Saturday, January 25, 2020
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€ ’43à ¢Ã‹â€ ’122+3 on the interval [à ¢Ã‹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€ ’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€ ’43à ¢Ã‹â€ ’122+3 has first derivative f(x) = = = 123à ¢Ã‹â€ ’122à ¢Ã‹â€ ’24x 12x(x2à ¢Ã‹â€ ’xà ¢Ã‹â€ ’2) 12x(x+1)(xà ¢Ã‹â€ ’2)  Thus, f(x) is increasing on (à ¢Ã‹â€ ’10)(2) and decreasing on (à ¢Ã‹â€ ’à ¢Ã‹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€ ’43à ¢Ã‹â€ ’122+3 is differentiable everywhere on [à ¢Ã‹â€ ’23], with f(x)=0 for x=à ¢Ã‹â€ ’102. These are the three critical points of f on [à ¢Ã‹â€ ’23]. By the First Derivative Test, f has a relative maximum at x=0 and relative minima at x=à ¢Ã‹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€ ’43à ¢Ã‹â€ ’122+3 is continuous on [à ¢Ã‹â€ ’23], f must have an absolute maximum and an absolute minimum on [à ¢Ã‹â€ ’23]. We simply need to check the value of f at the critical points x=à ¢Ã‹â€ ’102 and at the endpoints x=à ¢Ã‹â€ ’2 and x=3: f(à ¢Ã‹â€ ’2) f(à ¢Ã‹â€ ’1) f(0) f(2) f(3) = = = = = 35Â à ¢Ã‹â€ ’2 3Â à ¢Ã‹â€ ’29 30  Thus, on [à ¢Ã‹â€ ’23], f(x) achieves a maximum value of 35 at x=à ¢Ã‹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€ ’43à ¢Ã‹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.
Friday, January 17, 2020
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
Thursday, January 9, 2020
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
Wednesday, January 1, 2020
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,
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