Wednesday, March 18, 2020
Comparison Between the Jungle and Fast Food Nation The Jungle is a book written by a journalist who aimed at bringing out the kind of life experienced by those who had migrated to the United States. Although, the novel was then interpreted and used to expose the situation of the health that was being experienced in the industry (Sullivan 1996).Advertising We will write a custom essay sample on Comparison Between the Jungle and Fast Food Nation specifically for you for only $16.05 $11/page Learn More The book elaborates on the harsh conditions of people who migrated to the United States. The Jungle talks more of the problems that people both employed and unemployed faced out off the high level of corruption in by the leaders in the state. The writer of the book came up with suggestion on what ought to be done to eliminate the issue of minimal salaries among the citizens, hence, improve the living standards of the people in the United States. The writer of the novel brings out the picture of America as a natio n that is experiencing various food problems as a result of fast food cuffs, hence, threatening the lives of many citizens in the region (Cecilia 2004). Despite of their advantage of having numerous food supplies in the region, people in the state continue to suffer instead of enjoying the food resources. The writer feels that this is a problem that people created for themselves due to their unhealthy practices that lead to death as well as severe illnesses. There were numerous similarities in the United States on the way people were treated by the government as well as their day to day life. Both the state faced almost common problems as well as close kind of hardship despite of the differences in the kind of resources that were found in the two regions (Jacobs 2010). The writers did not only focus on the problems that the citizens were experiencing but also viewed at the possible solutions that could employed to diminish the problems in the two regions. In the writing, the writer elaborated on the problems that come hand in hand as a result of poor planning by the government, corruption as well as minimal wages among the employed citizen, thus, making the lives of the citizen difficult (Sullivan 1996). With the increased level of poverty in the United States, many people and mostly the immigrants face the greatest challenge in trying to meet their daily needs. However, there are some people who are concerned with the low living standard of the members of the states, for instance, the writer of fast food nations who went to an extra mile by organizing for supervision on the meat packing facilities but his effort fails to work out as people allocated to perform the duties were so reluctant to notice the poor hygiene in the plants. This does not help much in improving the already low level of living in the region reason being that the government was not concerned with its people (Tichi 2004). This could have been enhanced by the fact that the government in the region was also corrupt making the situation even worse.Advertising Looking for essay on comparative literature? Let's see if we can help you! Get your first paper with 15% OFF Learn More In both states, the major cause of health problems came as a result of cattle rearing by the citizen. However, the problem was not alls about the rearing of the animals but the processes that followed after the animals were ready for consumption. For instance, in the United States, the writer discovers that the level of cleanliness in the meat plant was low as those concerned could only clean them only when there was going to be an inspection. This, therefore, portrays how much the citizen consumed unhealthy meat from the company. Nevertheless, the efforts of those who tried to expose the kind of unhealthy activities taking place in the regions were cut short by those who wanted to hide the true face of the governance in the region. This was done by reporters who said that t he information given by the writer were just false allegation and that they were misleading since they were baseless. As well, in America, the state of the unhealthy conditions was also experience by the citizens. As in the United States, the hazardous practices were also taking place in the meat companies. The processing of meat was found to be unhygienic for human consumption but the people in the states consumed it without their knowledge. In the United States, the conditions were worse as compared to those in America (Jacobs 2010). The reason behind this was because the carcasses from pigs, chicken as well as from horse were incorporated into cattle feed. Thus, these kinds of practices were the major cause of diseases like bovine spongiform encephalopathy commonly known by the people in the regions as Mad Cow Disease. The consumption of the unhygienic meat not only caused diseases in those who consumed it but it also led to introduction of detrimental micro organisms like E. col i in the meat, hence, this could eventually lead to sickness among the citizens. In the same state, there were other diseases like obesity which was also a threat to the health as well as to the lives of the citizens. Teen labor was also a challenge to the immigrants of the United States as every member of the family was expected to work to aid in supporting the family. This was after the family became a victim of the con men as well as schemes that exhausted their saving. The situation was even more terrible as it diminished all what the citizens had saved to a point of not being in a position to pay for the cheap accommodation that could be easily acquired in the low standard houses known as slums (Sullivan 1996). This lead to eviction of the immigrants leaving the members frustrated.Advertising We will write a custom essay sample on Comparison Between the Jungle and Fast Food Nation specifically for you for only $16.05 $11/page Learn More The different o ccupations as well as lifestyle of the family change their lives from that juncture. This was greatly contributed by the harsh condition they faced at workplace. The situation at work place was made worse by the hardships they faced such as accidents that occurred at workplace as well as deaths of the members of their family that with proper conditions as well as planning could have been avoided. Sexual harassment is also a problem in the two states especially in the United States where the level of security was poor as well as the migrants lacking a government that would cater about their wellbeing (Tichi 2004). Also, the kind of government that was found in the region was so corrupt, hence, no one could promote the security in the region. Those who could not speak English in both novels, that is the immigrants, were discriminated as they could not acquire good jobs as well as they were not counted as important to the society and hence, they could not associate with the others in t he community. This led them be with no choice of the kind of jobs to do, hence, ending up doing the hard jobs which could only afford them minimal salaries thus making their living harder. Lack of well paying jobs, led to poverty that was elaborated by how people died of simple conditions that could have been prevented if the people earned enough. The writer narrates of how a character by the name Ona died when giving birth due to lack of capital to pay for medical attention. In addition to this their younger son drowned in the streets that were muddy. This elaborates on the low living states the immigrants experienced yet no one was concerned to improve their way of life (Jacobs 2010). The main characters of the novels are among those who went through these hardships as well as the disappointments in their lives. They were so much into promoting the way of lives of those who experienced similar difficulties. However, their efforts were cut short by those who dominated the region su ch that ay time they came up with a way of improving the state nobody could side with them making the situation hard for them. In conclusion, the novels bring out the kind of life that people face especially in a land that is not theirs. The people of the United States as well as America represent a vast number of people who live below the poverty levels. The kind of living they experience as well as the neglect from the government. The novels can, therefore, be used to bring out to the issues that the government ought to look into to improve peoples way of living. Jacobs, Jane. Introduction to The Jungle. NY: McGraw-Hill. 2010.Advertising Looking for essay on comparative literature? Let's see if we can help you! Get your first paper with 15% OFF Learn More Sullivan, Mark. The Condemned-Meat Industry. NY: Our Times, 1996. Tichi, Cecilia. From the Jungle to Fast Food Nation. Philadelphia: University of Pennsylvania Press. 2004.
Monday, March 2, 2020
Presidents Elected Without Winning the Popular Vote Five U.S. presidents have taken office without winning the popular vote. In other words, they did not receive a plurality regarding the popular vote. They were elected, instead, by the Electoral College- or in the case of John Quincy Adams, by the House of Representatives after a tie in the electoral votes. They were: Donald J. Trump, who lost by 2.9 million votes to Hillary Clinton in theÃ 2016 election.George W. Bush,Ã who lost by 543,816 votes to Al Gore in theÃ 2000 election.Benjamin Harrison,Ã who lost by 95,713 votes toÃ Grover ClevelandÃ in 1888.Rutherford B. Hayes, who lost by 264,292 votes to Samuel J. Tilden in 1876.John Quincy Adams,Ã who lost by 44,804 votes toÃ Andrew JacksonÃ in 1824. Popular vs. Electoral Votes Presidential elections in the United States are not popular vote contests. The writers of the Constitution configured the process so that only the members of the House of Representatives would be elected by popular vote. The Senators were to be selected by state legislatures, and the president would be selected by the Electoral College. The Seventeenth Amendment to the Constitution was ratified in 1913, making the election of senators to occur through popular vote. However, presidential elections still operate under the electoral system. The Electoral College is made up of representatives who are generally selected by the political parties at their state conventions. Most states except Nebraska and Maine follow a winner-take-all principle of electoral votes, meaning that whichever partys candidate wins a states popular vote for the presidency will win all of that states electoral votes. The minimum electoral votes a state can have is three, the sum of a states senators plus representatives: California has the most, with 55. The Twenty-Third Amendment gave the District of Columbia three electoral votes; it has neither senators nor representatives in Congress. Since states vary in population and many popular votes for different candidates can be quite close within an individual state, it makes sense that a candidate might win the popular vote across the entire United States but not win in the Electoral College. As a specific example, lets say the Electoral College is only made up of two states: Texas and Florida. Texas with its 38 votes goes entirely to a Republican candidate but the popular vote was very close, and the Democratic candidate was behind by a very small margin of only 10,000 votes. In the same year, Florida with its 29 votes goes entirely to the Democratic candidate, yet the margin for the Democratic win was much larger with the popular vote win by over 1 million votes This could result in a Republican win at the Electoral College even though when the votes between the two states are counted together, the Democrats won the popular vote. Calls for Reform In general, it is very rare for a president to win the popular vote yet lose the election. Although this has only happened five times in U.S. History, it has occurred twice in the current century. In 2016, Donald Trump lost the popular election by nearly 3 million votes, about 2% of the total votes cast. Discussion for reform of the Electoral College dates to the first years of the Constitution and has frequently been the topic of scholarly discussion. Defenders of the Electoral College argue that its violations of majority rule are an example of constitutional provisions that require super-majorities to take action. The Electoral College allows a minority to take an action- that is to select a president- and it is the only device of its kind in the Constitution. The primary way to change it is to amend the Constitution. Since the way states count votes can affect who wins and who loses, electoral reform is by its nature political: if a party is in power, the method used to get it there is not likely to be a target of change. Despite the current climate in that Democrats support change while Republicans do not, scholars believe that the situation is only temporarily one that swings towards one party or another: One proposal called the National Popular Vote Interstate Compact (NPVIC) is a sub-national reform to the Electoral College in which states agree to commit their electoral votes, as a unit, to the winner of the aggregate, national vote.Ã Sixteen states have signed on to date, some of which are Republican-controlled. One major purpose of the Electoral College was to balance the power of the electorate so that votes in states with small populations would not (always) be overpowered by larger-populated states. Bipartisan action is required to make its reformation possible. Sources and Further Reading Bugh, Gary, ed. Electoral College Reform: Challenges and Possibilities. London: Routledge, 2010.Burin, Eric, ed. Picking the President: Understanding the Electoral College. University of North Dakota Digital Press, 2018.Colomer, Josep M. The Strategy and History of Electoral System Choice. The Handbook of Electoral System Choice. Ed. Colomer, Josep M. London: Palgrave Macmillan UK, 2004. 3-78.Goldstein, Joshua H., and David A. Walker. The 2016 Presidential Election Popular-Electoral Vote Difference. Journal of Applied Business and Economics 19.9 (2017).Shaw, Daron R. The Methods Behind the Madness: Presidential Electoral College Strategies, 1988Ã¢â¬â1996. The Journal of Politics 61.4 (1999): 893-913.Virgin, Sheahan G. Competing Loyalties in Electoral Reform: An Analysis of the U.S. Electoral College. Electoral Studies 49 (2017): 38Ã¢â¬â48.
Friday, February 14, 2020
Introduction to decision making slp - Assignment Example My direct manager is a low level manager who is in charge of the employees and manages them. He makes operational decisions in the company which are small and deal with the day to day running of the low level affairs in the company. His main role is to report the needs and requirements of the employees to his manager. He is also in charge of the office supplies and his duty is to ensure that the supplies are enough in the office, well distributed, and that each employee maintains (Nutt & Wilson, 2010). His direct manager makes tactical decision and acts as a bridge between my manager and the senior level manager. He handles policy proposals on new products, new technology or new marketing tools. He then hands over the proposals and products to the senior level managers who then make the implementation. He sometimes acts as the human resource manager handling the problems of employees that are minute and do not require the attention of the senior managers. At the very top are senior managers who make strategic decisions which change the direction of affairs within the company and outside. Their decisions are influenced by both internal and external environment. They also work hand in hand with tactical manager to implement the product and project proposals, test new theories and technologies as well as liaise with other companies and the media in case announcements need to be made. The strategic managers are the face of the company as they are the known parties. 2. Give some specific examples of both programmed and non-programmed decisions that you have observed. Give at least one example of each type of decision, and explain why you think this decision falls into this category with references to the background materials. Programmed decisions are those that have been made severally until there is a clear guideline on what steps to take in order to make the decisions. These decisions can be passed down from one
Saturday, February 1, 2020
Reading Assessment - Assignment Example The current immigrants are different from the earlier immigrants in the sense that they form the source of terrorists and other ill intention arrivals with an objective of destroying the United States. John Amagoalik quote emphasizes relevance of Nunavut to original inhabitants of Australia and Greenland, Canada. Amagoalik argues that preserving Nunavut would assist Inuit and Aboriginals in protecting the erosion of their culture and history. Therefore, Nunavut existence remains symbolic of Aboriginals and Inuit historical reality. Gurston Dacks quote contradicts ideas advanced by Amagoalik. He argues that Nunavut have no particular value. Dacks, further states that Nunavut existence would only revoke postwar ideals rather than infer benefits to Austrian or Canadian community (Smith, 1971). Clash of civilization theory proposes that post-Cold War would result from peoplesÃ¢â¬â¢ religious and cultural identities. Media arguments and research findings support clash of civilizations theory. Media reports on continued disagreements between Islam community and the Western countries relates to the theory. Contemporary media news reports that Islamic communities have decided to form militant movements that reject Western values and way of living. Unfolding religious and cultural events prompts for analysis to support clash of civilizations theory. Cultural and religious differences between Islam and Christian community require insightful analysis and professional views to ascertain clash of civilizations
Friday, January 24, 2020
The topic of abortion is one of the most controversial issues in todayÃ¢â¬â¢s society. Thousands of abortions take place every single day, and yet public opinion remains at a standstill as to whether or not abortion is ethical or not. According to a poll in 2013, fifty-four percent of the American public believes that the practice of abortion should be legal in all or most cases (Ã¢â¬Å"Public Opinion on AbortionÃ¢â¬ ) Abortion has been defined as Ã¢â¬Å"the act of removing a human embryo or fetus from the uterus of a pregnant woman prior to the completion of the full term of pregnancyÃ¢â¬ (Rich, Wagner, and Geraldine). There are very strong opinions for and against this issue, but no one can deny the vast gray area of abortion. A personÃ¢â¬â¢s stance on the situation is often determined by how he views the fetus: a part of the motherÃ¢â¬â¢s body or as a human being. Abortion continues to be a moral issue because people have various views on the rights of the fetus and mo ther, the circumstances of the pregnancy, and their own religious convictions concerning the issue. In the most recent study, 1.21 million abortions took place in the United States in the year 2008 (Ã¢â¬Å"About AbortionÃ¢â¬ ). According to a study performed by the Guttmacher Institue, twenty-one percent of all pregnancies in the United States end in abortions (Ã¢â¬Å"Induced AbortionÃ¢â¬ ). Fifty percent of pregnancies in the United States are unplanned (Ã¢â¬Å"Induced AbortionÃ¢â¬ ). Of those unplanned pregnancies, four out of ten result in abortions (Ã¢â¬Å"Induced AbortionÃ¢â¬ ). When analyzing the statistics of women who receive abortions, one must become aware that outside circumstances might contribute to a womanÃ¢â¬â¢s decision to have an abortion. For instance, women who are one hundred percent below the federal poverty level ac... ...rtion Federation: History of Abortion. National Abortion Federation, n.d. Web. 17 Mar. 2014. "Public Opinion on Abortion." Public Opinion on Abortion. Pew Research Center, July 2013. Web. 17 Mar. 2014. Rich, Alex K. Wagner, Geraldine. "Abortion: An Overview." Points Of View: Abortion (2013): 1. Points of View Reference Center. Web. 16 Mar. 2014. Rubio, Marco. "Why Abortion is Bad for America." Human Life Review Winter 2012 2012: 19-26. ProQuest Health Management. Web. Voegeli Jr., William J. Ã¢â¬Å"A Critique of the Pro-Choice ArgumentÃ¢â¬ Review of Politics Vol. 43, no. 4 (Oct., 1981) , Pp. 560-571 Published by: Cambridge University Press for the University of Notre Dame Du Lac on Behalf of Review of Politics Stable Print. "World Abortion Laws 2009 Fact Sheet." Center for Reproductive Rights. Center for Reproductive Rights, 2013. Web. 18 Mar. 2014.
Thursday, January 16, 2020
http://sahatmozac. blogspot. com ADDITIONAL MATHEMATICS FORM 5 MODULE 4 INTEGRATION http://mathsmozac. blogspot. com http://sahatmozac. blogspot. com CHAPTER 3 : INTEGRATION Content Concept Map page 2 3Ã¢â¬â4 5 6 7 8Ã¢â¬â9 10 Ã¢â¬â 11 12 4. 1 Integration of Algebraic Functions Exercise A 4. 2 The Equation of a Curve from Functions of Gradients. Exercise B SPM Question Assessment Answer http://mathsmozac. blogspot. com 1 http://sahatmozac. blogspot. com Indefinite Integral a) o o a x n a dx = ax + c. xn+ 1 + c. n+ 1 b) x n dx = c ) o d x = a o x n d x = a n x + n + 1 1 + c . Integration of Algebraic Functions ) ) The [f (x) Ã ± g(x) ]dx = o f (x) dx Ã ± d o Equation of a Curve from Functions of Gradients o g(x)dx y = y = o f Ã¢â¬Ë( x ) d x c, f (x) + http://mathsmozac. blogspot. com 2 http://sahatmozac. blogspot. com INTEGRATION 1. Integration is the reverse process of differentiation. dy 2. If y is a function of x and = f Ã¢â¬Ë( x) then o f Ã¢â¬Ë( x)dx = y + c, c = c onstant. dx If dy = f ( x ), then dx o f ( x)dx = y 4. 1. Integration of Algebraic Functions Indefinite Integral a) b) o o a dx = ax + c. n a and c are constants xn+ 1 x dx = + c. n+ 1 n c is constant, n is an integer and n ? Ã¢â¬â c) o ax dx = a o ax n + 1 x dx = + c. n+ 1 n and c are constants n is an d) o [f ( x ) Ã ± g ( x ) ]dx = o f ( x) dx Ã ± o g ( x)dx http://mathsmozac. blogspot. com 3 http://sahatmozac. blogspot. com Find the indefinite integral for each of the following. a ) ? 5dx b) ? x 3 dx c) ? 2 x dx 5 d) ? ( x ? 3x 2 )dx Always remember to include Ã¢â¬Ë+cÃ¢â¬â¢ in your answers of indefinite integrals. Solution : a) ? 5dx ? 5x ? c b) 3 ? x dx ? x3? 1 ? c 3 ? 1 x4 = ? c 4 2 c) 5 ? 2 x dx ? 2 x5? 1 ? c 5 ? 1 2 x6 = ? c 6 1 = x6 ? c 3 d) ? ( x ? 3x )dx ? ? xdx ? ? 3x 2 dx = x 2 3 x3 ? ?c 2 3 x2 = ? x3 ? c 2 Find the indefinite integral for each of the following. a) ? ? x ? 3x ? dx 2 x 4 b) ?x ? x 2 4 ? ? ? 3 ? ? dx x ? ? a) Solution : x ? 3Ãâ"2 ? ? x 4 ?dx ? ? x 3Ãâ"2 ? ? ? x4 ? x4 ? dx ? ? b) 2 4? ? ? 2 4? ? 3 ? 4 ? dx = ? ? 3x ? 2 ? dx x ? x ? ? ? = ? 3Ãâ"2 ? 4 x ? 2 dx ? x ? 1 ? 3x 3 = ? 4? c 3 ? ?1 ? 4 = x3 ? ? c x ? ? x? 3 ? 3x? 2 dx ? x? 1 ? x? 2 = ? 3? c ? 2 ? ?1 ? 1 3 =? 2 ? ?c 2x x ? ? ? ? http://mathsmozac. blogspot. com 4 http://sahatmozac. blogspot. com 1. Find ? ? 3x 2 ? 4 x ? 10 dx. ? [3m] 2. Find ? ? x 2 ? 1 ? 2 x ? 3 ? dx. ? [3m] 1? ? 3. Find ? ? 2 x ? ? dx. x? ? 2 [3m] 4. Find ? ? 2x ? ? 3 ?x? 3 ? ? 2 ? dx. 4 x ? [3m] 6x ? 5 5. Integrate with respect to x. x3 [3m] 6. Find ? ?x 5 ? 4Ãâ"2 2x 4 ? dx [3m] 3 ? ? 7. Find ? x ? 6 ? 6 ? x . x ? ? 2 [3m] 8. Integrate x 2 ? 3x ? 2 with respect to x. x ? 1 [3m] http://mathsmozac. blogspot. com 5 http://sahatmozac. blogspot. com The Equation of a Curve from Functions of Gradients dy ? f Ã¢â¬Ë( x), then the equation of the curve is dx If the gradient function of the curve is y ? ? f Ã¢â¬Ë( x ) dx c is constant. y ? f ( x) ? c, Find the equation of the curve that has the gradient function 3x ? 2 and passes through the point (2, ? 3). Solution The gradient function is 3x ? 2. dy ? 3x ? 2 dx y ? ? (3x ? 2)dx y? 3Ãâ"2 ? 2x ? c 2 The curve passes through the point (2, ? 3). Thus, x = 2, y = ? 3. 3(2) 2 ? 3 ? ? 2x ? c 2 ? 3 ? 6 ? 4 ? c c ? 5 Hence, the equation of curve is y? 3x 2 ? 2x ? 5 2 http://mathsmozac. blogspot. com 6 http://sahatmozac. blogspot. com 1. Given that dy ? 6 x ? 2 , express y in terms of x if y = 9 when x = 2. dx 2. Given the gradient function of a curve is 4x ? 1. Find the equation of the curve if it passes through the point (? 1, 6). 3. The gradient function of a curve is given by dy 48 ? kx ? 3 , where k is a constant. dx x Given that the tangent to the curve at the point (-2, 14) is parallel to the x-axis, find the equation of the curve. http://mathsmozac. blogspot. com 7 http://sahatmozac. blogspot. com SPM 2003- Paper 2 :Question 3 (a) Given that y ? 2 x ? 2 and y = 6 when x = ? 1, find y in terms of x. dx [3 marks] SPM 2004- Paper 2 :Question 5(a) The gradient function of a curve which passes through A(1, ? 12) is 3 x 2 ? 6 x. Find the equation of the curve. [3 marks] http://mathsmozac. blogspot. com 8 http://sahatmozac. blogspot. com SPM 2005- Paper 2 :Question 2 A curve has a gradient function px 2 ? 4 x , where p is a constant. The tangent to the curve at the point (1, 3) is parallel to the straight line y + x ? 5 =0. Find (a) the value of p, [3 marks] (b) the equation of the curve. [3 marks] http://mathsmozac. blogspot. com 9 http://sahatmozac. blogspot. com 1.Find the indefinite integral for each of the following. (a) ? ? 4x 3 ? 3 x ? 2 dx ? (b) 3? x ? ? 2 2 ? 6? ? dx x3 ? 1 ? 2 ( c) (c) ? ? x 5 + 5 6x ? 3 ? ? dx ? ? x2 ? 3 (d) ? ? ? x2 ? ? ? 2 ? ? dx ? ? 2. If dy ? 4 x3 ? 4 x, and y = 0 when x = 2, find y in terms of x. dx http://mathsmozac. blogspot. com 10 http://sahatmozac. blogspot. com 3. If dp v3 ? 2v ? , and p = 0 when v = 0, find the value of p when v = 1. dv 2 4. Find the equation of th e curve with gradient 2 x 2 ? 3 x ? 1, which passes through the origin. 5. d2y dy dy Given that ? 4 x, and that ? 0, y = 2 when x = 0. Find and y in terms 2 dx dx dx of x. http://mathsmozac. blogspot. om 11 http://sahatmozac. blogspot. com EXERCISE A 1) 2) 3) 4) 5) 6) 7) 8) x ? 2 x ? 10 x ? c 3 2 SPM QUESTIONS 1) y ? x2 ? 2x ? 7 2) y ? x3 ? 3 x 2 ? 10 3) p ? 3, y ? x3 ? 2 x 2 ? 4 x4 ? x3 ? 3x ? c 2 4 3 1 x ? 4x ? ? c 3 x 4 2 x x 1 ? ? 3 ? 2x ? c 2 2 x 6 5 ? ? 2 x 2x 2 x 2 ? ?c 4 x 1 2 x3 ? 3 ? c x 2 x ? 2x ? c 2 ASSESSMENT 1) (a ) x 4 ? 3 2 x ? 2x ? c 2 2 3 (b) 3x ? ? 2 ? c x x 6 x 1 (c ) ? ?c 9 24 x 4 x3 9 (d ) ? 6x ? ? c 3 x y ? x4 ? 2 x2 ? 8 p? 7 8 2 3 3 2 x ? x ? x 3 2 2 3 x ? 2 3 EXERCISE B 1) y ? 3x 2 ? 2 x ? 1 3 x 2 24 ? 2 ? 2 2 x 2) 2) y ? 2 x 2 ? x ? 3 3) y ? 3) 4) y? 5) y? http://mathsmozac. blogspot. com 12 http://sahatmozac. logspot. com ADDITIONAL MATHEMATICS FORM 5 MODULE 5 INTEGRATION http://mathsmozac. blogspot. com 13 http://sahatmozac. blogspot. com CONTENT CONCEPT MAP INTEGRATION BY SUBSTITUTION DEFINITE INTEGRALS EXERCISE A EXERCISE B ASSESSMENT SPM QUESTIOS ANSWERS 2 3 5 6 7 8 9 10 http://mathsmozac. blogspot. com 14 http://sahatmozac. blogspot. com CONCEPT MAP INTEGRATION BY SUBSTITUTION un ? ax ? b ? dx ? ? du ? a n DEFINITE INTEGRALS If b d g(x) ? f (x) then dx b where u = ax + b, a and b are constants, n is an integer and n ? -1 OR (a) ? f (x)dx g(x)? ? g(b) ? g(a) a a (b) ? f (x)dx f (x)dx a a b b (c) ? f (x)dx f (x)dx ? ? f (x)dx a b a b c ? ax ? b ? ? ? ax ? b ? dx ? a ? n ? 1? n n ? 1 ? c, where a, b, and c are constants, n is integer and n ? -1 http://mathsmozac. blogspot. com 15 http://sahatmozac. blogspot. com INTEGRATION BY SUBSTITUTION un ? ? ax ? b ? dx ? ? a du n where u = ax + b, a and b are constants, n is an integer and n ? -1 O R ? ax ? b ? ? ? ax ? b ? dx ? a ? n ? 1? n n ? 1 ? c, where a, b, and c are constants, n is integer and n ? -1 Find the indefinite integral for each of the following. (a) ? ? 2 x ? 1? dx 3 (b) ? 4(3 x ? 5)7 dx 2 (c) ? dx (5 x ? 3)3 SOLUTION (a) ? ? 2 x ? 1? dx 3 Let u = 2x +1 du du ? 2 ? dx ? dx 2 3 3 ? du ? ? (2 x ? 1) dx ? ? u ? ? ? ? u3 = ? du 2 u 3 ? 1 = ? c 2(3 ? 1) u4 +c 8 (2 x ? 1) = +c 8 = Substitute 2x+1 and substitute dx with du dx = 2 OR (2 x ? 1) 4 ? c ? (2 x ? 1) dx ? 2(4) 3 = ? 2 x ? 1? 8 4 ?c Substitute u = 2x +1 http://mathsmozac. blogspot. com 16 http://sahatmozac. blogspot. com (b) ? 4(3 x ? 5) dx 7 (c) Let u ? 3 x ? 5 du du ? 3 ? dx ? dx 3 7 4u 7 du ? 4(3 x ? 5) dx ? ? 3 4u 8 = ? c 3(8) u8 ? c 6 (3u ? 5)8 = ? c 6 = 2 dx ? ? 2(5 x ? 3) ? 3 dx (5 x ? 3)3 Let u ? 5 x ? 3 du du ? 5 ? dx ? dx 5 ? 3 2u ? 3 du ? 2(5 x ? 3) dx ? ? 5 2u ? 3 = ? c 5(? 2) ? OR 4(3 x ? 5)8 ? c ? 4(3 x ? 5) dx ? 3(8) 7 u ? 2 ? c ? 5 1 = ? 2 5u 1 =? ?c 5(5 x ? 3)2 = = (3x ? 5)8 ? 6 DEFINITE INTEGRALS If d g ( x) ? f ( x) then dx b (a) (b) ? b a b f ( x)dx ? ? g ( x) ? ? g (b) ? g (a) a ? (c ) ? a b f ( x)dx ? ? ? f ( x)dx a b a f ( x)dx ? ? f ( x)dx ? ? f ( x)dx b a c c http://maths mozac. blogspot. com 17 http://sahatmozac. blogspot. com Evaluate each of the following ( x ? 3)( x ? 3) (a) ? 12 dx x4 1 1 (b) ? 0 dx (2 x ? 1) 2 SOLUTION (a) x2 ? 9 2 ( x ? 3)( x ? 3) ? c ? ?12 4 dx ? 1 x4 x 2 9 ? 2? x = ? 1 ? 4 ? 4 ? dx x ? ?x = ? 12 ( x ? 2 ? 9 x ? 4 )dx ? x ? 1 ? x ? 3 ? ? =? ? 9? ? ? 3 ? ?1 ? ?1 2 2 (b) ?0 1 1 1 dx ? ?0 (2 x ? 1)? 2 dx 2 (2 x ? 1) 1 = ? 0 (2 x ? 1) ? 2 dx ? (2 x ? 1) ? 1 ? =? ? ? ?1(2) ? 0 ? 1 = ? ? 2(2 x ? 1) ? 0 =? ? ? 1 1 ? 2[2(1) ? 1] ? 2[2(0) ? 1] ? 1 1 ? 1 3? = ? 3 ? ? x x ? 1 ? 1 3 ? ? 1 3? = ? 3 ? ? 3 ? ? 2 2 ? ? 1 1 ? 1 3 = ? ? ? (? 1 ? 3) 2 8 1 =? ?2 8 1 =? 2 8 1 ? 1? = ? ? 6 ? 2? 1 = 3 http://mathsmozac. blogspot. com 18 Distributed:18. 1. 09 Return:20. 1. 09 INTEGRATE THE FOLLOWING USING SUBSTITUTION METHOD. (1) ? ( x ? 1)3dx (2) ? ?4 ? 3 x ? 5 ? dx ? 5 (3) ? 1 ? 5 x ? 3? dx 4 1 ? ? (4) ? ? 5 ? x ? dx 2 ? ? ?3 1 ? ? (5) ? 5 ? 4 ? y ? dy 2 ? ? 4 3? 2 ? (6) ? ? 5 ? u ? du 2? 3 ? 5 19 http://sahatmozac. blogspot. com EXERCISE B 8 1. Evaluate ? 3 ( x3 ? 4)dx Answer : 1023. 75 2. Evaluate Answer: 3 ? ?3 1 2 x( x ? x ? 5)dx 8 83 96 ?2 ? 3. Integrate ? x ? 5 ? with respect to x ? 3 ? 4 4. Evaluate ? 1 3 1 ? ? ? 2 ? 3x ? 4 ? dx ? 1 x ? ? 1 Answer: 3 ? 2 ? ? x ? 5? ? c 10 ? 3 ? 5 Answer : 3 5. Evaluate ? 3 1 ? 2 x ? 1 2 x ? 1? dx 4 x2 6. Given that of 2 5 ? 5 2 f ( x)dx ? 10 , find the value 5 Answer: 1 6 ? ? 1 ? 2 f ( x)? dx Answer :17 http://mathsmozac. blogspot. com 20 http://sahatmozac. blogspot. com ASSESSMENT ?6 and 2. (a) ? 5(2 ? 3v) dv 4 (b) ? dx 5 3 ? 1 ? 5 x ? 1. Given that ? 2 2 1 f ( x)dx ? 3 ? 2 3 f ( x)dx ? ?7 . Find (a) the value of k if (b) ? ? kx ? f ( x)? dx ? 8 1 ? ? 5 f ( x) ? 1? dx 3 1 Answer : (a) k = (b) 48 22 3 3.Show that d ? x 2 ? 2 x 2 ? 6 x 4. . ? dx ? 3 ? 2 x ? ? 3 ? 2 x ? 2 4 Given that ? 4 0 f ( x)dx ? 3 and Hence, find the value of Answer : 1 10 ? ? 3 ? 2x ? 0 1 x ? x ? 3? ? 0 g ( x)dx ? 5 . Find 4 0 2 dx . ? f ( x)dx ? ? g ( x)dx (b) ? ?3 f ( x) ? g ( x)? dx (a) 0 4 0 4 Answer: (a) Ã¢â¬â 15 (b) 4 http://mathsmozac. blogspot. com 21 http://sahatmozac. blogspot. com SPM QUESTIONS SPM 2003 Ã¢â¬â PAPER 1, QUESTION 17 1. Given that ? SPM 2004 Ã¢â¬â PAPER 1, QUESTION 22 k n dx ? k ? 1 ? x ? ? c , 2. Given that 1 ? 2 x ? 3? dx ? 6 , where k ; -1 , find the value of k. [4 marks] ? 1 ? x ? find the value of k and n [3 marks] Answer: k = 5 5 Answer: k = ? =-3 3 5 4 SPM 2005 Ã¢â¬â PAPER 1, QUESTION 21 6 6 3. Given that ? 2 f ( x)dx ? 7 and ? 2 (2 f ( x) ? kx)dx ? 10 , find the value of k. Answer: k = 1 4 http://mathsmozac. blogspot. com 22 http://sahatmozac. blogspot. com ANSWERS EXERCISE A 1. 3 ( x + 1)4 + c 2. 60 (3 x +5) Ã¢â¬â 4 + c 3. ?20 EXERCISE B 1. 1023. 75 ? 5 x ? 3? 3 ?c 2. 3 83 96 5 4. 3? 1 ? ?5 ? x? ? c 2? 2 ? ? y? ?c ? 6 4 ?2 3 ? 2 ? 3. ? x ? 5? ? c 10 ? 3 ? 1 3 5 5. 1 6 6. 17 1 ? 5. ?10 ? 4 ? 2 ? 6. 4. 3 2 ? ? ? 5 ? 5 ? u ? ? c 3 ? ? ASSESSMENT 22 1. (a) k = 3 (b) 48 2. (a) 90(2 Ã¢â¬â 3v) +c ? 100 (b) (1 ? 5 x) ? 4 ? c 3 3. 1 10 -5 SP M QUESTIONS 1. k = ? 2. k = 5 3. = 1 4 5 3 n=-3 4. (a) Ã¢â¬â 15 (b) 4 http://mathsmozac. blogspot. com 23 http://sahatmozac. blogspot. com ADDITIONAL MATHEMATICS MODULE 6 INTEGRATION http://mathsmozac. blogspot. com 24 http://sahatmozac. blogspot. com CHAPTER 3 : INTEGRATION Content Concept Map 9. 1 Integration as Summation of Areas page 2 3 4Ã¢â¬â6 7Ã¢â¬â8 9 Ã¢â¬â 11 12 Ã¢â¬â 14 15 Exercise A 9. 2 Integration as Summation of Volumes Exercise B SPM Question Answer http://mathsmozac. blogspot. com 25 http://sahatmozac. blogspot. com a) The area under a curve which enclosed by x-axis, x = a and x = b is a) The volume generated when a curve is rotated through 360? bout the x-axis is ? ? b a y dx b) The area under a curve which enclosed by y-axis, y = a and y = b is b a Vx ? ? ? y 2 dx a b x dy b) The volume generated when a curve is rotated through 360? about the y-axis is c) The area enclosed by a curve and a straight line ? ? f ( x) ? g ( x)? dx b a Vy ? ? ? x 2 dy a b http://mathsmozac. blogspot. com 26 http://sahatmozac. blogspot. com 3. INTEGRATION 3. 1 Integration as Summation of Area y y = f(x) b a a b 0 The area under a curve which enclosed by x = a and x = b is x 0 x y = f(x) ? b a ydx The area under a curve which is enclosed by y = a and y = b isNote : The area is preceded by a negative sign if the region lies below the x Ã¢â¬â axis. ? b a xdy Note : The area is preceded by a negative sign if the region is to the left of the y Ã¢â¬â axis. The area enclosed by a curve and a straight line y y = g (x) y = f (x) a The area of the shaded region = = b b x ? ? ? f ( x) ? g ( x)? dx a b a a b f ( x)dx ? ? g ( x) http://mathsmozac. blogspot. com 27 http://sahatmozac. blogspot. com 1. Find the area of the shaded region in the diagram. y y = x2 Ã¢â¬â 2x 2. Find the area of the shaded region in the diagram. y y = -x2 + 3x+ 4 x -1 0 4 0 x http://mathsmozac. blogspot. com 28 http://sahatmozac. logspot. com 3. Find the area of the shaded region y y=2 4. Find the area of the shaded region in the diagram. y y = x2 + 4x + 4 0 x = y2 x -2 -1 0 2 x http://mathsmozac. blogspot. com 29 http://sahatmozac. blogspot. com 5. Find the area of the shaded region in the diagram y 1 x = y3 Ã¢â¬â y x 6. y y = ( x Ã¢â¬â 1)2 0 0 x x=k -1 Given that the area of the shaded region in 28 the diagram above is units2. Find the 3 value of k. http://mathsmozac. blogspot. com 30 http://sahatmozac. blogspot. com 3. 2 Integration as Summation of Volumes y y=f(x) The volume generated when a curve is rotated through 360? about the x-axis is 0 a b xVx ? ? ? y 2 dx a b y y=f(x) The volume generated when a curve is rotated through 360? about the y-axis is b a 0 x Vy ? ? ? x 2 dy a b http://mathsmozac. blogspot. com 31 http://sahatmozac. blogspot. com y y=x(x+1) Find the volume generated when the shaded region is rotated through 360? about the x-axis. x 0 Answer : x=2 ? ? ? y 2 dx 0 2 Volume generated ? ? ? x 2 ? x ? 1? dx 2 2 0 ? ? ? ( x 4 ? 2 x3 ? x 2 )dx 0 2 ? x 5 2 x 4 x3 ? ? ? ? ? 4 3 ? 0 ? 5 2 25 2(2)4 23 ? ? ? ? ? ? ? ? 0? 5 4 3? ? 256 1 ? ? @ 17 ? units 3 . 15 15 y y ? 6 ? x2 The figure shows the shaded region that is enclosed by the curve y ? ? x 2 , the x-axis and the y-axis. Calculate the volume generated when the shaded region is revolved through 360? about y-axis. 0 Answer : Given y ? 6 ? x 2 substitute x ? 0 into y ? 6 ? x Then, y ? 6? 0 y? 6 2 x Volume generated ? ? ? x 2 dy 0 6 ? ? ? ? 6 ? y ? dx 6 0 ? y2 ? ? ? ?6 y ? ? 2 ? 0 ? 62 ? ? 6(6) ? 2 ? 18? units 3 . ? ? ? ? 0? ? ? 6 http://mathsmozac. blogspot. com 32 http://sahatmozac. blogspot. com 1. y y = x (2 Ã¢â¬â x) 0 x The above figure shows the shaded region that is enclosed by the curve y = x (2 Ã¢â¬â x) and x-axis. Calculate the volume generated when the shaded region is revolved through 360? bout the y-axis. [4 marks] http://mathsmozac. blogspot. com 33 http://sahatmozac. blogspot. com 2. y R (0, 4) Q (3, 4) P (0, 2) y? = 4 (x + 1) 0 x=3 x The f igure shows the curve y ? ( x ? 2) 2 . Calculate the volume generated when the shaded region is revolved through 360? about the x-axis. http://mathsmozac. blogspot. com 34 http://sahatmozac. blogspot. com 3. y R (0, 4) x y ? ? 3? x 0 x=k The above figure shows part of the curve y ? ? 3 ? x and the straight line x = k. If the volume generated when the shaded region is revolved through 1 360? about the x-axis is 12 ? units3 , find the value of k. 2 http://mathsmozac. logspot. com 35 http://sahatmozac. blogspot. com SPM 2003- Paper 2 :Question 9 (b) Diagram 3 shows a curve x ? y 2 ? 1 which intersects the straight line 3 y ? 2 x at point A. y 3 y ? 2x 3y ? 2x x ? y2 ? 1 ?1 0 x Diagram 3 Calculate the volume generated when the shaded region is involved 360? about the y-axis. [6 marks] http://mathsmozac. blogspot. com 36 http://sahatmozac. blogspot. com SPM 2004- Paper 2 :Question 10 Diagram 5 shows part of the curve y ? y 3 ? 2 x ? 1? 2 which passes through A(1, 3). A(1,3) y? 0 a) b) Di agram 5 3 ? 2 x ? 1? 2 x Find the equation of the tangent to the curve at the point A. [4 marks] A egion is bounded by the curve, the x-axis and the straight lines x=2 and x= 3. i) Find the area of the region. ii) The region is revolved through 360? about the x-axis. Find the volume generated, in terms of ? . [6 marks] http://mathsmozac. blogspot. com 37 http://sahatmozac. blogspot. com SPM 2005- Paper 2 :Question 10 In Diagram 4, the straight line PQ is normal to the curve y ? straight line AR is parallel to the y-axis. y x2 ? 1 at A(2, 3). The 2 y? x2 ? 1 2 A(2, 3) 0 R Diagram 4 Find (a) (b) (c) Q(k, 0) x the value of k, [3 marks] the area of the shaded region, [4 marks] the volume generated, in terms of ? when the region bounded by the curve, the y-axis and the straight line y = 3 is revolved through 360? about y-axis. [3 marks] http://mathsmozac. blogspot. com 38 http://sahatmozac. blogspot. com EXERCISE A EXERCISE B 1. 1 1 ? unit 2 15 1. 1 1 units 2 3 5 units 2 6 2. 2. 20 3 6 ? unit 3 5 k ? ?2 3. 3. 2 2 units 2 3 2 units 2 3 SPM QUESTIONS SPM 2003 Volume Generated ? 52 ? units3 15 4. 24 SPM 2004 i) Area ? 1 units 2 5 49 ? units3 1125 5. 1 units 2 2 k? 4 ii) Volume Generated ? 6. SPM 2005 a) k ? 8 1 b) Area ? 12 units2 3 c) Volume Generated ? 4? units? http://mathsmozac. blogspot. com 39
Wednesday, January 8, 2020
Now that this paper has evaluated AquinasÃ¢â¬â¢s Summa Contra Gentiles, it will move on to evaluate his next important work. In the years 1265Ã¢â¬â1274 Aquinas wrote what is considered one of his most prominent works, The Summa Theologiae. In Summa Theologiae (also known as Suma Theologica or simply Summa), Aquinas gave five proofs for the existence of God. This paper will first tell why these proofs are necessary then describe the proofs in themselves. These proofs are necessary because Aquinas believed that the existence of God is not self-evident. A self-evident proposition is one in which the predicate forms part of what is meant by the subject (PUT, 103). Meaning that Ã¢â¬Å"God existsÃ¢â¬ is not self-evident because we cannot grasp divine essenceÃ¢â¬ ¦show more contentÃ¢â¬ ¦Therefore anything that is in the process of changing cannot change itself so one thing is changed by another which in turn is changed by yet another (Clark, 122). Eventually, this stream of chang e has to stop somewhere or else there would be no first cause of change and consequently no subsequent causes. So when we come to the first cause that is not changed by anything else, Aquinas believed it is what we understand to be God (Clark, 122-123). The second proof is derived from the nature of causation. Aquinas thought that in the natural world we find causes in a natural order of succession. We never see something causing itself because if we did then it would be pre-existing and this would be impossible (Clark, 123). Every first cause impacts an intermediate (there can be many intermediates) which then impacts a last. You cannot take out any one cause without getting rid of its effects (Clark, 123). So you cannot take out the first cause without losing the intermediates and last causes that follow. Thus Aquinas thought that we must suppose a first cause, which is God (Clark, 123). The third proof addresses the issue of what is unnecessary and what is unnecessary. Our experience has shown us that in life there are things that are necessary and things that are unnecessary. Things that are