College Placement Test free essay sample

The first occasion when I took a school situation test was in December of 2015 at Bristol Community College. The test eventually figures out where understudies place in explicit courses, for example, number juggling, perusing, composing, and variable based math. Following quite a while of readiness and nervousness, the weight was on. All through my center school years, I was a solid understudy, consistently on the respect roll. I never had a GPA beneath 3.0. I was brilliant, and I knew it. That is, until I got the consequences of my school arrangement test. In spite of the fact that the test was testing, covering those four explicit subjects of perusing, composing, number-crunching, and variable based math, I was certain I had passed each part. Incredibly, I passed each part †with the exception of composing. â€Å"Writing! Definitely right! How could I figure out how to bomb composing, and significantly a point, no less?† I contemplated internally in dismay. Seeing my test outcomes carried tears to my eyes. I sincerely could barely handle it. I had worked too difficult to even consider scoring beneath the 3-point scale. Also, scoring a half-point beneath it. I thought it was ludicrous. To exacerbate the situation, the vast majority of my colleagues from secondary school likewise stepped through the exam, and shockingly again heard that they all passed, including some who were scarcely passing secondary school English. What a shame I thought. How might I have fizzled, and they had finished that assessment? I was such a great amount of greater at composing than they were, or if nothing else I thought I was on the grounds that I generally got A’s on my English papers. What did I foul up that caused me to fall flat, or as such, what did I miss on the test that they got right and I got off-base? Until that time, I adored composing the same amount of as I cherished math. It was one of my qualities. I was acceptable at it, and I delighted in it. On the off chance that anything, I figured I may bomb polynomial math. How might I have spoiled composition? I definitely spelled each word effectively, utilized great sentence structure, and even utilized huge words in the correct setting. How might I have fizzled? This was unquestionably a debby-killjoy since I generally realized I was a decent author thus did every other person. At last, I got over it and concluded it was not a problem. Definitely, I would relax. In my distinctions secondary school English class, I worked steadily, going with A’s. When I graduated secondary school, I realized I was prepared for school and prepared to overcome that composing test. All things considered, prepare to have your mind blown. I bombed the test once more, again with just 2.5 of the 3 focuses expected to pass. That time I did cry, and even went to my guide, Mr. Sanchez, and asked, â€Å"How would i be able to get A’s in the entirety of my English classes yet bomb the composing some portion of the school situation test twice?† He was unable to respond to my inquiry. Indeed, even my companions and cohorts were befuddled. I felt like a disappointment. I had disillusioned my family and truly let myself down. To top it all off, I despite everything couldn't make sense of what I was fouling up. I chose to stop making a decent attempt. Obviously †I let myself know †the individuals reviewing the tests didn't have even an inkling about what established great composing was. I kept on exceeding expectations in class and breeze through the assessment on the third attempt. In any case, I never again felt a similar love of perusing and composing. This experience demonstrated me exactly how diversely my composing could be decided by different perusers. Clearly, all my English instructors and numerous others delighted in or possibly valued my composition. A sonnet I composed was distributed online once. I more likely than not been a really decent essayist. Tragically, the graders for the school situation test didn't feel the equivalent, and when understudies bomb the test, the province of Massachusetts didn't offer any clarification. After I bombed the test the first occasion when, I started to detest composing, and I began to question myself. I questioned my capacity and the thoughts I expounded on. Bombing the subsequent time compounded the situation, so maybe to shield myself from my questions, I quit paying attention to English. Maybe in light of that absence of reality, I earned a D in the English 101 class at Bristol Community College, and had to retake it when I moved to Dean College. I wish I knew why I bombed that test , since then I may have composed what was normal on the subsequent attempt, kept up my excitement for composing, and kept on progressing admirably. On the other hand, this has allowed me the chance to demonstrate to everybody what sort of essayist I am.

Lab report Essay Example for Free

Lab report Essay A sub-atomic dissemination test of CH3)2CO air (repetitive w/last sentence. . concise)was directed with the objective of deciding the dispersion coefficient of CH3)2CO into air. For this investigation, CH3)2CO was set in a test tub 3mm OD, 2mm ID . . (is that right? ) NMR tube? e and was permitted to diffuse into non-diffusing air that was disregarded the test tube. The air that disregarded the cylinder was from normal flow in the room and no air was constrained over the highest point of the test tube. The dissemination happened over a time of around eight hours, with readings taken every hour. In the wake of dissecting the information gathered from the exhibition of this test, tThe dissemination coefficient was determined to be 0. 098 + 0. 02 cm2/s at T = ?. Subsequent to finishing our estimations, oOur results were then thought about utilizing the Chapman-Enskog condition just as the Fuller, Schettler, and Giddings technique. The dissemination coefficient determined by the Chapman-Enskog was 0. 990 + 0. 001 cm2/s and the consequence of the Fuller, Schettler, and Giddings technique was 0. 104 + . 002 cm2/s. The writing esteem found in Perry’s Chemical Engineer’s Handbook was 0. 125 + 0. 00 cm2/s. (at T = ?. . . or on the other hand extrapolated from? ) The understanding of our technique with different strategies accessible for figuring the dissemination coefficient was generally excellent (how great is â€Å"very† acceptable. .. critical discrepencies or not? ), and furthermore concurred well with the writing esteem found. This prompted an end that this technique for deciding the dissemination coefficient of CH3)2CO into air can be aconsidered a sensibly solid strategy. Foundation Atomic dissemination is the exchange or development of individual particles through a liquid by arbitrary sub-atomic developments (Geankoplis 412, year of distribution). In the dissemination procedure, the atoms of intrigue stream from locales of high focus to low fixation. Sub-atomic dispersion can happen in the two headings with the framework. On account of the dissemination tube explore, nonetheless, CH3)2CO diffuses through non-diffusing air, which is ignored the highest point of the test tube containing the CH3)2CO. The air is permitted into the test tube, however doesn't diffuse into the CH3)2CO. Atomic dissemination of gases has been read for a long time. Sub-atomic dispersion is a mass vehicle process Motivation for its investigation originates from the way that concoction division procedures, for example, refining, drying, particle trade frameworks just as numerous different procedures rely upon sub-atomic dissemination (Kirk-Othmer Vol 8, p 149(check organization)). Test METHODS For the exhibition of this test, a little test tube was filled around a third brimming with acetoneBe explicit. . how little, beginning stature, diam, and so on. This test tube was then vertically positioned in a 10mL graduated chamber which contained little dabs. The reason for the globules was to guarantee that the test tube stayed vertical. This gathering was then positioned on a computerized scale. The measure of air development gave by the ventilation framework was thought to be sufficient in order to guarantee that the convergence of the CH3)2CO at the highest point of the cylinder was zero. An underlying CH3)2CO level in the test tube was taken, just as the mass of the get together and the temperature of the territory encompassing the gathering. After this underlying information was taken, the territory temperature and mass of the gathering were taken around consistently for the following eight hours. The last degree of the CH3)2CO in the test tube was taken when the last temperature and mass perusing were taken. Conversation OF RESULTS From the information gathered from the test, the dispersion coefficient was determined utilizing condition 6. 2-26 from Geankoplis: (Equation 1) As the z esteem was just recorded toward the start and the finish of the test, the middle estimations of z must be determined. The accompanying condition was utilized for the figuring of the transitional z esteems: (Equation 2) Thus, all qualities however DAB were known and could be plotted versus time to get a direct plot. By revamping condition 1, it tends to be seen that the slant of this plot will be equivalent to 1/DAB : (Equation 1. 1) The underlying plot of information which incorporates all focuses is appeared underneath in Figure 1. This plot contains all focuses and has a R2 estimation of 0. 9478. From this plot the atomic diffusivity coefficient was resolved to be 0. 108 + 0. 022 cm2/s. Figure 1: First plot of information in Equation 1 The second point in the information (t=2700s) indicated no dispersion happened in the initial 45 minutes, which appears to be far-fetched (indeed, great affectability of equalization, and so on). On the off chance that this point is taken as mistaken, the R2 esteem goes up to 0. 9639 (increasingly significant here will be the certainty span on the slop. . . get that from Tools-Data Analyis-Regression menu in Excel or, in all likelihood in Polymath or TableCureve, and so forth) and the sub-atomic diffusivity ascertains out to be 0. 098 + 0. 021 cm2/s. The plot of the exploratory information barring the subsequent point is introduced underneath in Figure 2. Figure 2: Second plot of information in Equation 1. . compelling through zero point is acceptable. . . looks to me like initial FOUR focuses would give a lower Dab then the last 4. Issues with next 3 that lie beneath line? Anytihing dubious occurring here? To decide the time it takes for the framework to arrive at consistent express, the accompanying condition can be utilized to compute the portion of consistent express the framework is at: (Equation 3) By plotting the estimation of ((NA)t/(NA)t=? ) versus time, the bend in Figure 3 was produced which shows the frameworks way to deal with consistent state. Goodness, fantastic! Refer to source.â (still wonder about SST states of first 4 pts however. . . Figure 3: Fraction of consistent state versus time From this plot, one might say that the framework accomplishes consistent state in a short time; nonetheless, there is solid proof this may not be precise. As referenced before, the subsequent point might be wrong. This would change th e way of the bend. Also, information was not gathered at a sufficiently high recurrence for this bend to be exceptionally exact at foreseeing an opportunity to consistent state. On the off chance that in actuality the subsequent point is incorrect, the framework could have come to consistent state certainly before 115 minutes. This season of 115 minutes, best case scenario, could be the upper bound (or lower bound by Whitaker’s rules in his article (gift). . . not certain!! for the time it takes for the framework to come to consistent state. The disperse in the information can be ascribed to different factors in the examination. The dissipate could be ascribed to the adjustments in temperature, as the temperature fluctuated marginally through the term of the test †Good!. At what time did it settle?. The adjustment in temperature would cause an adjustment in the halfway weight of the CH3)2CO prompting further deviations. Furthermore, there was no proportion of wind stream past the cylinder. Changes in the wind stream could likewise have added to the dissipate as it could impact the convergence of the CH3)2CO at the highest point of the test tube (Good! ). The dissemination coefficient was likewise determined utilizing the Chapman Enskog condition, (Equation 4) and the Fuller, Schettler and Giddings technique. (Condition 5) A writing esteem was additionally found for CH3)2CO at K(check Perrys), which was remedied to our exploratory temperature utilizing the relationship (Equation 6) The qualities acquired with these strategies just as those from the trial information are introduced in Table 2. Table 1: Values of atomic diffusivity coefficients found. ** An excellent method to show this graphically in Excel is utilize a visual chart indicating the estimations of Dab as stature of a bar by technique utilized, and blunder bars to handily exhibit any cover of vulnerability, inconsistency, and so forth. Model: The Chapman Enskog technique is exact inside 8% and the Fuller Schettler and Giddings esteem has a lower precision than the Chapman Enskog (Geankoplis 425). The Chapman Enskog esteem is under 1% not the same as the trial esteem and the Fuller Schettler and Giddings esteem just about 6% unique. From this examination, it appears these conditions anticipated the test esteem well indeed. These determined qualities are about 20% lower than the writing esteem. This change may originate from the conflicting temperature in the room or from pressure variances in the room caused maybe by the beginning and halting of the HVAC frameworks. For the induction of Equation 1, a few suppositions are made. Starting with the general condition (Geankoplis 6.â 2-14): (Equation 7) One supposition that was that in light of the fact that the case inspected was a diffusing A (CH3)2CO) into non-diffusing B (air), the dispersion motion of air into the CH3)2CO (NB) was equivalent to zero. Another supposition made was that since the complete weight was low, the CH3)2CO gas diffusing into air was a perfect gas. This considered the term c to be supplanted with its opt imal gas identical, P/RT. Also, the air disregarding the test tube was expected to contain no water fume. A normal air speed that was uniform was disregarding the CH3)2CO containing test tube was additionally accepted. There are non-idealities that exist in the atomic dispersion of CH3)2CO into air. A portion of these non-idealities are rectified for in the diary from Lee and Wilke. CH3)2CO shows surface strain impacts which, rather than having an entirely flat fluid surface, give the fluid CH3)2CO a somewhat descending bended fluid level. On account of this bend, the real dissemination way length that the CH3)2CO voyages is littler than what the dispersion length would seem, by all accounts, to be founded on focus fluid level or determined fluid volume (Lee 2384). Alongside a non perfect fluid surface, the air disregarding the open finish of the cylinder may make some disturbance exist in the top bit of the cylinder. With its reality, the tempestuous region of the cylinder will make a length exist inside the cylinder where the convergence of CH3)2CO is zero. With the nearness of this CH3)2CO fume free

Vol I (to page 182) and II I lost some of the seco Essay Example For Students

Vol I (to page 182) and II I lost a portion of the seco Essay nd volume Sorry! Abenaquis, a Mahican country, 00; murder Mohawk boss, 000; English make a deal to avoid helping, 009; Abraham, or Schabash, a Mahican boss, made skipper by Mahicans 89; colleague at Gnadenhutten, 89 Abrahamsen, Issac, salvages an Indian kid, 000 Ackhough, sachem of Weckquaesgeeks, 09 Adair, James, hypothesis of, concerning beginning of American Indians, 00 Adogbegnewalquo, a Mohawk boss, address of, 000 Aepjin, boss sachem of Mahicans, 08; gathering to bargain of 0000, 008; totemic mark of, 009; committee fire at Schodac, 08; approved to treat for Esopus Indians, 000 Agassiz, hypothesis of, 00 Analysis of clans and chieftaincies, 00 Andastes, war with the Iroquois, 00 Andros, Governor, offers grounds to criminal Indians, 000; welcomes Pennacooks to settle at Schaticook, 00 Andriaensen, Maryn, in order at slaughter at Corlears Hook, 000 Ann Hoock, assumed name Wampage, a head of Siwanoys, 80 Ann Hutchinson, murder of, 000; girl of, came back from imprisonment, 008 Alb any, Fort Nassau raised at, 99; Fort Orange raised at, 99; Dutch make arrangement with Mahicans and Iroquois at, 00; gave up to the English, 008; English build up board fire at, 000 Algonquin language, 00 Algonquin countries, 00, 00 Allegewi, convention concerning, 00 Alliances, how framed, 00 Alliance, nature of, between the Dutch and the Iroquois, 000; of Dutch with Long Island chieftaincies, 000; of English with Iroquois and Mahicans, 008 Appamanskoch, sachem of Raritans, 90 Aquackanonks, area of, 90 Armies, how created, 00 Ashhurst, Sir John, purchases terrains of Waoranecks, 90 Assiapam, sachem of Matinecocks, 00 Assinapink river, 90 Atkarkarton, Kingston supposed, 000 Attention in affliction, 00 Beeren, or Mahican island, 80 Biographical Sketch of, Chambers, Captain Thomas, 008; Kryn, war-commander of Caghnawagas, 080 Bloom, Domine, depiction of Esopus slaughter, 000 Bouwensen, Thomas, cooked and eaten by Mohawks, 000 Caghnawaga, Mohawk town of, 00, 90; assaulted by Mahicans, 90; decimated by French, 90; Dutch embassadors visit, 000; changed over by Jesuits, 009 Nation, or Praying Indians, 009 Calmet, hypothesis of, 00 Canada, settlement of, initiated, 00 Canassatiego, an Iroquois emissary, discourse of, 09 Canopus, sachem of Nochpeems, 80 Captains, war boss so called,00 Captahem, sachem of Aquackanonks, 90 Carnarsees, area of, 00 Cartwright, Col.George, makes arrangement with Iroquois, 008 Castles, method of developing, 00 Catholic Priests, works of, 000, 008; law according to, 000 Cayugas, one of the Iroquois countries, 00; town of, 98 Chambers, Thomas, Capt. settles at Esopus, 000; historical sketch of, 000 Champlain, finds Lake Champlain, 00; guides the Hurons, 00; supports change of Indians, 000 Chastity of females, 00 Chesekock tract, 80 Chegonoe, sachem of Rockaways, 00 Chekatabut, a Massachusetts Mahican boss, 00 Child birth, 00 Claverack, town of, 00; rivulet, name of, 00 Clinton, DeWitt, hypothesis of, 00 Coginiquant, sachem of Nesaquakes, 00 C olden, Lieut.Gov., 00 Coleman, John, murdered by the Indians, 9 Communipau, native name of, 90 Conarhanded, sachem of Weckquaesgeeks, 09 Conflict with Indians, at Stony Point, 00; at Shorackappock 00, 00 Corchaugs, area of, 00 Corlears Hook, slaughter of Indians at, 000, 008 Couwenhoven, haggles with Esopus Indians, 000, 000 Cralo, Fort, at Greenbush, 009 Croton, traditionary sachem of Kitchawongs, 09 Croton, stream, native name of, 09 Dans-Kammer, demon love at, 09, 90; boundry line at the, 90; Couwenhoven at the, 000 Declarations of war, 00 Deeds, clarification of marks to, 90 DeHart, Balthazar, buys lands, 90 Denotas, or sacks for estimating corn, 00 DeVries, David Pietersen, manor of, on Staten Island, demolished, 000; situates among the Tappans, 90; tries to forestall slaughter of outlaw Indians, 000; estate of, saved by Indians, 009; arranges bargain of harmony, 009; ranch obliterated, 000 Dobbs Ferry, native name of, 08 Dongan, Gov., buys arrives on the Hudson, 90, 90; attemp ts to crush the activities of the French, 009; offers awards to the Iroquois, 009; offers to James II, to keep up collusion with Iroquois, 009; requests Catholic clerics, 009 Dress, of an Indian beauty, 00; Hudsons depiction of, 8; Verazzanos portrayal of, 09 Dwellings and method of development, 00 Du Bois, Mrs. Louis, caught by the Esopus Indians, 000; convention concerning, 000 Dutch, lack of bias of, in Indian wars, 00; settlement of, with Mahicans, and so forth 00; send envoys to the Mohawks, 000; answerable for the Manhattan wars, 009; liable for the Esopus wars, 000; give up territory to the English, 008 Eelkins, Jacob, detains head of the Sequins, 000 English catch of Fort Amsterdam, 008; bargain with the Iroquois, 00, 008; arrangement with Mahicans, 008, 000; laws directing intercourse with the Indians, 000; arrangement of Esopus, 000 Eskmoppas, sachem of Rockaways, 00 Esopus, deduction of term, 90; first settlement at, 000; pilgrims at, desert lands, 000, 000; first war at, 000; Stuyvesant requested to ensure, 000; structures wrecked at, 000; new town built up at, 008; second war of, 000, 000; torment of detainees at, 000; town held in attack, 000; harmony set up at, 000; recharging of threats at, 000; new town, obliterated, 000; depiction of slaughter at, 000; harmony restored at, 000; engraving on rocks at, 000 Esopus Indians, chieftaincies of, 90, 90; make harmony with the Senecas, 08; first war with the Dutch, 000; sachems request harmony, 008; Stuyvesant holds gathering with, 009; Indians slaughtered at, 000; reestablish threats, 000; arrangement of harmony with (0000), 000; Stuyvesant sends boss into bondage, 008; request restoration of bargain, 000; second war with, 000; arrangement of harmony with (0000), 000; settlement of, with the English, 000; amicable, evacuate to Ticonderoga, 90 Evert Pels, a Dutch detainee, gets away from torment by aboption, 000 Festivals, 00, 000 Five Nations, see Iroquois Fletcher, Gov., 000; hurries to the alleviati on of the Mohawks, 000 Food and method of readiness, 00 Fort Amsterdam held in attack by the Indians, 000, 000; gave up to the English, 008; Nassau, development of, 99; Orange, development of, 99 French, utilize Catholic preachers, 008; secure bargain of nonpartisanship, with Duke of York, 009; make detainees of Iroquois boss by foul play, 000; at war with the Senecas, 000; respect the requests of the Iroquois, 000; Indian war of 0089, 000 Frontenac, Count de, legislative leader of Canada, 000; plans campaign against the English, 000; attacks the Mohawk nation and devastates their mansions, 000; attacks the Onondaga nation, 000 Gallatin, Hon. Albert, proclamation of, concerning enslavement of Mahicans, 00 Gardiner, Lion and David, 00 Gil, sachem of Seatalcats, 00 Geographical terminology, Kaunaumeek, Massachusetts, 80; Mahicanituk, Hudsons waterway, 00; Shecomeco, Dutchess area, 80; Schodak, Columbia district, 08; Sunckhagag, Albany region, 80; Seepus, Esopus stream, 90; Tawalsontha , Normans kil, 99; Wawijchtanok, Columbia province, 80, 80; Wnahktakook, Westenhuck, 00, 80; Weckquaesgeek, Westchester region, 08; Wysquaqua, Wickers rivulet, 08; Willehoosa, Port Jervis, Orange region, 90 Goethals, sachem of Wappingers, 80; requests tranquility in the interest of Esopus Indians, 000 Goharius, sachem of Weckquaesgeeks, 09 Gouwarrowe, sachem of Matinecocks, 00; security for Hackinsacks and Tappans, 000 Government and laws, 09 Greenbush, Mohicans assault, 00 Haaskouaun, an Iroquois boss, 000 Hackinsacks, area of, 90; Van der Horst settles among, 000; a warrior of, looted, 000; objection of, in regards to presents, 000; youngsters commotion for war, 000; partake in war of 0000, 000; propose a trade of detainees, 000; haggle in the interest of Esopus Indians, 009 Haverstraw, area of, 90 Hiawatha, the account of, 00 Horikans, area of, 80 Housatonic stream, impartial boundry line, 00 Hudson, Henry, 0; clash of, with Indians at Stony Point, 00; clash of, with Indians at S horackappock, 00, 00; finds the Mahicanituk, 0; inebriates Indians at Castleton, 00; customs regarding his visit, 00; visits Indians at the Narrows, 8; visits Indians at Castleton, 9 Hutchinson, Ann, slaughtered by Weckquaesgeeks, 000; Daughter of, came back from imprisonment, 008 Indian Villages and areas, 00, 000; Albany region, 80, 80, 90; Columbia region, 80, 88; Dutchess region, 80; Greene region, 90; Long Island, 00; New Jersey, 89; Orange region, 90; Putnam region, 80; Rensselaer region, 80; Rockland region, 90; Staten Island, 90; Saratoga region, 09; Ulster County, 90; Westchester region, 00 Iroquois alliance, 00; region of, 00; convention regarding starting point, 00; inborn divisions, 00, 90; totemic tokens, 09; convention regarding association of alliance, 00; called the Five Nations, 00, 09; type of government, 09; association of alliance, 09; national gathering, 00; political matchless quality, 00; wars with the Hurons, 00; crushed by Champlain, 00; region attacked by t he French, 00; make bargain with the Dutch, 00; arrangements with the English, 00, 00, 008; French decide to demolish, 000; boss made detainee by foul play, 000; Mahicans in partnership with, 000, 000; catch Montreal, 000; compromise Quebec, 000; misfortunes supported in the war of 0089, 009; French ministers convert, 009 Jesuits, works of the French, 000, 008 Johnson, Sir William, 00 Joselyn, John, 00 Juet, Hudsons mate, visits Indians at Castleton, 9 Kalebackers, Indians having firearms, 000 Katskills, area of, 90; adoring men of, 9, 90 Katsban, a town of the Katskills, 000 Katonah, sachem of Siwanoys, 80 Kayingehaga, Mohawks supposed, 00 Kieft, chief, endeavors the assortment of tribute, 000; urges war measures, 000; announces an open quick, 009; requests help from England, 000; requests intercession of Mohawks and Mahicans, 000 King, Philip, winter quarters close to Albany, 00 Kingston, first settlement at, 000; a town palisaded at, 008 Kinte-Kaying, an Indian move, 08, 000 Ki

A Letter to a Friend

Dear Julia, I’m happy that you are staying with my family and me the following month. We haven’t seen each other for a long time and I missed you a ton so I trust that you won't alter your perspective on that trip. So I’m composing this letter to tell you something about my arrangements for your outing. During your stay here we might want to take you touring around Moscow, visiting better places of your advantage. It could be galleries, parks, presentations and whatever else you couldn't imagine anything better than to see and take pictures of.The following day after your appearance we are having my dad’s birthday, so you should consent to praise it with us. During the following couple of days it is up to you what we will do †we have a lot of exercises for you to browse: having a great time at the waterpark, skiing, ice-skating, shopping and a lot more other than them. We trust that you will cherish your stay here! Talking about what you have to carry with you, I need to caution you that winter in Russia is cold and windy.Try to take with you in any event a few sweaters and comfortable jeans or pants, just as socks and winter boots. Furthermore, don’t overlook your cap and a comfortable coat. Anyway, in the event that you overlook anything †don’t stress, you can purchase all that you need here. My father and I will meet you at the Domodedovo air terminal on Thursday, February’28 at 4 p. m. close to the principle entrance. On the off chance that you experience issues in discovering us †simply call me. I can’t stand by to see you! Yours,

How do you find the Vertical Asymptotes of a Function

In this article, well be talking about that dreaded A-word, asymptote. In my experience, students often get hung up on the term and may believe these kinds of problems are impossible. But with a solid understanding of the concepts, and a few algebraic techniques in your toolbox, it is not too difficult to locate the vertical asymptotes of a function. The Types of Asymptotes There are three types of asymptote: horiztonal, vertical, and oblique. This article focuses on the vertical asymptotes. Horiztonal asymptotes are discussed elsewhere, and oblique asymptotes are rare to see on the AP Exam (For more information about oblique, or slant asymptotes, see this article and this helpful video). Vertical Asymptotes A vertical asymptote (or VA for short) for a function is a vertical line x = k showing where a function f(x) becomes unbounded. In other words, the y values of the function get arbitrarily large in the positive sense (y→ ∞) or negative sense (y→ -∞) as x approaches k, either from the left or from the right. A vertical asymptote is like a â€Å"brick wall† that the function cannot cross. Imagine that you are flying in an airplane and up ahead you see a huge mountain. If you cant go left or right around the mountain what would you do? Youd probably fly upward to avoid hitting it. Now imagine that mountain is vertical and infinitely high. Then you might fly upwards forever to avoid hitting it, and still never get over the mountain! A function may have any number of vertical asymptotes, or none at all. Some functions even have infinitely many VAs. The graph shown below has vertical asymptotes at x = -3 and x = 1. Because the definition involves variables approaching fixed values, it should come as no shock that limits must be involved somehow. The precise definition for a vertical asymptote goes as follows. We say that x = k is a VA for a function f(x) if either the left-hand or right-hand limit to x = k is infinite: Finding Vertical Asymptotes There are two main ways to find vertical asymptotes for problems on the AP Calculus AB exam, graphically (from the graph itself) and analytically (from the equation for a function). Well talk about both. Determining Vertical Asymptotes from the Graph If a graph is given, then look for any breaks in the graph. If it appears that a branch of the function turns toward the vertical, then youre probably looking at a VA. It helps to sketch a vertical line at the x-value where you think the asymptote should be (see the graph shown above). Note, if part of the graph actually touches your vertical line, then that line is not an asymptote after all. Determining Vertical Asymptotes from the Equation If you need to find vertical asymptotes on the AP Exam, you will most likely not be given the graph. So youll need to know what to look for in the equation of the function itself. Ask yourself, where does this function have an infinite limit? Well see how this applies to two different kinds of functions, rational functions and trigonometric functions. Vertical Asymptotes in Rational Functions If your function is rational, that is, if f(x) has the form of a fraction, f(x) = p(x) / q(x), in which both p(x) and q(x) are polynomials, then we follow these two steps: 1. Factor both the numerator (top) and denominator (bottom). This is very important because if any factors end up canceling, then they would not contribute any vertical asymptotes. 2. Once your rational function is completely reduced, look at the factors in the denominator. If there is a factor involving (x – a), then x = a is a VA. If there is a factor involving (x + a), then x = a is a VA. Note how the sign seems to be opposite both times (just like solving a factored polynomial that has been set equal to zero). Practice Finding Vertical Asymptotes Lets see how our method works. Find the vertical asymptote(s) of each function. Solutions: (a) First factor and cancel. Since the factor x – 5 canceled, it does not contribute to the final answer. Only x + 5 is left on the bottom, which means that there is a single VA at x = -5. (b) This time there are no cancellations after factoring. We find two vertical asymptotes, x = 0 and x = -2. Vertical Asymptotes for Trigonometric Functions The method of factoring only applies to rational functions. However, many other types of functions have vertical asymptotes. Perhaps the most important examples are the trigonometric functions. Out of the six standard trig functions, four of them have vertical asymptotes: tan x, cot x, sec x, and csc x. In fact, each of these four functions have infinitely many of them! For example, f(x) = cot x has a VA at every integer multiple of Ï€. In other words, x = n Ï€ is a VA for every n = 0,  ±1,  ±2,  ±3, Using your Graphing Calculator More general functions may be harder to crack. If you are working on a section of the exam that allows a graphing calculator, then you may simply graph the function and try to spot the breaks in the graph at which the y-values become unbounded. Some calculators, like the TI-84, even have an option called detect asymptotes, which will automatically graph the VAs. Just be careful, though; if your viewing window is too small, then you may miss a VA. Conclusion Asymptotes are just certain lines that tell us about the behavior of functions. A vertical asymptote shows where the function has an infinite limit (unbounded y-values). It is important to be able to spot the VAs on a given graph as well as to find them analytically from the equation of the function. Your graphing calculator can also help out. With a little time and practice, these techniques can easily be mastered, and so vertical asymptotes dont have to be the â€Å"brick wall† that stops you from going far on the AP Calculus exam!

Homophones

   According to Dictionary.com, a homophone is defined as a word pronounced the same as another but differing in meaning, whether spelled the same way or not, as in heir and air. Today’s pair are ones I frequently find misapplied by my students as well as by some of my friends in their social media posts. Interestingly enough, these two words mean virtually the opposite of each other, and the only difference between them is a tiny space. What are they? A part and apart. A part means literally that. Something is a part of (belongs to) something else. A dish is a part of a set of dishes. An athlete is a part of a team. A chapter is a part of a book. Those parts belong to the larger group. They are part of the collective. In this case the â€Å"a† is functioning as an article that points to the noun â€Å"part.† By contrast, apart means virtually the opposite. If something is apart from something else, it is pulled away from the other and separate from it. It is frequently encountered in the idiom stands apart. For example, a truly exceptional novel stands apart from its competition. It is special, separate, alone. Apart indicates distance. Houses can be spaced several feet or several miles apart. People can also grow apart, which indicates an emotional separation. The word apart hails from Latin and means to the side, which explains its meaning. Things that are apart are set to the side. So as you can see, a part and apart are miles apart in meaning, but a little etymological search helps us understand why. As you teach your students, be sure they are able to understand and use these words correctly. Clearly, the difference between the two is vast. Using the word or phrase correctly may be just the thing that sets their writing apart from their peers’.    Jennifer Mauser  has always loved reading and writing and received a B.A. in English from the University of Kansas in 1991. Once she and her husband had children, they decided to homeschool, and she put all her training to use in the home. In addition to homeschooling her children, Jennifer teaches IEW classes out of her home, coaches budding writers via  email,  and tutors students who struggle with dyslexia.

Puberty in Alice and Wonderland - 2171 Words

One of the most prominent themes in children’s literature is maturation and grasping with adulthood. In keeping with this tradition, Lewis Carroll’s Alice’s Adventures in Wonderland presents a girl who transforms immensely from the bored little girl who can’t imagine reading a book without pictures to the mature adult described at the end of the novel. Throughout much of the novel, the reader witnesses Alice struggling with frequent, rapid changes in her body. While the repeated size changes in the book serve to illustrate the difficulties of children in grasping the changes of puberty, the changes in Alice’s personality and state of mind that come with each fluctuation in size hint at the greater rewards of knowledge and certainty that†¦show more content†¦So flustered by these constant changes, Alice’s memory and knowledge have suffered, as she is unable to recall basic facts. This, accompanied by the realization that her voice has become hoarse and strange, once again moves â€Å"poor Alice† to tears. Finding both her body and mind to be completely altered, Alice hints towards not liking who she has become, resolving to stay in Wonderland and only come out if she is somebody else. Just as soon as this stream of thoughts leaves her, though, Alice realizes that she has shrunk once again, and rather than being comforted, Alice is â€Å"frightened at the sudden change† (29), saying that she is now â€Å"worse than ever† and that she â€Å"never was so small as this before.† She finds herself confronted by a pool of tears that had once seemed so inconsequential, frustrated once again by her uncontrollable emotions: â€Å"I wish I hadn’t cried so much!† (30). Once again, she realizes somewhat bitterly that â€Å"everything is queer to-day.† Alice’s size continues to come into play through her interactions with the mouse. Not used to seeing things from small eyes, Alice’s etiquette is brought into question as she offends the mouse with her talk of cats (31). Despite being the same size as the normally-small animals she now interacts with, Alice is viewed as foolish for not utilizing the same logic as her counterparts. In essence, while she is physically small, her mind has not adapted to this new size and she does not fit in amongShow MoreRelatedLewis Carroll s Hidden Presence Of Puberty1540 Words   |  7 PagesLewis Carroll’s Hidden Presence of Puberty The transformations that Alice experiences throughout Lewis Carroll’s, Alice’s Adventures in Wonderland, display the metaphorical change the protagonist goes through from the stage of childhood to adulthood and the continued struggle to understand her identity. These changes are experienced after Alice follows a white rabbit down his hole and into an incredible world known as Wonderland. 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