Жас ғалымдардың VII халықаралық Ғылыми конференциясының материалдары 25-26 сәуір 2011 жыл
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Торебекова А.О. Проектирование и разработка программных продукт «Компьютерная система формирование приказов»………………………………………………………….............. 243 86. Тюрина Е.Л. Информационные технологии в экономике………………………….…………….. 245 87. Уайсова М.С. Информационные технологии в маркетинге………………………….................... 248 88. Увалиев Б.М. Компоненты систем управления обучением ........................................................... 251 89. Утепова А.Т. Разработка робототехнических систем с искусственным интеллектом........................................................................................................................................... 253 90. Хаймуллина Б.Ш. Значение информационных технологий в современном обществе……………………………………………………………………………………………… 256 91. Шангытбаева Г., Бектурганова С. Оқу-әдістемелік кешендер клиент-серверлі бағдарламасын қҧру.............................................................................................................................. 258 92. Шаңғытбаева Г.А., Медеуова А.Б. Реляциялық деректер базасының негізгі ҧғымдары............................................................................................................................................... 260 ПОДСЕКЦИЯ 1.4 ФИЗИКА 93. Абдрахметова А.А. Импульсная катодолюминесценция активированного кристалла LiF-OH при комнатной температуре................................................................................................................ 265 94. Амангалиева Р.Ж. Зарядка пылевых частиц в плазме стратифицированного тлеющего разряда постоянного тока………………………………………………………………………......... 269 95. Аханова А.Е. Анализ схемотехники и надежность преобразователей частоты……………….…….…………………………………………………………………………. 271 96. Баданова Ж.К. Потоки Риччи для трехмерной части горловинного решения……………………………………………………………………………………………….. 278 97. Бауыржанкызы Г. Спектральные свойства полимерных композиций на основе поливинилового спирта…………………………………………………………………………….... 281 98. Даулетбакова О.Б. Решение уравнений Эйнштейна для модели кротовой норы….................................................................................................................................................... 285 99. Демидова Д. А., Русакова А.В., Машенцева А.А., Здоровец М. В. Влияние добавки этанола на скорость формирования трековых мембран………………………………………...................... 287 100. Есимов М.М., Рыскулов А.Е. Изучение процесса нанофрезерования с помощью атомно- силового микроскопа………………………………………………………………............................ 291 101. Есликовский К.А., Касимова Б.Р. Потери электроэнергии на электростанциях…………………………………………………………………………………….... 293 9 102. Жоламан А., Куандык Г. Перколяция, маркировка перколяционных кластеров и задача сфер........................................................................................................................................................ 295 103. Забелин С.А. Интерактивная очистка спутниковых снимков от атмосферной дымки…………………………………………………………………………………………………. 298 104. Иванов И.А. Реализация методики HIPIXE на ускорителе тяжелых ионов ДЦ- 60……………………………………………………………………………………………………… 300 105. Ильясов Б.М., Абдыгапаров А.К. Разработка новой фотоакустической камеры для анализа конденсированных сред. (ЕНУ им. Л.Н. Гумилева, г. Астана)…………………………………... 302 106. Имангалиева А.Н. Автоматизированная система расчетов оплаты телекоммуникационных услуг…………………………………………………………………………………………………... 304 107. Карманова А.Т. Изотропты диэлектрик және анизотропты диэлектрик шекарасындағы электромагниттік толқындардың шағылу және сыну есептерін матрицант әдісімен шешу...................................................................................................................................................... 306 108. Касымканова Р.Н. Модифицированный метод решения краевых задач электродинамики на основе задачи Зоммерфельда……………………………………….................................................. 310 109. Кемельбекова Г.М., Исабаева Г., Жанботин А. Дилатометрические исследования монокристаллов цезий-литиевого молибдата CsLiMoO 4 ……………………………………......... 311 110. Кенежанова А.Т. Физикалық процестерді Excel электрондық кесте кӛмегімен модельдеу.......................................................................................................................................... 313 111. Козловский А.Л. Оценка активности ядерных процессов на Солнце с применением мультифрактального анализа динамики солнечных пятен……………………………………....... 319 112. Кутумова Ж.Б. Использование мультифрактального анализа для классификации поверхностей твердых тел. (ЕНУ им. Л.Н. Гумилева, г. Астана)……………………………….... 322 113. Кустубаева Л.Е. Уравнение движения частиц в кротовой норе…………………….…………………………………………………………………………….... 325 114. Мирзанова Б.Б. Радиоқабылдағыш қҧрылғылар туралы жалпы мәліметтер. Радиоқабылдағыш қҧрылғының арналған мақсаты, қҧрылымы және әрекет ету принципі................................................................................................................................................ 328 115. Мухышбаева А.К., Абдукадиров Б.З. Электронно-микроскопические исследования кристаллов фторида лития и магния................................................................................................... 337 116. Онгарқызы А. Қазақстан республикасы ауылшаруашылығы саласында жерді қашықтықтан зондтау әдісін қолдану......................................................................................................................... 340 117. Рамазанова Р.К. Природа термолюминесценции в облученных сульфатах щелочно- земельных металлов............................................................................................................................. 343 118. Сауханбек Е.Н. Исследование процессов с 4b-струями в эксперименте ATLAS……………………………………………………………………………………………….... 344 119. Темиргалиева А.Е. Разработка системы управления с повышенным потенциалом робастной устойчивости на примере летательного аппарата............................................................................. 344 120. Толстойқызы А. Қазақстан республикасына жерді қашықтықтан бақылаудың GEONETCAST мҥмкіншіліктерін қолдануды енгізудің маңыздылығы......................................... 347 121. Шлимас Д.И., Машенцева А.А., Здоровец М.В. Изучение прочности трековых мембран на основе ПЭТФ в химически агрессивных средах……………………………………………………………………………………………......... 349 122. Шмелѐв И.А. Моделирование схем и печатных плат приборов ядерной электроники в Altium Designer…………………………………………………………………………………...................... 351 ПОДСЕКЦИЯ 1.5 ХИМИЯ 123. Әуелханқызы М., Лесбаев Б.Т., Нажипқызы М. Образование гидрофобной сажи в углеводородных пламенях................................................................................................................... 354 124. Буланаева А.М. Влияние серосодержащих соединений на свойства нефтепродуктов и физико-химические основы демеркаптанизации нефти и нефтепродуктов................................... 360 10 125. Ерметова Ю.Р. Қазақстанның эфирмайлы ӛсімдіктерін зерттеудің ӛзектілігі............................. 362 126. Зулхарнай Р.Н. Химическое изучение Artemisia Lercheana Kar.Et Kir.......................................... 366 127. Рахишева А.Д. Исследование теплопроводности и теплоты сгорания углей различных пластов разреза (Северный Экибазтузский бассейн)........................................................................ 371 128. Салыкбаева А.С., Исаева С.Х. Экстракция ионов никеля (ІІ) расплавом стеариновой кислоты.................................................................................................................................................. 372 11 СЕКЦИЯ 1 ЕСТЕСТВЕННО-НАУЧНОЕ НАПРАВЛЕНИЕ ПОДСЕКЦИЯ 1.1 МАТЕМАТИКА УДК 517.6 PROJECT TITLE APPLICATIONS OF THE CONCEPT OF DERIVATIVE OF A FUNCTION IN PRODUCTION ECONOMICS Abdirasev Omirzak Koptileuuly Student, Gumilyov Eurasian National University, Astana Scientific teacher - Baimadieva Galiay Abilakimkyzy Abstract Originality. Organization of production within firms is the backbone of a market economy. Increasing the efficiency of such organization constitutes an important objective of any management effort. If the organization of production can be described in the mathematical language, mathematics can be exploited to discover new ways of optimizing production. Thus, the subject of this project is a highly actual in today‘s economy. Project objectives. The objective of the project is to apply some of the properties of the derivative of a function to find out results for production economics applicable in real life. The first objective is to show how the derivative can be applied to determine the most technically efficient level of production of a firm. The second objective is to apply some of the rules of differentiation to finding the relationship between different cost functions of the firm. Scientific novelty. Scientific novelty of the project lies in applying the well-known and standard mathematical properties of derivatives of a function to explore novel results for production economics. Theoretical and applied value. The results found in the project have an important theoretical value for the economic theory of a firm and a key applied value for finding most technically efficient and least costly way of organizing the production within the firm. Research methods. The method used in the project consists of building a simple general mathematical model of the organization of production within a firm. Next, we analyze the functions used in the model and exploit mathematical theory to find several interesting properties of these functions. Then, we translate these properties back into economic language. Finally we present examples of practical application of these properties for the efficient organization of the firm production. PRODUCTION FUNCTIONS Consider a technology of production with one input only (e.g. hours of labor). Mathematically, let us describe this technology with a function ) (q f y where q measures the quantity of input and y measures the quantity of output produced. For example, if , 75 ) 10 ( , 5 ) 1 ( f f it means that using this technology of production, 1 hour of labor can produce 5 units of output and 10 hours of labor can produce 75 units of output. In this example (and in general), the relationship f does not have to be linear, because workers can specialize and produce more efficiently at a larger scale. Let us also assume that the function f has the following properties: 12 (1) f is twice differentiable continuous; (2) f is increasing; (3) there exists some a such that for , 0 a q f is convex and for q>a, f is concave. From the economic point of view, property 1 means that both input and output is divisible quantities and that at any level of production, small changes in the quantity of input imply small changes in the quantity of output. Property 2 means that, naturally, increasing the quantity input cannot decrease the output. Property 3 means that up to some level of input a, the technology of production becomes more and more efficient (output increases more than proportionally when input increases), and that beyond this level, the technology becomes less and less efficient as the scale of production keeps increasing (output increases less than proportionally when input increases). This latter assumption is plausible, because as the scale of production increases beyond some point, the firm has to incur additional management costs. For example, take first a small firm run by its owner. When he increases the number of workers, they can start to specialize in producing different parts of the final product (think of an assembly line) and this specialization implies that the production becomes more efficient. However, if the owner keeps increasing the number of workers, at some stage, he will be unable to manage the firm alone, and will have to hire one or several managers. This means that the cost of coordinating the workers starts to increase, and therefore the technology of production becomes less efficient. An example of such a function is given on Figure 1. Figure 1. An example of a production function 1) Using the concept of derivative of a function, we can rewrite the properties 2 and 3 in a very simple and mathematically handy form as follows: (2‘) f’(q)>0 for any q ; (3‘) there exists some a≥0 such that: f”(q)>0 for 0≤q The usefulness of this application of the concept of the second derivative in applied work can be illustrated as follows. From Property 3, we can see that given that the second derivative of f changes its sign before and after the level q = a, at this level the second derivative of f must be zero. Suppose now that we have data, based on past production of the firm, on different levels of input and the corresponding quantities of output produced. Using simple statistical techniques, we can find a twice-differentiable continuous function (e.g. a third-order polynomial) that describes best these data. For example, let this function be f(q) = 5q 3 – q 2 + 8q – 25 What is the level of production at which the efficiency of production is the highest? It is difficult to answer this question by only looking at the data. However, using our method, is sufficient to find at which q the second derivative f”(q) becomes zero: f”(q) = 30q – 2 = 0 q = 15. Q y f(q) 13 We have thus very easily found the most efficient level of production, using the concept of the second derivative of a function. In most economic models, the following class of two-parameter production functions is used: y=kq b ; here, k and b are positive constants. It is easy to see that, depending on the value of the parameter b determines, the shape of the production function is either everywhere convex or everywhere concave. In fact, let 0 f” = kb(b-1) < 0, which means that for any q, the function f is concave. On the other hand, let b>1. Then, f” = kb(b-1) >0, which means that for any q, the function f is convex. COST FUNCTIONS If the prices of inputs are known, we can also describe the financial side of production, using a cost function. Let us describe the total cost of producing the quantity of output y by C(y). Normally, function C has the following properties: (1) C is twice differentiable continuous; (2) C is increasing. The following two related concepts are important for financial and economic decisions of the firm: marginal cost m(y) and average cost c(y). Marginal cost m(y) indicates how much an additional unit of output costs to the firm. Mathematically, it is simply the first derivative of the C(y) total cost function: m(y) = C’(y) Average cost c(y) indicates, for any level of production y, how much each unit of output produced costs for the firm, on average. Mathematically, c(y) = C(y)/y How are these two functions related to each other? We can find it out using the properties of the derivative. In particular, we can prove the following proposition. Proposition. If m(y) > c(y), then c(y) is increasing. If m(y) < c(y), then c(y) is decreasing. At the minimum of c(y), m(y) = c(y). Proof: Using the rules of differentiation, we have y y c y m y y y C y C y y C y y C y y C dy d y c ) ( ) ( ) ( ) ( ' ) ( 1 ) ( ' ) ( ) ( ' 2 Then, if m(y) > c(y), then c’(y)>0, i.e. c(y) is increasing. If m(y) < c(y), then c’(y)<0, i.e. c(y) is decreasing. Denote the point at which m(y) = c(y) by y 0 . We see that at this point, c’(y 0 ) = 0. This implies that y 0 is an extremum point of the function c(y). Therefore, at the minimum of c(y), m(y) = c(y). QED. These relationships between the different cost functions are depicted on Figure 2. Figure 2. Relationship between cost functions 2) y m, c c(y) m(y) 14 Normally, the cost of function of the firm has the form C(y) = A + B(y) where A is a constant. A thus denotes the fixed cost: for example, even before starting to produce the first unit of output, the firm has to install the machines, rent an office, etc. All these expenditures are independent of the level of output. Even if the firm produces zero units of output, C(0) = A > 0. B(y) denotes the variable cost of production, normally these are expenditures associated with hiring workers, buying raw materials, etc. Let‘s suppose, for example, that A = 125 and that B(y) is a quadratic function: B(y) = 5y 2 How can we use the proposition derived above to find the output level that has the lowest average cost of production? The marginal cost is m(y) = d[125+5y 2 ]/dy = 10y The average cost is c(y) = [125+5y 2 ]/y = (125/y) + 5y From the Proposition above, it is sufficient to find the level of output such that marginal and average costs are equal: (125/y) + 5y = 10y y = 5. We have thus very easily found the level of production with the lowest average cost, thanks to our proposition that exploits some of the differentiation rules. жүктеу/скачать 8,59 Mb. Достарыңызбен бөлісу:
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