Maxima and Minima of Functions

Maxima and Minima of Functions This term paper presents concise explanations of the subjects general principles and uses worked examples freely to expand the ideas about solving the problems by suitable methods. Each example shows the method of obtaining the solution and includes additional explanatory techniques. For some topics, where it would have been difficult to understand a solution given on a single problem, the solution has been drawn in step-by-step form. All the figures used have been taken from Google Book search. The term paper covers the necessary definitions on MAXIMA AND MINIMA OF THE FUNCTIONS and some of its important applications. It covers the topic such as types of other method for solving the big problem in a shortcut method known . The aspects of how to develop some of the most commonly seen problems is also covered in this term paper. The motive of this term paper is make the reader familiar with the concepts of application of maxima and minima of the function and  where this is used. Focus has been more on taking the simpler problem so that the concept could be made clearer even to the beginners to engineering mathematics. MAXIMA AND MINIMA The diagram below shows part of a function y = f(x). The point A is a local maximum and the point B is a local minimum. At each of these points the tangent to the curve is parallel to the X axis so the derivative of the function is zero. Both of these points are therefore stationary points of the function. The term local is used since these points are the maximum and minimum in this particular Region. The rate of change of a function is measured by its derivative. When the derivative is positive, the function is increasing, When the derivative is negative, the function is decreasing. Thus the rate of change of the gradient is measured by its derivative, Which is the second derivative of the original function? Functions can have hills and valleys: places where they reach a minimum or maximum value. It may not be the minimum or maximum for the whole function, but locally it is. You can see where they are, but how do we define them? Local Maximum First we need to choose an interval: Then we can say that a local maximum is the point where: The height of the function at a is greater than (or equal to) the height anywhere else in that interval. Or, more briefly: f(a) à ¢Ã¢â‚¬ °Ã‚ ¥ f(x) for all x in the interval In other words, there is no height greater than f(a). Note: f(a) should be inside the interval, not at one end or the other. Local Minimum Likewise, a local minimum is: f(a) à ¢Ã¢â‚¬ °Ã‚ ¤ f(x) for all x in the interval The plural of Maximum is Maxima The plural of Minimum is Minima Maxima and Minima are collectively called Extrema Global (or Absolute) Maximum and Minimum The maximum or minimum over the entire function is called an Absolute or Global maximum or minimum. There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum. Assuming this function continues downwards to left and right: The Global Maximum is about 3.7 The Global Minimum is -Infinity Maxima and Minima of Functions of Two Variables Locate relative maxima, minima and saddle points of functions of two variables. Several examples with detailed solutions are presented. 3-Dimensional graphs of functions are shown to confirm the existence of these points. More on Optimization Problems with Functions of Two Variables in this web site. Theorem Let f be a function with two variables with continuous second order partial derivativesfxx, fyyand fxyat a critical point (a,b). Let D = fxx(a,b) fyy(a,b) fxy2(a,b) If D >0 and fxx(a,b) >0, then f has a relative minimum at (a,b). If D >0 and fxx(a,b) If D If D = 0, then no conclusion can be drawn. We now present several examples with detailed solutions on how to locate relative minima, maxima and saddle points of functions of two variables. When too many critical points are found, the use of a table is very convenient. Example 1:Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by f(x , y) = 22+ 2xy + 2y2- 6x Solution to Example 1: Find the first partial derivatives fxand fy. fx(x,y) = 4x + 2y 6 fy(x,y) = 2x + 4y The critical points satisfy the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously. Hence. 4x + 2y 6 = 0 2x + 4y = 0 The above system of equations has one solution at the point (2,-1). We now need to find the second order partial derivatives fxx(x,y), fyy(x,y) and fxy(x,y). fxx(x,y) = 4 fxx(x,y) = 4 fxy(x,y) = 2 We now need to find D defined above. D = fxx(2,-1) fyy(2,-1) fxy2(2,-1) = ( 4 )( 4 ) 22= 12 Since D is positive and fxx(2,-1) is also positive, according to the above theorem function f has a local minimum at (2,-1). The 3-Dimensional graph of function f given above shows that f has a local minimum at the point (2,-1,f(2,-1)) = (2,-1,-6). Example 2:Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by f(x , y) = 22- 4xy + y4+ 2 Solution to Example 2: Find the first partial derivatives fxand fy. fx(x,y) = 4x 4y fy(x,y) = 4x + 4y3 Determine the critical points by solving the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously. Hence. 4x 4y = 0 4x + 4y3= 0 The first equation gives x = y. Substitute x by y in the equation 4x + 4y3= 0 to obtain. 4y + 4y3= 0 Factor and solve for y. 4y(-1 + y2) = 0 y = 0 , y = 1 and y = -1 We now use the equation x = y to find the critical points. (0 , 0) , (1 , 1) and (-1 , -1) We now determine the second order partial derivatives. fxx(x,y) = 4 fyy(x,y) = 12y2 fxy(x,y) = -4 We now use a table to study the signs of D and fxx(a,b) and use the above theorem to decide on whether a given critical point is a saddle point, relative maximum or minimum. critical point (a,b) (0,0) (1,1) (-1,1) fxx(a,b) 4 4 4 fyy(a,b) 0 12 12 fxy(a,b) -4 -4 -4 D -16 32 32 saddle point relative minimum relative minimum A 3-Dimensional graph of function f shows that f has two local minima at (-1,-1,1) and (1,1,1) and one saddle point at (0,0,2). Example 3:Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by f(x , y) = x4- y4+ 4xy . Solution to Example 3: First partial derivatives fxand fyare given by. fx(x,y) = 43+ 4y fy(x,y) = 4y3+ 4x We now solve the equations fy(x,y) = 0 and fx(x,y) = 0 to find the critical points.. 43+ 4y = 0 4y3+ 4x = 0 The first equation gives y = x3. Combined with the second equation, we obtain. 4(x3)3+ 4x = 0 Which may be written as . x(x4- 1)(x4+ 1) = 0 Which has the solutions. x = 0 , -1 and 1. We now use the equation y = x3to find the critical points. (0 , 0) , (1 , 1) and (-1 , -1) We now determine the second order partial derivatives. fxx(x,y) = -122 The First Derivative: Maxima and Minima Consider the function f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3   on the interval [à ¢Ã‹â€ Ã¢â‚¬â„¢23]. We cannot find regions of which f is increasing or decreasing, relative maxima or minima, or the absolute maximum or minimum value of f on [à ¢Ã‹â€ Ã¢â‚¬â„¢23] by inspection. Graphing by hand is tedious and imprecise. Even the use of a graphing program will only give us an approximation for the locations and values of maxima and minima. We can use the first derivative of f, however, to find all these things quickly and easily. Increasing or Decreasing? Let f be continuous on an interval I and differentiable on the interior of I. If f(x)0 for all xI, then f is increasing on I. If f(x)0 for all xI, then f is decreasing on I. Example The function f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3 has first derivative f(x)  =  =  =  123à ¢Ã‹â€ Ã¢â‚¬â„¢122à ¢Ã‹â€ Ã¢â‚¬â„¢24x  12x(x2à ¢Ã‹â€ Ã¢â‚¬â„¢xà ¢Ã‹â€ Ã¢â‚¬â„¢2)  12x(x+1)(xà ¢Ã‹â€ Ã¢â‚¬â„¢2)  Ãƒâ€šÃ‚   Thus, f(x) is increasing on (à ¢Ã‹â€ Ã¢â‚¬â„¢10)(2) and decreasing on (à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢1)(02). Relative Maxima and Minima Relative extrema of f occur at critical points of f, values x0 for which either f(x0)=0 or f(x0) is undefined. First Derivative Test Suppose f is continuous at a critical point x0. If f(x)0 on an open interval extending left from x0 and f(x)0 on an open interval extending right from x0, then f has a relative maximum at x0. If f(x)0 on an open interval extending left from x0 and f(x)0 on an open interval extending right from x0, then f has a relative minimum at x0. If f(x) has the same sign on both an open interval extending left from x0 and an open interval extending right from x0, then f does not have a relative extremum at x0. In summary, relative extrema occur where f(x) changes sign. Example Our function f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3 is differentiable everywhere on [à ¢Ã‹â€ Ã¢â‚¬â„¢23], with f(x)=0 for x=à ¢Ã‹â€ Ã¢â‚¬â„¢102. These are the three critical points of f on [à ¢Ã‹â€ Ã¢â‚¬â„¢23]. By the First Derivative Test, f has a relative maximum at x=0 and relative minima at x=à ¢Ã‹â€ Ã¢â‚¬â„¢1 and x=2. Absolute Maxima and Minima If f has an extreme value on an open interval, then the extreme value occurs at a critical point of f. If f has an extreme value on a closed interval, then the extreme value occurs either at a critical point or at an endpoint. According to the Extreme Value Theorem, if a function is continuous on a closed interval, then it achieves both an absolute maximum and an absolute minimum on the interval. Example Since f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3 is continuous on [à ¢Ã‹â€ Ã¢â‚¬â„¢23], f must have an absolute maximum and an absolute minimum on [à ¢Ã‹â€ Ã¢â‚¬â„¢23]. We simply need to check the value of f at the critical points x=à ¢Ã‹â€ Ã¢â‚¬â„¢102 and at the endpoints x=à ¢Ã‹â€ Ã¢â‚¬â„¢2 and x=3: f(à ¢Ã‹â€ Ã¢â‚¬â„¢2)  f(à ¢Ã‹â€ Ã¢â‚¬â„¢1)  f(0)  f(2)  f(3)  =  =  =  =  =  35  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢2  3  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢29  30  Ãƒâ€šÃ‚   Thus, on [à ¢Ã‹â€ Ã¢â‚¬â„¢23], f(x) achieves a maximum value of 35 at x=à ¢Ã‹â€ Ã¢â‚¬â„¢2 and a minimum value of -29 at x=2. We have discovered a lot about the shape of f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3 without ever graphing it! Now take a look at the graph and verify each of our conclusions. APPLICATION AND CONCLUSION The terms maxima and minima refer to extreme values of a function, that is, the maximum and minimum values that the function attains. Maximum means upper bound or largest possible quantity. The absolute maximum of a function is the largest number contained in the range of the function. That is, if f(a) is greater than or equal to f(x), for all x in the domain of the function, then f(a) is the absolute maximum. For example, the function f(x) = -162 + 32x + 6 has a maximum value of 22 occurring at x = 1. Every value of x produces a value of the function that is less than or equal to 22, hence, 22 is an absolute maximum. In terms of its graph, the absolute maximum of a function is the value of the function that corresponds to the highest point on the graph. Conversely, minimum means lower bound or least possible quantity. The absolute minimum of a function is the smallest number in its range and corresponds to the value of the function at the lowest point of its graph. If f(a) is less t han or equal to f(x), for all x in the domain of the function, then f(a) is an absolute minimum. As an example, f(x) = 322 32x 6 has an absolute minimum of -22, because every value of x produces a value greater than or equal to -22. In some cases, a function will have no absolute maximum or minimum. For instance the function f(x) = 1/x has no absolute maximum value, nor does f(x) = -1/x have an absolute minimum. In still other cases, functions may have relative (or local) maxima and minima. Relative means relative to local or nearby values of the function. The terms relative maxima and relative minima refer to the largest, or least, value that a function takes on over some small portion or interval of its domain. Thus, if f(b) is greater than or equal to f(b  ± h) for small values of h, then f(b) is a local maximum; if f(b) is less than or equal to f(b  ± h), then f(b) is a relative minimum. For example, the function f(x) = x4 -123 582 + 180x + 225 has two relative minima (points A and C), one of which is also the absolute minimum (point C) of the function. It also has a relative maximum (point B), but no absolute maximum. Finding maxima or minima also has important applications in linear algebra and game theory. For example, linear programming consists of maximizing (or minimizing) a particular quantity while requiring that certain constraints be imposed on other quantities. The quantity to be maximized (or minimized), as well as each of the constraints, is represented by an equation or inequality. The resulting system of equations or inequalities, usually linear, often contains hundreds or thousands of variables. The idea is to find the maximum value of a particular variable that represents a solution to the whole system. A practical example might be minimizing the cost of producing an automobile given certain known constraints on the cost of each part, and the time spent by each laborer, all of which may be interdependent. Regardless of the application, though, the key step in any maxima or minima problem is expressing the problem in mathematical terms.

Department of Homeland Security Essay

Returning from a vacation to Germany in February, freelance journalist Bill Hogan was selected for additional screening by customs officials at Dulles International Airport outside Washington. Agents searched his luggage, he said, â€Å"then they told me that they were impounding my laptop. † Shaken by the encounter, Hogan examined his bags and found the agents had also inspected the memory card from his camera. â€Å"It was fortunate that I didn’t use [the laptop] for work,† he said, â€Å"or I would have had to call up all my sources and tell them that the government had just seized their information. † When customs offered to return the computer nearly two weeks later, Hogan had it shipped to his lawyer. How common Hogan’s experience is remains unclear. But an April ruling by the U. S. Ninth Circuit Court of Appeals found that the Department of Homeland Security, which oversees Customs and Border Protection, does have full authority to search any electronic devices without suspicion in the same way that it can inspect briefcases. Now, businesses and other organizations are pushing back, Congress is investigating, and lawsuits have been filed challenging how the program selects travelers for inspection. The ninth circuit ruling was the result of more than 20 lawsuits involving electronics seized from travelers who were nearly all of Muslim, Middle Eastern, or South Asian descent. Citing the lawsuits, customs officials decline to say how many computers, storage drives, cellphones, and BlackBerrys they have confiscated or what happens to them afterward. Officials declined to testify at a recent Senate hearing, although they wrote in a prepared statement that officers â€Å"have the responsibility to check items such as laptops and other personal electronic devices to ensure that any item brought into the country complies with applicable law and is not a threat to the American public. † But congressional investigators say that copies of drives are sometimes made, meaning customs could be duplicating corporate secrets, legal and financial data, personal E-mails and photographs, along with stored passwords for accounts with companies ranging from Netflix to Bank of America. The practice of storing and duplicating material might be something that both opponents and supporters of seizure could agree to regulate, says Kansas Republican Sen. Sam Brownback, an otherwise staunch supporter of customs’ authority. Larry Cunningham, an assistant district attorney from New York, told the hearing: â€Å"I am aware of no authority that would permit the government, without probable cause to believe it contains contraband, to keep a person’s laptop or to copy the contents of its files. † Whatever the case, the controversial practice has prompted some businesses to change their policies about traveling with corporate information. Many now require employees to access data remotely to avoid confiscations. â€Å"[Seizure] immediately deprives an executive or company of the very data–and revenue–a business trip was intended to create,† says Susan Gurley, head of the Association of Corporate Travel Executives, which is lobbying for greater transparency and government oversight of the confiscations. â€Å"As a businessperson returning to the U. S. , you may find yourself effectively locked out of your electronic office indefinitely. † Indeed, while Hogan’s computer was returned within two weeks, others say they have had theirs held for months. Customs insists that terrorism and child pornography are sufficient justification for electronics searches. And even civil libertarians agree it makes sense for customs to search luggage, which could pose immediate dangers to aircraft and passengers. But, says Marc Rotenberg, executive director of the Electronic Privacy Information Center, â€Å"customs officials do not go through briefcases to review and copy paper business records or personal diaries, which is apparently what they are now doing in digital form. These PDA’s don’t have bombs in them. † And then there are the precedents that critics say the program could set. Imagine, they say, if other nations began seizing the laptops of U. S. travelers. â€Å"We wouldn’t be in a position to strongly object,† Rotenberg says. Indeed, U. S. officials have advised visitors to this summer’s Olympics in Beijing that their laptops may be targeted for duplication or bugging by the Chinese. [Illustration] [Picture omitted]: Bill Hogan, who had his laptop seized at an airport, waits for a Senate subcommittee hearing. -CHARLIE ARCHAMBAULT FOR USN&WR

William Shakespeare s King Lear - 1306 Words

â€Å"All...shall taste the wages of their virtue...the cup of their deservings. (5.3.317-320)† King Lear is frequently regarded as one of Shakespeare’s masterpieces, and its tragic scope touches almost all facets of the human condition: from the familial tensions between parents and children to the immoral desires of power, from the follies of pride to the false projections of glory. However, one theme rings true throughout the play, and that very theme is boundless suffering, accentuated by the gruesome depictions of suffering our protagonists experience . There is no natural (nor â€Å"poetic†) justice depicted in this pre-Judeo-Christian world Shakespeare presents, as the relatively virtuous individuals (Kent, Gloucester, and Cordelia) in this†¦show more content†¦The â€Å"gods† are indifferent to the suffering because humans, not gods, are the main perpetrators of the profound cruelty found in this play. Because man has the power to both undermine societal â€Å"nature† and restore it, whether through Edmund’s Machiavellian mora l transgressions in his quest for power or Edgar’s actions to combat them, these characters take the place of their respective, self-created deities. Pagan Gods King Lear is set in a time where even though swords and kings existed, and knights still roamed the land, people still believed in the pagan gods. This is elucidated by the various mentions of the gods (plural) throughout the play, and the lack of a single entity (God). When King Lear disowns Cordelia, he does so by invoking â€Å"the sacred radiance of the sun† and â€Å"the mysteries of Hecate and the night.† (I.i.110-111) He later swears â€Å"by Apollo† to warn Kent, in which Kent rebukes by saying â€Å"Thou swear’st thy gods in vain.† (I.i.164) Lastly, when France proclaims his love for Cordelia he blames the â€Å"Gods† for possessing in him a quality that allows him to be so attracted Cordelia’s virtues. (I.i.263) However, this Pagan world contains the same â€Å"Slave morality† that Judean metaphysics claims, that Nietzche himself criticizes. As Wilson Knight states, the frequent pleas to the gods show at most an insistent need in humanity to cry for justification to something beyond its horizon (188). In fact he extrapolate, These

Comparing Cliges, Erec And Enide And The Three Couples

Cecilia Rivas Ms. McMillan English 11 March 20, 2017 The comparison of Cliges, Erec and Enide and the three couples. Alexander, Cliges, and Erec all have beautiful maidens; Enide, Soredamors, and Fenice. The women have been described to be the most beautiful and virtuous maidens of the land, all having gorgeous faces with the most perfect features. These men have never laid their eyes on something more beautiful until they met their loves, whom nature created so finely. Chretien de Troyes does a great job on describing the beauty and features of each individual. These three couples have a lot in common, and Chretien knows how to write them. Fenice was so beautiful that nature was never able to make her like again. She was so beautiful,†¦show more content†¦Alexander described Soredamors as fair and full of charm, so dear and precious. Her well-shaped nose and radiant face, and her laughing mouth which God shaped with great skill. Soredamors’ neck beneath her hair was so fair it was four times as white as ivory. Alexander saw her bosom exposed and whiter than snow. There is so much to say about every part of Soredamors that it will not be strange if he passed over anything. She had lovely hair, extremely straight and perfect parting of her hair and eyes. Her hair, parting, forehead, and eyes were so beautiful they were nevertheless describable. That is not the case when it comes to her nose, cheek and mouth. Mouth’s are described as a direct connection to God’s and Nature’s creation. Soredamors states that it is not without significance she is called by the name. She is destined to lo ve and be loved in return and intends to prove it by her name. The first part of Soredamors’ name is in significance of golden color, for she believes golden is the best. The end of her name calls love to mind, for whoever calls her by her right name always refreshes her with love. Soredamors has the meaning of ‘one gilded over with love.’ â€Å"If his beauty allures my eyes, and my eyes listen to the call, shall I say that I love him just for that?†(p.10, L.21) Soredamors felt as if that her eyes have betrayed her, she accuses her eyes of treason and succumbs to love. One cannot love with the eyes alone,

Deaf Children and Learning Essay Example

Essays on Deaf Children and Learning Essay The paper "Deaf Children and Learning" is a good example of an essay on education. Few researches have been conducted on multicultural Deaf children and adults in education because of the perception people have about deafness. In relation to this, Johnson and McIntosh (2009, p.74, line 23-29) argue that different  discussion and treatment of deafness from 12 other categories of disability outlined by IDEA 2004 makes researchers perceive Deafness as a cultural as well as linguistic minority instead of a disability. In other words, researchers fail to draw a conclusion about considering Deafness as a disability or not. In tandem with this, Johnson and McIntosh (2009, p.75, para.2, line 8-10) claim that the opposition of researchers’ understanding of a Deaf person as part of the cultural community contributes to researchers researching little on multicultural Deaf people. CLD deaf children in an education setting learn sign language in school that varies from the one used at home (Parasnis, 1997, p.74, para. 5, line 1-6). Deaf children in an education setting also utilize adaptive equipment as well as special services like hearing aids, FM systems in addition to ASL interpreters in communicating effectively with people in the society (Parasnis, 1997, p.73, para. 3, line 19-24). A CLD deaf person in a setting other than K-12 often uses gestures to communicate. The gestures used by such people correlate natural language. Such people also introduce language-like structure into the gestures they use in communication (Goldin-Meadow Mylander, 1998, p.279, para.3, lines 1-5 and para.4, lines 2-3). Additionally, deaf children from different cultures pass their messages via gesture sentences instead of single gestures (Goldin-Meadow Mylander, 1998, p.279, para.5, line 2-4).

Externalities, Pollution and Global Warming Free Essays

Topic 4: Externalities, Pollution and Global Warming ECON 1210B Economics and Society 1 Introduction Recall: Markets are usually a good way to organize economic activity In the absence of market failures, the market outcome is efficient, maximizes total surplus One major type of market failure: externalities Externality: the uncompensated impact of one person’s actions on the well-being of a bystander 2 Externalities and Efficiency In the presence of externality, market equilibrium is no longer efficient Individual’s estimates of resources value (or cost) are not correct (from the society’s point of view) Traditional belief: Government to step in to ensure efficient resource allocation And to protect the interest of bystanders as well 3 Negative Externality Negative Externality: the effect on bystanders is adverse Example: the neighbor’s barking dog talking on cell phone while driving makes the roads less safe for others health risk to others from second-hand smoke noise pollution from construction projects 4 Pollution: A Negative Externality Firms burn huge quantities of fossil fuels (coal, natural gas, oil) that cause acid rain and global warming Firms dump toxic waste into rivers, lakes, and oceans These environmental issues are simultaneously everybody’s problem and nobody’s problem 5 Pollution: A Negative Externality Example of negative externality: Air pollution from factory Firm does not bear the full cost of its production, so will produce more than the socially efficient quantity How govt may improve the market outcome: Impose a corrective tax on the firm equal to the external cost of the pollution it generates 6 Recap of Welfare Economics P $5 4 3 $2. 50 2 1 0 The market for gasoline The market eqm maximizes consumer + producer surplus. Supply curve shows private cost, the costs directly incurred by sellers Demand curve shows private value, the value to buyers (the prices they are willing to pay) 0 10 20 25 30 Q (gallons) 7 Analysis of a Negative Externality Key: distinguish private and social costs Private costs and social costs diverge in the presence of externality Producer concerns private cost, which neglect the external cost (pollution cost) Social cost represents the resource cost to a society social cost = private cost + external cost 8 Analysis of a Negative Externality P $5 4 3 2 1 0 The market for gasoline Social cost =private+ external cost external cost 0 External cost = value of the negative impact on bystanders = $1 per gallon (value of harm Supply (private cost) from smog, greenhouse gases) 10 20 30 Q (gallons) 9 Analysis of a Negative Externality P $5 4 3 2 D 1 0 The market for gasoline Social cost S The socially The socially optimal quantity optimal quantity is 20 gallons. We will write a custom essay sample on Externalities, Pollution and Global Warming or any similar topic only for you Order Now is 20 gallons. At any Q 20, At any Q 20, value of additional gas value of additional gas exceeds social cost exceeds social cost At any Q 20, At any Q 20, social cost of the social cost of the last gallon is last gallon is greater than its value greater than its value 10 0 10 20 25 30 Q (gallons) Analysis of a Negative Externality P $5 4 3 2 D 1 0 The market for gasoline Mkt eqm (Q = 25) Social cost is greater than social optimum S (Q = 20) overproduction resulted in DWL (red triangle) One solution: impose a corrective tax of $1/gallon on sellers, shift supply curve up $1 11 0 10 20 25 30 Q (gallons) Internalizing the Externality Internalizing the externality: altering incentives so that people take account of the external effects of their actions previous example: $1/gallon tax on sellers makes sellers’ costs equal to social costs When market participants must pay social costs, the market eqm matches the social optimum. Imposing the tax on buyers would achieve the same outcome: market Q will equal optimal Q 12 Summary For Pollution: A Negative Externality With negative externality, QMarket QSocial Optium firms over-produce (DWL exist) Remedy: The government can internalize the externality by imposing corrective tax Price tax S’ S Q = Qmarket = initial eqm Q’ = QSocial Optium = eqm after tax D Q’ Q Quantity 13 Externality in Consumption Consumption of alcohol, tobacco, and gasoline (private driving) all create negative externality to the society Got impose a heavy corrective tax on these goods to alter the incentives of customers, in order to mitigate of negative externality 14 Corrective Tax Rate (Levy / Charges) in HK Alcohol: 100% tax rate for alcohol with strength of more than 30% by volume Cigarettes: $1. 7 / each cigarette, tax for a pack of 20-stick cigarettes = $34 70% of the selling price of $50 / pack Leaded petrol: $6. 823/ litre, unleaded petrol: $6. 6/ litre About 40% of the selling price of each litre of gasoline 15 Example: Gasoline Tax Targets 3 Negative Externalities Congestion: the more you drive, the more you contribute to congestion Accidents: larger vehicles cause more damage in an accident Pollution: burning fossil fuels produces greenhouse gases 16 Case Study: Environmental Levy on Plastic Shopping Bags in HK Survey indicates that some 8 billion (8,000,000,000) plastic shopping bags are disposed of at landfills every year in HK This translates into more than 3 plastic shopping bags per person per day, which apparently go beyond our needs 7 Case Study: Environmental Levy on Plastic Shopping Bags in HK Address the problem of abuse, gov introduced an levy of $0. 5 HKD on each plastic shopping bag at the retail level Estimated negative externality of each plastic bag = ? 18 Positive Externality Positive Externality: the effect on bystanders is beneficial Example: When you get a flu vaccination, everyone you come into contact with benefits Research and Development (RD) creates knowledge others can use Renovating your house increases neighboring property values Restores of historical building 19 Positive Externalities from Education A more educated population benefits society: lower crime rates: educated people have more opportunities, so less likely to rob and steal better government: educated people make better-informed voters People do not consider these external benefits when deciding how much education to â€Å"purchase† 20 Positive Externalities from Education Result: market eqm Q of education too low How govt may improve the market outcome: subsidize cost of education In the presence of a positive externality, the social value of a good includes private value: the direct value to buyers external benefit: the value of the positive impact on bystanders 21 Analysis of a positive externality P The market for flu shots $50 40 30 20 10 0 0 10 20 30 S D External benefit = $10/shot Draw the social value curve. Find the socially optimal Q. What policy would internalize this externality? Q 22 Analysis of a positive externality P The market for flu shots $50 40 30 20 10 0 0 10 20 25 30 external benefit S Mkt eqm Q = 20 Social optimal Q = 25 underproduction resulted in DWL (red triangle) Social value = private value + external benefit D To internalize the externality, use Q subsidy = $10/shot. 23 Case Study: Innovation and Technology Policy in HK Should government subsidize high tech companies? Pros: Spillover effects International competitiveness Cons: Potential misallocation of public resource Potential problems of unfairness corruption 24 Case Study: Innovation and Technology Policy in HK Eg: Cyberport IT project? 25 Summary: Corrective Tax and Subsidy to Deal With Externalities If negative externality market produces a larger quantity than is socially desirable If positive externality market produces a smaller quantity than is socially desirable 6 Summary: Corrective Tax and Subsidy to Deal With Externalities remedy the problem: â€Å"internalize the externality† tax goods with negative externalities ideal corrective tax = external cost subsidize goods with positive externalities ideal corrective subsidy = external benefit 27 Private Solutions to Externalities? Government intervention is always controversial Major concerns of government intervention include fairne ss and efficiency The market does develop some possible solutions to externality over time 28 Private Solutions to Externalities? Social norms / moral codes Eg: littering Mergers Eg: MTR as a property developer Contracts between market participants and the affected bystanders However: If an externality affects many people, contract negotiation is virtually impossible 29 Public Policies Toward Negative Externalities Market-based policies provide incentives so that private decisionmakers will choose to solve the problem on their own Corrective Tax Tradable Pollution Permits 30 Public Policies Toward Negative Externalities Command-and-control policies: Regulation regulate behavior directly. Examples: limits on quantity of pollution emitted requirements that firms adopt a particular technology to reduce emissions 31 Policy Option: Example â€Å"Ace Electric† and â€Å"Billy Power† both are running coal-burning power plants Each emits 40 tons of sulfur dioxide per month SO2 causes acid rain other health issues Policy goal: reducing SO2 emissions 25% to 60 tons/month 32 Policy Option: Regulation Vs Corrective Tax Policy options 1. regulation: require each plant to cut emissions by 25% 2. corrective tax: make each plant pay a tax on each ton of SO2 emissions. Set tax at level that achieves goal. 33 Policy Option: Regulation Vs Corrective Tax Under Policy option 1, regulation, firms have no incentive to reduce emissions beyond the 25% target Suppose cost of reducing emissions is lower for â€Å"Ace Electric† than for â€Å"Billy Power† Socially efficient outcome: â€Å"Ace Electric† reduces emissions more than â€Å"Billy Power† 34 Policy Option: Regulation Vs Corrective Tax Corrective tax is a price on the right to pollute Under policy option 2, tax on emissions gives firms incentive to continue reducing emissions as long as cost of doing so is less than the tax If a cleaner technology available, tax gives firms incentive to adopt it Tax payment is money! So, corrective taxes enhance efficiency by aligning private with social incentives 35 Policy Option: Tradable Pollution Permits Recall: â€Å"Ace Electric† and â€Å"Billy Power† each emit 40 tons SO2, total of 80 tons. Goal: reduce 25% emissions to 60 tons/month Policy option 3: Tradable Pollution Permits issue 60 permits, each allows its holder to emit one ton of SO2 give 30 permits to each firm establish market for trading permits 36 Policy Option: Tradable Pollution Permits Each firm can choose among these options: emit 30 tons of SO2, using all its permits emit 30 tons, sell unused permits buy additional permits so it can emit 30 tons 37 Policy Option: Tradable Pollution Permits A system of tradable pollution permits achieves goal at lower cost than regulation Firms with low cost of reducing pollution (Ace Electric) sell whatever permits they can Firms with high cost of reducing pollution (Billy Power) buy permits Result: incentive to reduce pollution: permit = money 38 Tradable Pollution Permits in the Real World Emission of greenhouse gases causes the global warming The primary greenhouse gas in the atmosphere is the emission of carbon dioxide Carbon emissions trading is a form of emissions trading that specifically targets carbon dioxide 39 Tradable Pollution Permits in the Real World Carbon emissions permits traded in Europe since January 1, 2005 Recall: permit = money Firms will have strong incentive to reduce carbon emissions 40 How to cite Externalities, Pollution and Global Warming, Papers

Giotto di Bondone Example For Students

Giotto di Bondone Biography Outline1 Biography2 Key Ideas in painting3 Famous Paintings3.1 â€Å"The Lamentation†3.2 â€Å"The last judgement† Biography Giotto is an Italian painter, founder of protorenesissa. Originally, he is from Cole da Vespignano in Tuscany. He studied at the Ciampayo workshop (between 1280 and 1290), he worked mainly in Florence (where since 1334 he directed the construction of the Cathedral of Santa Maria del Fiore and the city fortifications) and Parma. At the beginning of the 14th century, Giotto visited Rome. The name of Giotto is connected with a new stage of the development of Italian and European art (Protoranesians), the discontinuance of Italian Florence with medieval artistic canons and the traditions of the so-called Italian-Byzantine art, as well as frescoes. Giotto is rightly considered to be the father of European painting, the founder of realism. Numerous disciples, followers, and imitators of the renowned master expanded its influence to many cultural centers of Europe. His biography was commented by Boccaccio, who believed that he could by right be called one of the lights of Florentine glory. The painter is not only wonderful but also beautiful, whose glory is great among the artists, – Boccaccio said. Giotto di Bondone was born in Vespignano, near Florence, probably in 1267, and as it’s known, he has died in 1337 at the age of seventy. According to Vasari, Giotto was the son of a farmer. There is one story known about him. Once, glorified painter Cimabue, who was walking along the road from Florence to Vespignano, saw the herds of a boy who drew a sheep on a smooth stone. Chimabye took him to his students, and, after some time, the student surpassed the teacher. Together with Cimabee, Giotto went to Rome for the first time around 1280 and then visited Assisi. Apparently, at the end of the 80s, a young artist married Jundo di Lapo del Spialla, who had four daughters and four sons. Giotto had the most average, vague look small growth, ugly. In the stories of that time, Giotto acts as a wise man and a joker. Painter spent the next years in Assisi, where he participated in the painting style of the Upper Church of San Francesco. His earliest works include some frescoes on the theme of the Old and New Testaments, in the upper zone of the walls of the longitudinal nave. These paintings were made in the first half of the nineties. It’s believed that during that time Giotto traveled to Rome for a while, where he was impressed by the Cavalline’s paintings and early Christian mosaics. Key Ideas in painting Giotto di Bondone didn’t understand the laws of the prospect, didn’t study the human anatomy, the figures on its frescoes didn’t correspond to their size landscapes. But the three-dimensional world voluminous and tangible is open again, triumphantly approved by the brush of the artist. The symbolism of Byzantine art was rejected. The painter used the simplicity. Nothing superfluous, the attention of the artist in famous paintings is focused on the main and gives a synthesis, a grand generalization. The acquaintance of Giotto with the late antique artwork and the works of Pietro Cavallini contributed to the development of his creative method, the creation of such an outstanding monument of Protoniansansan painting, like the artwork of the Capella del Arena (Skroveniy) in Padua (1304-1308). The Giotto Frescoes, dedicated to the life of Mary and Christ, located on the walls of three horizontal rows, are distinguished by the dramatic and vital convincing images, the bold construction of space, the almost sculptural mold of plastic volume, the simplicity and at the same time the expressiveness of gestures and angles, bright, festive coloring and unique style. .u73575921255a8601354f7a91c361e212 , .u73575921255a8601354f7a91c361e212 .postImageUrl , .u73575921255a8601354f7a91c361e212 .centered-text-area { min-height: 80px; position: relative; } .u73575921255a8601354f7a91c361e212 , .u73575921255a8601354f7a91c361e212:hover , .u73575921255a8601354f7a91c361e212:visited , .u73575921255a8601354f7a91c361e212:active { border:0!important; } .u73575921255a8601354f7a91c361e212 .clearfix:after { content: ""; display: table; clear: both; } .u73575921255a8601354f7a91c361e212 { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u73575921255a8601354f7a91c361e212:active , .u73575921255a8601354f7a91c361e212:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u73575921255a8601354f7a91c361e212 .centered-text-area { width: 100%; position: relative ; } .u73575921255a8601354f7a91c361e212 .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u73575921255a8601354f7a91c361e212 .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u73575921255a8601354f7a91c361e212 .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u73575921255a8601354f7a91c361e212:hover .ctaButton { background-color: #34495E!important; } .u73575921255a8601354f7a91c361e212 .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u73575921255a8601354f7a91c361e212 .u73575921255a8601354f7a91c361e212-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u73575921255a8601354f7a91c361e212:after { content: ""; display: block; clear: both; } READ: Ivan ShishkinThe work of Giotto was distinguished by the enormous graphics power and dramatic expressiveness of his images. The impact on the viewer was stronger, the more natural and realistic were the artistic techniques of Giotto, who set himself the task of representing a real three-dimensional space constructed on a linear perspective basis. Without a knowledge of the scientific perspective, it would seem that the means of Giotto, seemingly primitive, had reached the unprecedented effect of the art of his time, one of the first to enter the artwork of the interior. At the same time, Giottos second conquest was the creation of plastic, rounded human figure, and each of his depicted figures possessed purely material merit and, by virtue of this profound vitality, realistic conviction. The most famous paintings of the master are: â€Å"Adoration of the Magi†, â€Å"Dante Alighieri†, â€Å"Madonna in Glory†, â€Å"Massacre of the Innocents†, â€Å"Nativity: Birth of Jesus†, as well as â€Å"The Lamentation† and â€Å"The last judgement† you’ll read about below. Famous Paintings â€Å"The Lamentation† The frescoes of the Italian painter and architect Giotto marked the birth with the discovery of a completely different page in the narrative of art history. Until the heyday of his work, Italian sculptors followed the rules of medieval and Roman antique drawings traditions. Giottos architecture breaks the connection with the technique of painting the past up to the complete refusal to follow its laws. Florentines believe that the famous master has crossed the threshold in the new era of art. The general emotional background of the painting The Lamentation is the ingenious artistic reception of Giotto. All participants of the event are involved in the occurrence, and the position of their bodies, the details are drawn by the folds of clothing, the ratio of dimming and illuminated areas create an incredibly realistic spatial depth of the artwork. In the focus of the drawing, there are the reduced faces of the murdered Christ and the Mother. It’s precisely in this center of the greatest emotional stress that all objects and characters of the canvas are comfortably directed. The views of the heavenly and terrestrial participants of the sorrowful scene are visually and mentally immersed in the events. â€Å"The last judgement† In the center of â€Å"The last judgement,† in an oval frame supported by angels, Jesus sits in all his splendor. To the left and to the right sit the apostles, each on a separate throne (the most luxurious throne belongs to the apostle Peter). There are the angels arms above them, the slender ranks of the heavenly host. The most interesting are two separate angelic figures under the very vault. They proclaim the discovery of the New Jerusalem and, as it were, wrap the edges of the canvas or parchment, emphasizing the illusory nature of what is happening. It can be interpreted in two ways. Either all the earthly manifestations are illusory before the higher heavenly reality. Or Giotto alludes to us that everything he wrote on the wall of the Scrovegni Chapel, with all the naturalism, is just art of architecture and Florence.