Curve Sketching

Lg

Description

• exposé

L E S S O N 25 25-1 Curve Sketching Objectives 1. To learn how to use the first and second derivatives to find a good sketch of the graph of a function f. 2. To identify the points needed to frame the graph of a function f. 3. To learn how to determine the horizontal and the vertical asymptotes for a function. 4. To learn how to use infinite limits to locate asymptotes.

• find intervals of upward concavity
• horizontal asymptote of the curve
• inflection points
• 25.4 find
• appropriate viewing window
• intervals
• find
• graph
• function

Sujets

• Exposé
• Function
• Interval
• Concave
• Inflection
• View
• Sample

Informations

• Publié par: nosheng
• Nombre de lectures: 18
• Langue: English

Extrait

Ruben Seroussi Abel Ehrlich (3.ix.1915 – 30.x.2003): Now that he is gone Abel Ehrlich is no longer with us. In the face of this undeniable fact a troubling thought arises: when he was still among us, even among his close friends who met with him from time to time or who kept in touch by frequent telephone calls, was he then more ‘with us’ than he is now? More ‘with us’ than at those times when we listen to or perform his works? Ehrlich gave himself totally to creativity. There was a kind of covenant within him – between his own living organism and the living organism of his creation; between his uniqueness, even capriciousness, and his neverending capacity to surprise us in his music. Can we say that such a person has ever truly been ‘with us’? Alongside the natural grief that his death caused his close friends and his admirers (his demise was, considering his physical state, as surprising as would be that of a young man), his presence in my consciousness brings with it that whiff of thrilling promise of a new discovery in the bottomless treasury of his compositions. There is the everincreasing feeling of certainty that the covenant mentioned above ensured not only a mirrorlike relationship between his organic and creative activities, but bestowed also a private yet genuine kind of eternity. Thus his presence remains very real. Ten years ago I wrote that, if Israeli music is to take root, it must come to terms with Ehrlich’s work. Today, in view of the enormous corpus of works he has left us, there is another aspect to this essential ‘initiation rite’: the urgent need to preserve Ehrlich’soeuvre, to catalogue the source materials and to make them easily accessible to performers and researchers. After his death I participated in arranging two concerts devoted to his music (for details of the second concert see Calendar, December 2004). In order to organize both of them I had to go through the material (approximately 1500 works) currently housed at the offices of the Israel Composers’ League, following the conservation and cataloguing work begun by the musicologist and friend, Dushan Mihalek. With so many works available for any given instrumental combination, our choice of works for both concerts, according to titles and dates of composition, was almost random. 1

Time and again we faced the same phenomenon: unknown pieces revealed themselves as veritable gold mines. Of course, to gain insight into the true value of these works, performing musicians have to analyse the music in profound detail, carrying out the ‘detective’ work which is indispensable for properly serious performance of any piece of music, whatever its style or period. Unfortunately, such attention, essential for the future of Ehrlich’s music, cannot be taken for granted. Having listened to performances of his music for many years, I have come to realise that many performers do not even begin to investigate the nature and quality of these works. Apparently, they consider it an achievement simply to play the notes as they are written, without searching for the links between the notes and without trying to understand the work’s message. In my experience, Ehrlich creates a whole new syntacticfunctional mechanism in each and every one of his works. This always reminds me of the image of the boy who plays with sticks and strings, drawn from Ehrlich’s own childhood stories. He told this story at one of his public appearances, at a concert at which one of his works was performed (Israeli Music Days, Jerusalem Theatre, 1999). Much to the orchestra’s surprise and halfhearted tolerance, they found themselves listening to an old man speaking of what was really important, disregarding completely the formal constraints that the situation demanded. When he was just three years old, he saw at the playground the very thing he wanted most of all: a game of sticks and strings which could be shaped into any pattern you may wish. Since then, he found in musical notation a substitute for those sticks and strings; to him they represent, quite clearly, the dimension of time and melody in music. Every work is a new ‘game’ and it must be interpreted in a very specific way, for the rules vary from game to game. If we add to that Ehrlich’s lifetime of musical experience, strewn as it was with the profound quests and challenges which characterise his development, we will come to realize how much we stand to gain by proper research and study of his works. In his workFrom the Diaryfor solo guitar, which he dedicated to me in 2001, he reacted wittily to my usual demand for a polyphonic composition for the instrument, “without any inferiority feelings”. He created a complex, intricate polyphonic texture (or even more than just that) – even though there are very few instances when two notes are played simultaneously. The use of silence is essential for separating the various strata in this ‘imaginary’ polyphonic texture, creating the

2

illusion that several different lines are being played simultaneously. He also managed to merge, with his extraordinary compositional virtuosity, pitched notes and un pitched sounds (as well as equaltempered and microtonal pitches) in an organic, motivic fashion. Yet all these features could easily be lost if the performers do not do their job: without seeking and discovering a specific and appropriate way to analyse the musical text, we would probably have perceived the work as one melodic line, undefined and unexplained. When I performed the work during my latest tour of Buenos Aires and Montevideo, it so greatly impressed one of the leading classical guitarists of our time, the Uruguayan Eduardo Fernández, that he later sat with me for a long time studying a copy of the manuscript and asked to see the mapping of events as well as the special means I had chosen to produce the sounds. This did not surprise me at all, but I would have been much happier (to put it mildly) if more Israeli musicians had shown such enthusiasm and had paid such close attention to Ehrlich’s music. In my 1995 article (A clear labyrinth: Reflection on Abel Ehrlich’s music, in a special issue ofIMI Newsdedicated to Abel Ehrlich), I attempted to explain why Ehrlich’s music is not accepted in Israel. I mentioned his prolific output – several thousands of compositions – which was alltoofrequently associated with lack of quality; his whimsical titles, which belie the profundity and intensity of his music; and his clear avoidance of heroic gestures or extroverted pathos, which made it harder to grasp his music immediately, without attentive analysis. I also described the lack of avantgarde sophistication or pseudointellectual wit in his music, features that would have made it easier for musicians specialising in contemporary music to classify these works as ‘relevant’. At the time, I spoke of Ehrlich’s “noble aspiration to insignificance”. This special concept was not quite clear to him; but for me, it remains the sign of the creative artist who has risen above the petty strategic considerations of everyday life, and could do so because of the breadth and clarity of his vision in assessing the situation. This was the source of the internal covenant I described above. Time is the basic material of music and of real life alike; and Ehrlich used almost all of his own time for music. He genuinely treated creativity as a way of life. I clearly recall Ehrlich’s own reaction (both spoken and unspoken) to this analysis. Without saying so, he made it clear that just as one does not argue with

3

success, one does not dissect or explain failure. This, of course, is quite right. The ability to accept the incomprehensible without attempting to explain it and, worse still, trying to justify it, is the first condition for maintaining your sanity. On the other hand, we can still pinpoint specific factors in this tale of ‘failure’. It will suffice to mention the most obvious one. Here is a composer who lived in this country from 1939 until his death in 2003, whose music has become wellknown, who was part of the ‘canon’ already in the early 1950s, who wrote over a hundred orchestral works – and yet none of them has ever been performed by our Philharmonic Orchestra. How much ridicule and silly supercilious smirks did Ehrlich have to endure? He has spoken to me only of a small part of his prolonged, uneasy relations with the established performing bodies and the academia. Today, we are well aware (and whoever is not, had better look into the matter again, even if he feels that things are going well for him…) that Israeli music still waiting to be recognised as a respectable entity on the strength of its own intrinsic value – and not because of the functional aspects on which it is doomed to depend. Ten years ago I wrote that the history of art music in Israel is still waiting to be written; at present I can see clearly the honourable place Ehrlich’s music is destined to take in this narrative, but I still wonder whether it will be possible to contain and digest this limitless world within any defined framework. I concluded my 1995 article by quoting Borges’s poemSpinoza, whose setting for voice and guitar by Abel Ehrlich remains, sadly, unperformed. I found, and still do, that Borges’s words capture the essence of Ehrlich’s art and personality. SPINOZA/ J. L. Borges The Jew’s hands, translucent in the dusk, Polish the lenses time and again, The dying afternoon is fear, is Cold, and all the afternoons are the same. The hands and the hyacinthblue air That whitens at the Ghetto edges Do not quite exist for this silent Man who conjures up a clear labyrinth – Undisturbed by fame, that reflection 4

Of dreams in the dream of another Mirror, nor by maiden’s timid love. Free of metaphor and myth, he grinds A stubborn crystal: the infinite Map of the One who is all his stars. Translators: Richard Howard & Cesar Rennert From Jorge Luis Borges,Selected Poems, 19231967(Boston: Seymour Lawrence, 1972) Ruben Seroussi is a composer, guitar player and lecturer at the BuchmannMehta School of Music, Tel Aviv University IMI News2005/1, pp. 23 © Israel Music Institute, 2005

5