AP Calculus Exam Review Limits and Continuity

Limits and continuity are topics that show up frequently on both the AP Calculus AB and BC exams. In this article, well discuss a few different techniques for finding limits. Well also see the three-part definition for continuity and how to use it. Keep in mind this is just a short review. Much more can be said about limits and continuity in general. Limits What is a limit anyway? In simple terms, a limit for a function is a particular y-value that the function approaches as x is allowed to approach a specified value. We use the following notations for a limit: This notation means: The y-value of f(x) approaches L as x approaches a. Its important to realize that the limit value does not depend on the actual value of f(a). For example, in the graph below, the limit exists at x = 3 even though the function has a hole in the graph at that point. Finding Limits Algebraically When graphs are not given, it can be tricky to find limits. Usually, you have to simplify the expression in some way and then plug in the x-value. Common techniques include: Factor and cancel Combine fractions / LCD Rationalize a radical LHospitals rule Lets see how these techniques might be used in the following examples. Factoring Example Problem 1 is a typical case in which factoring helps out. If the numerator and denominator are both polynomials, then try factoring first. Once the common factor is canceled, then you can plug in the x value to find the limit. Fractions and Least Common Denominator Because problem 2 has fractions on top of fractions, we look for ways to simplify or get rid of some of the fractions. In my experience, the easiest way to do this is by finding the least common denominator (LCD). Find the denominators of the small fractions, x and 3 in this case. Then the LCD would be 3x. Multiplying the LCD to both the top and bottom allows you to simplify the expression. Rationalizing Radicals In algebra, rationalizing means changing the form of a number to avoid radicals in the denominator. The situation is different in calculus. Sometimes the radical is in the numerator, and we rationalize the expression anyway. The reason for this is that rationalization often changes the form in just the right way so that the limit problem can be solved. To rationalize an expression, multiply both the numerator and denominator by the conjugate. The conjugate of a radical expression is found by changing the sign in front of the radical. In other words, the conjugate of (a + b) is equal to (a b). LHospitals Rule After you learn about derivatives in calculus, then you can use one of the most powerful tools for finding limits, LHospitals Rule (LHospital, or LHpital, is a French name, pronounced Low-pee-tal. NOT Le Hoss-pih-tal). In order to use LHospitals Rule, you must first check to see that your limit has the right form. First of all, it must be a fraction of two functions, p(x) / q(x). If it isnt a fraction, there still may be ways to use LHospital as long as you can convert the given expression into a fractional form. Secondly and this is crucial! when you plug in the given x-value, the fraction must either evaluate to 0/0 or /. If both criteria are met, then LHospitals Rule states that the limit of a fractional expression is the same as the limit after taking derivatives of the numerator and denominator. That is, Problem 4 fits the bill. It has the form 0/0, so we can use LHospitals Rule. Infinite Limits Sometimes the limit expression involves infinity ( or -). This can happen in two different ways. If the values of f(x) seem to get bigger and bigger without bound as x approaches a number, then we say the value of the limit is . Conversely, if the values drop without bound (getting bigger in the negative sense), then the value of the limit is -. In either case, the function has a vertical asymptote at that point. Read here for more information about vertical asymptotes. The other way that infinity may play a role is that you might have a limit in which x or x -. Either case leads to a horizontal asymptote, which you may read more about in this article. Further Resources for limits Check out this Magoosh article or this video for more practice and explanation about limits. Continuity Now that weve discussed limits, we can talk about continuity. Intuitively, we say a function is continuous if we can draw its graph without lifting the pencil. That is, there are no jumps, holes, or vertical asymptotes in the graph. However, we need a more precise definition! The Three-Part Definition for Continuity A function f(x) is continuous at a point x = a if the following limit statement is true: But wait, that doesnt look like a three-part definition! Well, in fact, there are three major ideas wrapped up in that limit equation. 1. The limit must exist. This means that the y-values of the graph must be tending toward some finite number as x a. So the function cannot be continuous at a vertical asymptote or a jump, places where the limit fails to exist. 2. The function value must exist. There must be a value for f(a). This means that a function is not continuous at a missing point of the graph (open circle). 3. The limit must agree with the function value. Even if a function passes both conditions (1) and (2), it could still fail to be continuous. We want the data (function value) to fit the trend (limit). If that does not happen, then the function is not continuous at that point. A Graphical Example Lets see how the three parts of the continuity definition play a role in the following problem. A graph of f(x) is shown below. Determine the largest set of x-values at which f is continuous. Write your answer in interval notation. Ok, we can see three obvious issues. In order from left to right, there are discontinuities at x = -3.5, -1, and 3. At x = -3.5, the function value fails to exist. At x = -1, the limit value fails to exist. At x = 3, the function value, f(3) = 2, doesnt agree with the limit value, 1. f is continuous everywhere else. Therefore, the answer is: (-, -3.5) U (-3.5, -1) U (-1, 3) U (3, ). Final Thoughts on Limits and Continuity Limits and continuity problems on the AP Calculus exams may be very easy or may be quite challenging. However, by keeping a few tools, definitions, and examples in mind, you can take your score to the limit!

Economic Growth Essay Example Pdf - Free Essay Example

Introduction Reich (2010, p. 1) argues that economic growth leads to increased prosperity in the developed, emerging and developing world. The argument is focused on the negative effects of slow economic growth for the world and the environment. This article provides a unique perspective regarding the impact of growth on environment and approves of economic growth as an indicator of improving environmental and economic conditions of people throughout the globe. This essay provides a critique of the argument made by the writer in support of economic growth. Critical Review of the Article The importance of economic growth should be considered in context of the impact of growth on environment. It is observed that economic growth tends to negatively impact the environment during its initial phase but growth leads to positive outcomes over the long-term. The focus of governments is to first improve the economic well being of their citizens and as the economic conditions improve the regulatory authorities direct their attention towards improvement of the environmental conditions. This article correctly suggests that the negative impact of growth on the environment is the result of inadequate and poor implementation of international law regarding the protection of environment (Troy 2011, pp. 1-6). The economic growth resulted in widespread damage to the environment in the developed world including the United States, Japan and the Western Europe during its initial phase; however, the introduction of strict environment regulation in these regions has improved the e nvironmental impact of growth. The spread of education across the developed world has also raised awareness regarding the need for environmental protection and businesses have responded to consumer awareness by becoming increasingly concerned about their footprint on the environment. The consumers in developed world are not willing to purchase goods and services from businesses that fail to demonstrate corporate responsibility towards the environment. This does not imply that the negative impact of growth on the environment has reversed but there is substantial effort from the business community to develop processes and technologies that can result in a reduction in the carbon emissions produced by the business in order to support sustainable growth. Therefore, the article rightly suggests that government improve environmental laws after the population has reached a certain minimum standard of living based on economic terms (Yueh 2003, pp. 3-8). The unprecedented economic growth over the past three decades has resulted in severe consequences for environment due to the carbon footprint of business on the surroundings. The growth of the airline industry has offered several benefits but at the expense of the environment due to the carbon emissions in the air. The airline industry is trying to reduce its carbon footprint but growth does imply increase in the demand for travel and increase in the overall emissions. This implies that, even when businesses aim to be socially responsible, growth continues to have a negative impact on the environment. Therefore, the proposition of the writer regarding the strengthening of the international law would require airlines to reimburse the damages to the environment and this process can be costly. It is also argued that economic growth automatically slows down in response to strict international law for the preservation of the environment (Kniivila 2007, pp. 23-26). The regulatory authorities have been relu ctant to adopt strict environmental legislation as this will lead to a slowdown in economic growth. The efforts to protect the environment are reduced by a slowdown in economic growth if examined in the context of the recent global financial crisis of 2008. This does not imply that the impact of business on the environment was not reduced due to a decline in economic growth over the period. This leads to the suggestion that a decline in the economic growth leads to a reduction in the damage on the environment but at the same time creates pressures that can reduce the efforts by the business community to protect the environment against their activities. It is argued that economic growth should be maintained at sustainable levels and unmonitored economic growth does not lead to benefits to the society. Therefore, the focus of the economies and business frameworks should be diverted to the adoption of growth strategies that create a sustainable environment (IMF 2011, pp. 18-26). The critical review of the assertions made by the article and the economic literature on growth and sustainable environment lead to the suggestion that economic growth and environment tend to be competing objectives; however, the use of sustainable technologies is a step forward in the direction of reducing the impact of growth on the environment. The governments need to adopt a model of growth that offers sustainability to the environment and as suggested by the article the developed world has the resources and the capability to facilitate the protection of environment by promoting strict international laws and standards regarding the environment (ODI 2008, pp. 12-18). A slowdown in economic growth is linked to the development of competing element between economic growth and the environment due to the reduced willingness of the middle class to contribute towards the protection of the environment through the payment of taxes. This argument mentioned in the article is con flicting as it is the wealthy capitalist and corporations that are causing damage to the environment and it is the purpose of the regulatory authorities to ensure that the burden of the taxation to support environmental rehabilitation to fall on the capitalists (United Nations 2011, pp. 33-37). Conclusion The critical review of the article and the literature regarding growth and environment leads to the suggestion that these are competing objectives and that it is the responsibility of the regulatory authorities to strengthen international law for the preservation of environment. The article argues that growth leads to the enabling of the government and the society to pay for the environmental needs of the country; however, this assertion fails to recognise the competing goals of growth and environment. The objective is to ensure the growth is responsible and regulatory authorities in the developed world should rely on a strong mechanism for ensuring that growth is sustainable. Bibliography IMF. (2011) World Economic Outlook: April 2011. Washington DC: International Monetary Fund. Kniivila, M., (2007). Industrial development and economic growth: Implications for poverty reduction and income inequality. New York: United Nations. ODI., (2008). Pro-poor growth and development. London: Overseas Development Institute. Reich, R., (2010). Why growth is good? Huffingtonpost.com., [internet] 17 Aug. Available from: https://www.huffingtonpost.com/robert-reich/why-growth-is-good_b_685311.html [cited 23/03/2012] Troy, M., (2011). Working Paper: New indicators for tracking growth in real time. Washington DC: International Monetary Fund. United Nations., (2011). World Economic Situation and Prospects. New York: United Nations. Yueh, L., (2003). Chinas economic growth with WTO accession: Is it sustainable? London: Royal Institute of National Affairs.

A Visit At The Museum Of Fine Arts - 1210 Words

Museum Assignment On November 8th, 2016 I took a trip to the Museum of Fine Arts in Boston, Massachusetts, to conduct research on different kinds of art. When I got there I went straight to The Ancient World exhibit to find a piece of artwork that I found fascinating. The moment I took a step through those giant glass doors, I saw the Colossal Statue of King Menkaura. This vast piece of artwork had so much history and detail from one particular era, it was almost breathtaking. As I continued on through The Ancient World exhibit there were many works of art that stood out to me. However it wasn’t until I reached the Greek Mythology portion of the museum when I really found the piece that literally made me gasp. The sculpture of Cupid (Eros) was exactly the type of artwork I was hoping to discover. The sculpture of Cupid (Eros) was simple, yet it also had a complex composition to it. The Colossal Statue of King Menkaura was built around 2,490 B.C and is the largest statue from the pyramid age. This piece of artwork fits into the category of an â€Å"An image or object related to a ruler, whether king, queen, emperor or other†. King Menkaura fits into this catergory because he was a ruler of the 4th dynasty during the Old Kingdom period, and was the creator of the three pyramids in Giza. One pyramid which he built for himself, was the Pyramid of Menkaura. This Egyptian statue is made out of Travertine, which is a type of limestone that is deposited by mineral springs. When IShow MoreRelatedA Visit to the Boston Museum of Fine Arts Essay867 Words   |  4 PagesA group of us Arts and Music students used the day on Wednesday to go into the city to visit the Boston Museum of Fine Arts. It was a cold, breezy day so we were all bundled up in jackets and hats. As we got off the E train and walked toward the museum, its appearance was not what I expected it to be. 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The Effects of Alcohol on Social and Emotional Development...

Adolescent Social Development The social development of adolescents is very much affected by the social world. Peer relationships, family relationships, school, work, and community play a critical role in an adolescent’s social development as well as culture. Adolescence begins around the age of eleven and lasts into the early twenties. As a child enters into adolescence, many changes are taking place, including physical changes in appearance, sexual maturity, hormonal changes, and the ability to reflect on one’s identity of self (Broderick Blewitt, 2010). As adolescents begin to experience these changes; they also experiment with new behaviors to help them transition from childhood to adulthood. Risk taking is a normal way that†¦show more content†¦Higher rates of substance abuse in young adolescents are linked to neighborhood instability (Leventhal Brooks-Gunn, 2000). Emotional Development According to Erickson (1968), developing identity is the central task of adolescents. Developing a sense of self or identity involves relating to others and learning to manage many emotions. This identity includes not only what individuals might become but also who they would like to become (Markus Nerius, 1986). Identity consists of two concepts, self-concept and self-esteem. Self-concept includes beliefs about one’s traits, roles and goals, and interest, values, and beliefs. Self-esteem involves assessing how one feels about self-concept (APA, 2000). Adolescent Risk Taking Behavior According to Arnett and Balle-Hjensen (1993) there are several theories that may explain why adolescents engage in risky behaviors. They proposed that need for excitement, fun, and unique, powerful sensations outweigh the potential dangers involved in an activity. Jesser (1991) proposed another theory and stresses that risk behaviors occur in a group context and involve peer acceptance and status within the group. 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The Accomplishments Of Edmund Kean - 1513 Words

Edmund Kean, one of the greatest and most historic British actors to ever live. Kean might have been born on March 17th in the year of 1789,and I say might have because no one really knows when he was born. The exact date of his birth has been a mystery for what seems like forever now, and as the years keep going, the chance of this mystery being solved becomes less and less possible. We’re at the point now where we’ll probably never find out when Edmund Kean was born, all we have is a rough estimate at the moment. He had a turbulent life and therefore he also used that to his advantage by being a turbulent genius and was recognized for his megalomania and his erotic behavior which caused him being a villain it mostly every Shakespearean play he took part in. (Britannica). Something that has actually been proven was that his birth mother was a lady named Ann Carey, she was actually an actress. His father was Edmund Kean the first, He died at a very young age after havin g his son. He was a reckless youth and ended up killing himself at a young age of twenty two years of age. After Kean’s death, the young boy was pretty much adopted by his fathers oldest brother, Moses Keen and his wife Charlotte Tidswell. He was a part time member of the Drury Lane Theater Comapany, and work the play named The 11th Duke of Norfolk.(Brittanica).Charlotte Tidswell really worked hard at trying to make young Kean to grow up and be a very well educated and mannered person, but her attempts

Harry And Sally Relationship Essay - 1033 Words

Harry and sally’s relationship didn’t start like most conventional relationships do. The two are living life and going to through the motions of a developing relationship right before their eyes. Love can make you act in ways you didn’t think were possible. At first glance, Harry is dating Sally’s friend and this is how the two met. They initiated the new relationship while Sally was traveling to New York. Harry tagged along because the college he was attending was on the way to Sally’s destination. During this ignition contact, Harry tried to make small talk which gave him a little insight to Sally’s life. She told him she was moving to New York to become a journalist. During the small talk Harry took a chance and told Sally that she was†¦show more content†¦She accepts his offer and does so. Their growing friendship begins to intensify by having dinner together more frequently and sharing one another’s day becomes a part of their everyday life. They are becoming closer as friends. Although they were bonding, becoming more reliant on each other, being only friends was all that Sally had on her mind. Still the two celebrated holiday’s together, purchasing a Christmas tree, counting down the seconds until the new year rang in, and even talking on the phone every night before they went to bed. People assumed they were a couple but instead of a relationship they had a very strong friendship. Since most of their everyday lives are integrated, sharing each other’s lives day in and day out, Harry tries to take their friendship to a new level by trying to kiss her at midnight. Sally was against there being anything more and she was stern about it. He made it clear he wanted to have more with her and to move forward in life but she just pushed away. Until one night, she learned her ex-boyfriend was getting married. She was beside her self and wanted Harry by her side. He rushed over to her rescue. She confided in him and let her feelings win. Little did both of them know that this would be the beginning of their relationship coming apart. One thing quickly leads to another and before they knew it they were kissing. The moment seemed to be heating up even faster and they both had made a decision that was filled with regret, theyShow MoreRelatedF. Scott Fitzgerald s The Great Gatsby2385 Words   |  10 Pageswho encouraged Fitzgerald to follow his dreams as a writer. (Werlock, Fitzgerald, F. Scott. The Facts On File Companion to the American Novel). Moreover, Fitzgerald was the class of 1917 in Princeton, but due to an unfortunate ending to his relationship with Ginevra King, his social habits and health, caused a decrease in his GPA and overall attitude towards learning. 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Classical Theories of Hierarchical Management free essay sample

Classical Theories of Hierarchical Management The purpose of this essay will be to argue the assumptions of classical theories regarding hierarchical management and how it is essential in modern day times. The ranks within a hypothetical pyramid determine the position of an employee within any organisations. Classical management still takes place in today’s management concepts. Max Weber studied bureaucratic organisations, Henri Fayol created the administrative principles and Fredrick Winslow Taylor researched scientific management. These influential people took apart in the formation of hierarchy. The concept of classical management, Weber’s theories and hierarchy will be argued further. In a hierarchical organisation, ranks are what determine a position for an employee. It is a straightforward process as it follows the layout of a pyramid (Meehan, 2012). The organisational structure indicates the method that an organisation employs to delineate lines of communication, authorities, policies and responsibilities. It determines the extent and nature of how leadership is disseminated throughout the organisation as well as the method by which information flows. A flat or hierarchical structure commonly adapts to an organisation (Goessl, 2010). Max Weber, a German sociologist that purposed different characteristics found in effective bureaucracies that would effectively conduct decision-making, control resources, accomplish organisational goals and protect workers (Business Mate, 2009). Ian Clark defines hierarchy as: ‘A social arrangement characterized by stratification in which, like angles, there are orders of power and glory and society is classified in successively subordinated grades’ (1989, pg. 2). Clark’s quote describes hierarchy as having â€Å"orders of power† above others, which refers to the superior of the organisation. Below the superior are the next employees in line which are referred to in the quote as ‘medium powers and smaller powers. ’ Everyone within this hierarchy works as a group to reach a goal. Weber produced the idea of hierarchism when he was making observations regarding his organisation. He became concerned with the fact that people were in positions of authority not because of their job experience and capabilities, but because of their social status in the German society. For this very reason he believed that organisations failed to reach their performance potential. (Shermerhorn et al, 2011, Pg. 91) According to Astley (1988), Weber came up with the ideal type of hierarchy resonates with common experience in established domestic policies (pg. 202-203), which reflected in expression of support by citizens for their political leaders even though they may not respect them as individuals. (Lake, 2006, pg. 36) Weber founded on the principles of logic, order and legitimate authority. The characteristics of Weber’s bureaucratic organisation include; clear division of labour, formal rules of authority and clear hierarchy of authority (Shermerhorn et al, 2011, Pg. 91-92). Clear division of labour is a form of specialization in which the production of a product or service is devised into several separate tasks, each performed by one person (Cengage, 2001). According to Weber’s design, the knowledge within the division of labour defines each employee’s job, giving them a â€Å"sphere of competence†, and the authority to persevere with individual tasks without interrupting others. Large scales are broken down into small manageable units and specialties that are then appointed to each individual (Shafritz Ott, 2001, pg. 134). This is an example of how different departments within a hierarchical system work in organisations. Formal rules of authority apply in a particular system such as an organisation where authority must be used to stabilize employees and guide them into the direction of the leaders. Authority helps prevent anarchy, and helps to define a clear hierarchy of decision-making. A clearly defined hierarchy will potentially lead to an effective organisation, consisting of strong and legitimate authority relations between leaders and followers. Weber distinguished three different types of authority, traditional, rational-legal and charismatic (Business Mate, 2004). Rational and traditional authorities are stable enough to provide the fundamentals for permanent administrative structures such as a business organisation. Where as charismatic authority will need to evolve into a more stable form of authority. Clear authority of hierarchy is designed to benefit the company and employees. This is the same as the overall concept of hierarchy as discussed above. Organisations are methodically ordered into a clear chain of command. The hierarchical structure effectively delineates the lines of authority and the subordination of the lower levels to the upper levels (Cutajar, 2010). The further the chain of command, the more power, authority and responsibility is needed to run the organisation. This usually depends on the size of the business. This merit principle is the most commonly applied in bureaucratic or semi-bureaucratic organisations as they are constructed most fundamentally on hierarchy and divisions of labour (Martin, 1985). According to Weber (1998), the purest form in which legal domination can be governed is as a bureaucracy. Officials in a bureaucracy obey orders predominantly because of demands for their sense of status. Employees are specially trained, have an assigned domain of competence and develop a strong sense of duty in relation to their work. They have a care based on superiority and achievement and can only be completed under certain circumstances (Pg. 63). The creation of a bureaucratic organizational structure is a means of exerting power and is best used in large-scale organisations having both advantages and disadvantages. Advantages include facilitating the management of people and situations and predicting the outcome of actions by others, giving them structure within their organisation (John, 2011). Despite having many disadvantages, it is believed that the degree of control given to the superior over employees is possibly the most damaging outcome. The supervisor will receive almost total control over their subordinates as they are given demands and instructions to follow (Rabie. 2004). Another disadvantage that can effect the organisation significantly is communication. This can be lost through the chain of hierarchy called horizontal communication. The top-down structure keeps different departments from communicating which can produce poor productivity and inefficiency (Handel, 2003, pg. 326) In 1916, Henri Fayol published Administration Industrielle et Generale which contains the original foundation of his theories for management (Sapru, 2006, pg.  101; Pryor Tenaja, 2010, pg. 493). The book outlines his views on the proper management of organisations and the people within them. (Schermerhorn et al, 2011, pg. 90). Fayol synthesized various tents or principles of organisations and management. He argued that the principles needed to exist in order for organisations to work efficiently. Fayol’s five functions are stil l relevant to today regarding management roles and actions; to forecast and plan, to organize, to command, to co-ordinate and to control (Docstoc, 2011). Fayol synthesized 14 principles for organisational design and effectiveness, which reflect closely to Weber’s idea of hierarchism. The science of administration is based on laws and principles. Fayol’s 14 Principles of Management refer to hierarchism within an organisation; two of these principles include Unity of Command and Unity of Decision. They both support the idea of having only one superior above many employees. Unit of Command promotes the idea of subordinates receiving orders of command only from one person (the superior). (Sapru, 2006, pg. 108; Lucey, 1991). This classical idea is still supported in today’s management. Unity of Decision relates to Unity of Command as the superior of the hierarchy (chief or executives) makes the decisions and agrees or disagrees over the purpose and objectives of the plan (Cliffs Notes, 2012). This person has total power and control over all subordinates in the organisation. Additional relevant principles include the Scalar Chain/Life of Authority and Order, which has strong reference to Weber’s concept of chain of authority. The scalar chain of command of reporting relationships from top executives to ordinary shop operators or drivers need to be sensible, clear and understood (Docstoc, 2011). Lower level managers should always keep the upper level managers informed of progress regarding their work and activities. For this chain to be structural, order needs to be implied. Basically an organisation needs to provide an orderly environment for each individual member who is required to understand how his or her roles fit confidently into the organisation (Management Study Guide, 2012; Onkar, 2008). Consequently policies, actions, instructions and rules need to be put into place for employees to follow in a structural order otherwise management wouldn’t work sustainably. The concept of the administrative principles works well within organisations, as it is an extremely comprehensive way to deal with management techniques and has been proven to work. Its advantages cover majority of tasks that a manager may need to know in order to achieve success (Mitchell, 2011). Disadvantages to the theory include that humans are naturally going to make mistakes. The theory works best when having the foundation of harmony among employees. However, when mistakes are made it can weaken the strength of the team (Kramer, 2010). Fredrick Winslow Taylor, an American engineer who â€Å"worked on promoting efficiency in manufacturing enterprises around the turn of the century†(Brown, 1980, pg. 288). Scientific management is referred to as Taylorism; this is a philosophical system of testing and analyzing techniques used to increase the productivity of an organisation (Cannon, 2011, p9, 3-4). He drew on the idea that they should have a detailed analysis of each job, using the techniques of method of study in order to find the method of working that would bring the largest average rate of production – this was called ‘one best way’ (Kermally, 2004, pg.10- 14). Taylor touched on the concept that the reference to hierarchism depended on the amount of knowledge and responsibility a person holds. In a study of scientific management in practice, C. Bertand Thompson (1915) discussed the responsibilities of planning in relation to managers. Thompson stated that the responsibility of management is ‘functional foremanship’ by which details of administration and determination of the sequence of operations, tools and methods used, importance of order and instructions of employees are all important as it determines each individuals knowledge and expertise (pg.271). This refers to the concept that the duties prearranged in management contain a high level of education and leadership skills to effectively coordinate the various tasks associated with management. Employees need to have complete knowledge of the functions performed and implemented training before being placed in their hierarchical level of expertise. Taylor found that workers w ere deliberately working at less than full capacity. They worked as slow as they could without getting into trouble (Business Knowledge Source, 2010). The advantage of scientific management meant that workers would increase their performance, changing everything from how they did their jobs to the tools they used. Responsibilities were divided up between hierarchies of management so that managers would be responsible for planning work methods using scientific principles and workers would be responsible for executing the work accordingly. Another advantage is that superiors trained workers and allocated them in the right department of the organisation. This way, employees would know â€Å"the one best way† rather then relying on their own various rules of thumb (Bagad, 2009, pg. 22-23). This concept related with Weber’s theory of ‘clear division of labour’ as discussed above. Taylorism has a number of disadvantages, which affected the workers; just as easily could it be abused, exploit human beings and conflict with their labour and unions (Scribd, 2010). Another recognised issue ‘was’ the individual differences; this would cover the fact that it may work only for certain employees and not for others. This could increase production for one person but decrease for another. It does not address the fact that the economic interests of each worker and manager are the same; therefore employees would most likely resent and attempt to sabotage Taylor’s methods of measuring process and retraining (Business Knowledge Source, 2010). From this literature, it is evident that hierarchy plays a major role within management structure today. With reference to Max Weber, Fredrick. W. Taylor and Henri Fayol, all contributed to the concept of hierarchy. Max Weber’s idea of bureaucracy is very formal and rigid; this will perhaps only apply in organisations with high standards, as employees are to obey orders predominantly due to the demands made by their superiors. Bureaucracy works well under the influence of leadership and motivation, with well-defined procedures within a productive business framework. Issues can arise from the superior as they are given the most control over their employees, which is very demanding over the departments below. Fayol’s theory of administrative principles works well in most organisations, as it is an extremely comprehensive way to deal with management techniques. It covers majority of tasks that a manager may need in order to achieve success. Although, a factor that Fayol didn’t take into consideration was the fact that humans will always make mistakes as this can weaken the strength of the team. Lastly, Taylor’s theory produced the idea that workers could increase their performance by dividing their responsibilities between hierarchies of management so that managers would be responsible for planning work methods and employees would be responsible for executing the work as instructed. This method could easily fail as each employee has individual differences. Increased performance in one employee may not work for another. It doesn’t address the economic interests of each worker and manager, as they are not the same. Overall, hierarchy works productively for a number of organisations.