office in recent years, and the rapid development of ICT is enabling an increasing number of paperless practices. The relationship between paperless work styles and ICT is intimate and interdependent; a paperless work and lifestyle cannot be implemented without ICT, and the use of ICT should naturally lead to becoming paperless. Paradoxically, however, the consumption of paper has increased exponentially since the advent of personal...

Words: 2014 - Pages: 9

Free Essay

...TRAINING: •To know about the company’s management and functions of various departments. •To know how the company is working and the types of financial transactions it deals with. Tamil Nadu Newsprint and Papers Limited (TNPL) were formed by the Government of Tamil Nadu in April 1979 as a Public Limited Company under the provisions of the Companies Act, 1956. The primary objective of the company is to produce newsprint and printing & writing paper using bagasse, a sugarcane residue, as the primary raw material. The company is in the business of manufacturing and marketing of Newsprint and Printing & Writing Papers. The products are being marketed throughout the country and also being exported to 20 countries around the world. The factory is situated at Kagithapuram in Karur District of Tamil Nadu. The initial capacity of the plant was 90,000 tpa of Newsprint and Printing & Writing paper which commenced production in the year 1984. The Company was incorporated on 16th April, with a capacity of manufacture 50,000 tpa. Of newsprint, and 40,000 tpa of printing and writing paper. It was promoted by the Government of Tamil Nadu for the manufacture of Newsprint and Printing and Writing Papers using bagasse as the primary raw material. It manufactures newsprint, writing and printing paper. TNPL has obtained the ISO 9001-2000 certification from RWTUV of...

Words: 310 - Pages: 2

Premium Essay

...Learning Letter To be honest I’ve never been an excellent writer. When it comes to writing a paper for high school classes, scholarships, and basically everything else I’ve always had trouble with starting my paper and figuring out what to write about my topic. However, choosing a topic has never been a problem for me because I’m passionate about many different things. Whenever I would write a paper in high school I usually wouldn’t spend much time on it because of a couple different reasons, either the teacher chose a topic for me and I simply wasn’t very interested, and also because of procrastination. I believe this class will help me become interested in writing which will motivate me to do the work. As a person I’ve always been more of a reader than a writer. I started reading fantasies like the Lord of the Rings novels at a young age. During my freshmen year of high school I was introduced to writers and poets like Charles Bukowski, Allen Ginsberg, and Hunter S Thompson, and I’ve been reading similar works ever since. One thing that I’ve always wanted to do with writing is being able to write poetry similar to Bukowski. I’m hoping this class can help with that. Even though this class is obviously required to take I’m excited to be in it so I can improve on the things that I struggle with in writing. By the end of this quarter I want to be able to choose a topic, start the paper with ease, and also be able to generate ideas about the topic easily. I’m excited to see......

Words: 291 - Pages: 2

Free Essay

...Peer review for Zunwang Liu’s Draft By Guanyi Pan Summary: -the author analyzed the EJBR, and talk about its characteristics such as the length of the article, design of each journal, the audience of the journal, the tones of the articles and so on. Then she perorates that EBR is a example of text that can help us to learn the characteristic of discourse community with readers of JEBR actively share goals and communicate with others to pursue goals. Major point: Observation: the main point of the introduction is unclear. The analyzing parts in the paper is OK. The whole paper is talking about the EJBR. But it is hard to find a conclusion about them. 2. Do not have page number. 3. Observation: lack of the purpose of analyzing Location: page:page 2 Suggestion: After analyzing the length and other formats of EJBR, the author does not give a conclusion of them. So I am confused about why she wrote this, and what is the purpose of it. 4.Observation: unclear object Location: page 3 Suggestion: When the author talks about the audience of the journal, she only wrote “expert members”. I think she should point out what kind of the experts they are. 5. Observation: Need more examples in details. Location: page 5 Suggestion: I think there should be some examples to define about the gatekeeping of this journal. Minor Point: 1.There are some grammar problems and most of them have been corrected by last peer viewer. 2. The in-text citation format is not total correct. 3....

Words: 262 - Pages: 2

Premium Essay

...match the genre of the writing that the position would involve. For example, if you are applying for journalism positions, submit “clips”—actual articles that have been published in a campus newspaper, blog, or other publication. For a research position, submit an in-depth analysis of an issue or a topic. For a PR position, submit a press release that you have written from a previous internship or as the marketing chair of a campus group. If you don’t have any, you can write a press release for an upcoming event (just make sure you specify that it has not been published). Submit your best writing. If you are deciding between two papers you have written, and one is better written than the other but your weaker paper is topically more relevant, then choose the paper that is better written to submit. The other option is to rewrite the relevant paper to be stronger before you submit it. Remember, it’s your writing skills that the employer is assessing, and being topically relevant is just an added bonus. Provide excerpts if your samples are long. Most employers will specify how many pages...

Words: 475 - Pages: 2

Premium Essay

...and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop your cheating and write your paper. Stop......

Words: 596 - Pages: 3

Free Essay

...free account Copy & PaCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste Your PaperCopy & Paste...

Words: 419 - Pages: 2

Free Essay

...students will reflect on what they are thankful for, and visually present it by creating a placemat to use on their Thanksgiving table. Materials Pencil Paper Construction paper with leaves Construction paper with lines Large construction paper in various colors Glue Scissors Butcher paper Procedure: Beginning Teacher will instruct students to write a list of things they are thankful for. Once the list is written, the students will be handed a sheet of construction paper with the outlines of four different shapes of leaves on it. The students will cut out the leaves, and choose four things they are thankful for to copy down onto the leaves. Middle Once the leaves are finished, the students will be given three more sheets of construction paper; one large sheet, and two with lines on it to cut into strips. Students will be instructed to fold the long sheet in half, and cut from the fold to one inch away from the edge. The teacher will model this so there are few errors. Students will cut the other sheets of paper into strips along the drawn lines. Students will weave the strips of paper into the large sheet of paper, creating a placemat Once all strips are woven in, the students will glue the four leaves with what they are thankful for on them. End The students will place their placemats on a sheet of butcher paper in the back of the room to dry Once all students have finished, teacher will lead a discussion with the students to talk about what they are......

Words: 620 - Pages: 3

Premium Essay

...‘ My Reflection Letter” I feel like my writing has come along way however this class has given Me the opportunity to see that I need a lot of improvement in my grammar. But it as help me learn to take better notes while reading .I feel that I have learned a lot thus far in English- 090. However in the past, I have always felt afraid to express myself when writing. This I know is a very important aspect of composing and have been very critical of myself. I have always expected to strive to do my best . I put effort and thought into each assignment. However writing the first paper that was given , It really helped me to understand that most people don’t get it right their first try. Initially I would approach it as preparing my writing down note. Next, I proof read my work and correct the grammar and punctuation. Often, I will have someone read it for composition and clarification of my sentences. Finally, I would prepare my final copy. I have felt so much less pressure knowing that my writings don’t have to be perfect the first time. This is why I really like how you give us the opportunity to revise our essays as many times as we need to get them to our satisfaction. I know that I’m never content Often it reaches the point when I get frustrated and think, “Okay, I need to stop stressing over this. My biggest Road blocks does not allow me to think of ideas fast enough. As writing, one thing I really need to work on is organizing my thoughts...

Words: 421 - Pages: 2

Free Essay

...This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read This essay is a great essay that we get to read......

Words: 759 - Pages: 4

Premium Essay

...Technology(IJAET) ISSN 2231-1963 CALL FOR PAPER IJAET is a carefully refereed international publication. Contributions of high technical merit are to span the breadth of Engineering disciplines; covering the main areas of engineering and advances in technology. IJAET publishes contributions under Regular papers, Invited review papers, Short communications, Technical notes, and Letters to the editor. Book reviews, reports of and/or call for papers of conferences, symposia and meetings could also be published in this Journal Author Benefits : • • • • • • Rapid publication Index Factors and Global education Index Ranking Inclusion in all major bibliographic databases Quality and high standards of peer review High visibility and promotion of your articles Access of publications in this journal is free of charge. PUBLICATION CHARGES: A small publication fee of INR3500 upto 10 pages is charged for Indian author and for foreign author is USD 100 upto 10 pages for every accepted manuscript to be published in this journal. All the transaction Charges will be paid by Author (Inter Banking Charges, draft). Submission Guidelines: Guidelines Authors are kindly invited to submit their full text papers including conclusions, results, tables, figures and references. • The text paper must be according to IJAET Paper format and paper format can download from our website (www.ijaet.org).The Full text papers will be accepted in only .doc format. • The papers are sent to......

Words: 367 - Pages: 2

Premium Essay

...This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is a paper. This is......

Words: 441 - Pages: 2

Premium Essay

...How to Make Charcoal from Paper By Karren Doll Tolliver, eHow Contributor Homemade paper charcoal briquettes can be used in backyard grills. Commercial charcoal for grilling food is expensive and can be harmful to the environment. However, industrious do-it-yourselves can make their own "charcoal" from newspaper. This reduces the amount of newspaper refuse as well as the amount of commercial charcoal consumed. In addition, no lighter fluid is needed with the homemade charcoal paper. Therefore, petroleum-based products are also conserved. Making your own charcoal takes only water and a washtub. The time spent forming the charcoal paper briquettes is negligible, although they need to dry for a couple of days in the sun. Things You'll Need • Washtub • Water • Old newspaper Instructions 1 Tear the old newspaper into pieces about the size of your hand or smaller. 2 Place all the torn newspaper pieces in the washtub. Cover with water and let sit for at least one hour. The newspaper will be ready when it is thoroughly saturated with water and is mushy to the touch. 3 Grab a large handful of the mushy newspaper. Form it into a ball about the size of a golf ball or ping pong ball, squeezing out as much water as you can. Repeat until all the mushy newspaper is in ball form. Discard the water. 4 Place the wet newspaper balls in the sun for at least two days. Do not let them get rained on. They must be completely dry and brittle. At this point they are ready for use in the......

Words: 1319 - Pages: 6

Free Essay

...ANALYSIS PAPERS INSTRUCTIONS You are required to complete 3 separate analysis papers over the selected topics. Each paper must be 3–5 pages in length not including the title page, abstract page, and reference page. Each paper must be supported by at least 3 scholarly sources cited in current APA format. You will use references from both textbooks and outside peer-reviewed sources. The peer-reviewed sources must be chosen from PsycArticles or a similar search format from the Liberty University Online Library. Please consult the Analysis Papers Grading Rubric for this assignment. Analysis Paper 1 Consider yourself within the context of the “ecological model” and discuss how this model has influenced your own development. You will recall from your reading of McWhirter et al. that development is the result of “interconnected and embedded ecological systems”. Refer to Bronfenbrenner’s “ecological model” and the “at-risk tree” to complete your discussion. You will use your textbook and the Clinton et al. text as your references in addition to 2 other scholarly sources. Submit this assignment by 11:59 p.m. (ET) on Sunday of Module/Week 2. Analysis Paper 2 Manuel is a 15-year-old “at-risk” adolescent, with whom you have been working at your practicum site. One day when you are talking with him, he reports that he is extremely depressed and just doesn’t care about living anymore. List 4 of the “Legal Do’s and Don’ts” outlined in the text that you will want to consider when...

Words: 455 - Pages: 2