Методическая разработка урока математики на тему «Определенный интеграл, его вычисление и свойства»

Государственное областное бюджетное

профессиональное образовательное учреждение

«ЛИПЕЦКИЙ ПОЛИТЕХНИЧЕСКИЙ ТЕХНИКУМ»

Методическая разработка

урока математики

«Определенный интеграл, его вычисление и свойства».

Выполнил: преподаватель математики

Заварзина В.Г.

Липецк 2016

Тема: «Определенный интеграл, его вычисление и свойства».

Цели урока:

Образовательные: 1) Систематизировать практические и теоретические знания, выработать умение находить неопределенный и определенный интегралы.

2) Развивать культуру устного вычисления определенных интегралов.

3) Содействовать развитию у студентов умений осуществлять самоконтроль, самооценку.

Развивающие: Развивать мышление и речь студентов, развивать навыки самостоятельного мышления, интеллектуальные навыки (анализ, синтез, сравнение, сопоставление), внимание, память.

Воспитательная: Содействовать воспитанию интереса к математике, активности, мобильности, аккуратности и внимательности.

Задачи урока:

1.Развитие познавательного интереса к предмету.

2.Воспитание самостоятельности, настойчивости при достижении конечного результата.

3.Формирование культуры учебной деятельности и информационной культуры.

Тип урока: комбинированный.

Методы обучения: фронтальный, индивидуальный, наглядно-практический.

Формы организации взаимодействия на уроке: учебная, групповая работа, индивидуальная работа.

Оборудование и наглядные средства обучения: мультимедийный проектор, интерактивная доска, презентация, демонстрационный и раздаточный материал.

Методическая цель: активизировать мыслительную деятельность студентов.

Ход урока.

1.Организационный момент.

Подготовка студентов к уроку (проверка отсутствующих на уроке, наличие тетрадей)

2. Актуализация опорных знаний.

Преподаватель.

Давайте вспомним историю возникновения интегралов.

Прошу послушать историческую справку об интеграле.

Историческая справка.

Сообщение студента.

Понятие интеграла и интегральное исчисление возникли из потребности вычислять площади (квадратуру) любых фигур и объёмы (кубатуру) произвольных тел.

Предыстория интегрального исчисления восходит к древности. Ученый, создавший интеграл. Евдокс Книдский (живший около 408-355 гг. до н.э.) – древнегреческий учёный. Он дал полное доказательство теоремы об объёме пирамиды; теоремы о том, что площади двух кругов относятся как квадраты их радиусов. При доказательстве он применил так называемый метод «исчерпывания», который нашёл своё использование (с некоторыми изменениями) в трудах его последователей.

Через две тысячи лет метод «исчерпывания» был преобразован в метод интегрирования, с помощью которого удалось объединить самые разные задачи – вычисление площади, объёма, массы, работы, давления, электрического заряда, светового потока и многие, многие другие.

Что представляет собой «метод исчерпывания» рассмотрим на простом примере.

Предположим, что надо вычислить объём лимона, имеющего неправильную форму, и поэтому применить какую-либо известную формулу объёма нельзя. С помощью взвешивания найти объём также трудно, так как плотность лимона в разных частях его разная.

Поступим следующим образом. Разрежем лимон на тонкие дольки. Каждую дольку приближённо можно считать цилиндром, радиус основания, которого можно измерить. Объём такого цилиндра вычислить легко по готовой формуле. Сложив объёмы маленьких цилиндров, мы получим приближенное значение объёма всего лимона. Приближение будет тем точнее, чем на более тонкие части мы сможем разрезать лимон.

Вслед за Евдоксом метод «исчерпывания» и его варианты для вычисления объёмов и площадей применял древний учёный Архимед. Успешно развивая идеи своих предшественников, он определил длину окружности, площадь круга, объём и поверхность шара. Он показал, что определение объёмов шара, эллипсоида, гиперболоида и параболоида вращения сводится к определению объёма цилиндра. Выражаясь современным языком, Архимед определил интегралы.

Что же такое интеграл? Слово «интеграл» произошло от латинского integer — целый, то есть целая, вся — площадь. Термин был предложен в 1696 г. Иоганном Бернулли. 

Современное обозначение неопределенного интеграла было введено Лейбницем в 1675 году. Он адаптировал интегральный символ , образованный из буквы S — то есть от сокращения слова латинского  summa (сумма). Преподаватель.Давайте повторим первообразные некоторых элементарных функций.

Преподаватель.

А теперь вспомним правила нахождения первообразных и формулу Ньютона-Лейбница.

Преподаватель.

Сейчас 4 студента выполнят тест на время. 30 секунд на обдумывание.

Ответы писать на листах теста.

1.Если для любого х из множества Х выполняется равенство F´(x) = f(x), то функцию F(x) называют … для функции f(x) на данном множестве.

А) производной; В) первообразной; С) обратной; D) непрерывной.

2.Совокупность всех первообразных функций F(x) + С для данной функции f(x) называется … функции f(x)

А) область определения; В) производной; С) область значения; D) неопределенным интегралом.

3.С помощью формулы Ньютона – Лейбница находят…

 А) определенный интеграл; В) производную; С) обратную функцию;

D) неопределенный интеграл.

4.    Найдите множество первообразных для функции f(x) = 2

А) 0;  В) 2х + С;   С) 2х;   D) 2.

5. Криволинейной трапецией называется фигура, ограниченная сверху графиком функции у = f(x), снизу осью … , с боков прямыми …    .

А) непрерывной функции; Ох; х = а, х = b;    В) непрерывной, неотрицательной; у = а, у = b; Ох. С) непрерывной, неотрицательной; х = а, х = b; Оу.  

Ответы:

В это время студенты устно выполняют следующие задания.

Устная работа: (задания записаны на доске; цена ответа 1 балл)

1) Исправить ошибки в записи _;

Ответ: ( 2x + C)

_.

Ответ: (– 5x + C)

2) Найти интеграл:_dx;

Ответ: (_

_ ;

Ответ: ( _ + C = - _

_ .

Ответ: ( 5 ln │x│)

3) Вычислить: _ ;

Ответ: (1)

_

Ответ: ( sinπ - sin0 = 0 ).

3.Закрепление знаний.

Преподаватель.

Давайте решим следующие задания:

(Студенты выполняют задания.)

Задание 1.

Найдите неопределенные интегралы:

1. _; 2) _; 3) _; 4) _

Ответы: 1) _; 2) _; 3) _ - 6x + C; 4)_

Задание 2.

Решаете на этих же листах.

1) _; 2)_; 3)_;

4) _.

Ответы:

1) _

2)_;

3)_

4)_ + cos x+ C.

4. Самостоятельная работа.

Преподаватель.

А теперь сделайте задания самостоятельно.

Студенты выполняют задания по вариантам.

I Вариант.

Найдите определенные интегралы:

1)_ 2) _ 3)_ 4)_

Ответы: 1)_; 2)_; 3)_ 4)_

II Вариант.

Найдите определенные интегралы:

1)_ + 5)dx; 2)_ 3)_; 4)_

Ответы: 1)_ 2)_ 3)_ 4)_

5.Домашнее задание.

Преподаватель.

Вычислите интегралы:

1)_ 2) _ 3) _ + sinx)dx; 4) _

Ответы: 1) _ 2) – 3 cosx + _ 3) _ 4) 5sinx + _

6.Подведение итогов.

Какие вычисления мы производили сегодня на уроке?

Рефлексия деятельности студентов на уроке.

-Что понравилось на уроке?

-Какой материал был наиболее интересен?

-Оцените свою работу на уроке: плохо работал, хорошо, отлично. Поднимите руки, кто работал плохо? Почему? И т.д.

- Как вы думаете, пригодятся ли вам знания данной темы в вашей будущей профессии?

Выставление оценок .

Список литературы:

1. Дидактические материалы по алгебре и началам анализа для 11 класса /Б.М. Ивлев, С.М. Саакян, С.И. Шварцбурд. – М.: Просвещение, 2003.

2. Алгебра и начала анализа: Учеб. для 10–11 кл. общеобразовательных учреждений / А.Н. Колмогоров, А.М. Абрамов, Ю.П. Дудницын и др.; Под. ред. А.Н. Колмогорова. – М.: Просвещение, 2004.

3. Алгебра и начала математического анализа. В 2 ч. Ч. 2. Задачник для учащихся общеобразовательных . учреждений (профильный уровень)/А.Г. Мордкович и др. ; под редакцией А.Г. Мордковича—7-е изд., стер.—М.: Мнемозина, 2014.

4. Задачи по алгебре и началам анализа: Пособие для учащихся 10–11 кл. общеобразоват. учреждений /С.М. Саакян, А.М. Гольдман, Д.В. Денисов. – М.: Просвещение, 2003.

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Определенный интеграл, его вычисление и свойства

Цели урока :

Систематизировать практические и теоретические знания, выработать умение находить неопределенный и определенный интегралы.

Свойства определенного интеграла

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Свойства определенного интеграла

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7. ∫ a b f ( x ) dx ≥ 0 size 12{ Int cSub { size 8{a} } cSup { size 8{b} } {f left (x right ) ital "dx" = 0} } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAcQJAAAd BQAATgAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAMMJAAAcBQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAAAAAAAGgCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAATwMAAAEAAAArImQBAAAAAP// AQArIowAAQAAAAAAiwABAAIAAAAfAIoAAQBEAAAAAwA+AAAAEgBOaW1idXMgUm9tYW4gTm85 IEwAAAAAAAAZAQAAAAADAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEA hwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAA1AAAANwEAAABAGGNAAAAAAD//wEA YQCMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBM AAAAAAAAGQEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAANQAAAAJAQAAAQBijQAAAAAA//8BAGIA jAABAAAAAACLAAEAAgAAAB8AigABAEQAAAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAA AAAAAKYBAAAAAAMAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEA BQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAABGAgAAGgMAAAEAZnUAAAAAAP//AQBmAIwA AQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAADiAAAA1gEAAP// AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCG AAEABAAAAAAAAAByAAIAGgAAADQDAAAaAwAAAQAAACgATQAAAAAA//8BACgAjAABAAAAAACL AAEAAgAAAB8AigABAEQAAAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAAKYBAAAA AAMAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8A hgABAAQAAAAAAAAAcgACABcAAADTAwAAGgMAAAEAeLsAAAAAAP//AQB4AIwAAQAAAAAAiwAB AAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAADiAAAA1gEAAP//AAAAAAUAAAAA AAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAA AAByAAIAGgAAAMEEAAAaAwAAAQAAACkATQAAAAAA//8BACkAjAABAAAAAACLAAEAAgAAAB8A igABAEQAAAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAAKYBAAAAAAMAAAAFAAAA AAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAA AAAAcgACABoAAABFBQAAGgMAAAIAZHiOAQAAAAD//wIAZAB4AIwAAQAAAAAAiwABAAIAAAAf AIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAAAAAAAApgEAAP//AAAAAAUAAAAAAAAA/wMA AAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIA GgAAAO0GAAAaAwAAAQAAAGUiTwEAAAAA//8BAGUijAABAAAAAACLAAEAAgAAAB8AigABAEQA AAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAAKYBAAAAAAMAAAAFAAAAAAAAAP8D AAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgAC ABcAAABfCAAAGgMAAAEAMNMAAAAAAP//AQAwAIwAAQAAAAAAlQABAAQAAAAAAAAAlgABAAIA AAAJAIwAAQAAAAAA , если f ( x ) ≥ 0 size 12{f left (x right ) = 0} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAVAGAAAw AgAAMgAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAE8GAAAvAgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJv bWFuIE5vOSBMAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAALkAAACnAQAAAQBmdQAA AAAA//8BAGYAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wA AOIAAADWAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwAB AAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAApwEAAKcBAAABAAAAKABNAAAAAAD//wEA KACMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBM AAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAACwCAACnAQAAAQB4uwAAAAAA//8BAHgA jAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAOIAAADWAQAA //8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD///// AIYAAQAEAAAAAAAAAHIAAgAaAAAAGgMAAKcBAAABAAAAKQBNAAAAAAD//wEAKQCMAAEAAAAA AIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKYBAAD//wAAAAAF AAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQA AAAAAAAAcgACABoAAACEAwAApwEAAAEAAABlIk8BAAAAAP//AQBlIowAAQAAAAAAiwABAAIA AAAfAIoAAQBEAAAAAwA+AAAAEgBOaW1idXMgUm9tYW4gTm85IEwAAAAAAACmAQAAAAADAAAA BQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAE AAAAAAAAAHIAAgAXAAAA3AQAAKcBAAABADDTAAAAAAD//wEAMACMAAEAAAAAAJUAAQAEAAAA AAAAAJYAAQACAAAACQCMAAEAAAAAAA== .

Вычисление определенного интеграла

Теорема .

Пусть F ( x ) size 12{F left (x right )} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAASAEAAAw AgAAJAAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAB8EAAAvAgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJv bWFuIE5vOSBMAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAACnAQAAAQBGAgEA AAAA//8BAEYAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wA AOIAAADWAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwAB AAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAAwgEAAKcBAAABAAAAKABNAAAAAAD//wEA KACMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBM AAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAEYCAACnAQAAAQB4uwAAAAAA//8BAHgA jAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAOIAAADWAQAA //8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD///// AIYAAQAEAAAAAAAAAHIAAgAaAAAANAMAAKcBAAABAAAAKQBNAAAAAAD//wEAKQCMAAEAAAAA AJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA== - первообразная функции f ( x ) size 12{f left (x right )} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAQQEAAAw AgAAJAAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAAMEAAAvAgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJv bWFuIE5vOSBMAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAALkAAACnAQAAAQBmdQAA AAAA//8BAGYAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wA AOIAAADWAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwAB AAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAApwEAAKcBAAABAAAAKABNAAAAAAD//wEA KACMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBM AAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAACwCAACnAQAAAQB4uwAAAAAA//8BAHgA jAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAOIAAADWAQAA //8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD///// AIYAAQAEAAAAAAAAAHIAAgAaAAAAGgMAAKcBAAABAAAAKQBNAAAAAAD//wEAKQCMAAEAAAAA AJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA== . Тогда ∫ a b f ( x ) dx = F ( b ) − F ( a ) size 12{ Int cSub { size 8{a} } cSup { size 8{b} } {f left (x right ) ital "dx"=F left (b right ) - F left (a right )} } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAYYQAAAd BQAAhgAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAIUQAAAcBQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv bAAAAAAAAGgCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAATwMAAAEAAAArImQBAAAAAP// AQArIowAAQAAAAAAiwABAAIAAAAfAIoAAQBEAAAAAwA+AAAAEgBOaW1idXMgUm9tYW4gTm85 IEwAAAAAAAAZAQAAAAADAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEA hwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAA1AAAANwEAAABAGGNAAAAAAD//wEA YQCMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBM AAAAAAAAGQEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAANQAAAAJAQAAAQBijQAAAAAA//8BAGIA jAABAAAAAACLAAEAAgAAAB8AigABAEQAAAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAA AAAAAKYBAAAAAAMAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEA BQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAABGAgAAGgMAAAEAZnUAAAAAAP//AQBmAIwA AQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAADiAAAA1gEAAP// AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCG AAEABAAAAAAAAAByAAIAGgAAADQDAAAaAwAAAQAAACgATQAAAAAA//8BACgAjAABAAAAAACL AAEAAgAAAB8AigABAEQAAAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAAKYBAAAA AAMAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8A hgABAAQAAAAAAAAAcgACABcAAADTAwAAGgMAAAEAeLsAAAAAAP//AQB4AIwAAQAAAAAAiwAB AAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAADiAAAA1gEAAP//AAAAAAUAAAAA AAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAA AAByAAIAGgAAAMEEAAAaAwAAAQAAACkATQAAAAAA//8BACkAjAABAAAAAACLAAEAAgAAAB8A igABAEQAAAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAAKYBAAAAAAMAAAAFAAAA AAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAA AAAAcgACABoAAABFBQAAGgMAAAIAZHiOAQAAAAD//wIAZAB4AIwAAQAAAAAAiwABAAIAAAAf AIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAAAAAAAApgEAAP//AAAAAAUAAAAAAAAA/wMA AAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIA GgAAAO0GAAAaAwAAAQAAAD0ATwEAAAAA//8BAD0AjAABAAAAAACLAAEAAgAAAB8AigABAEQA AAADAD4AAAASAE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAAKYBAAAAAAMAAAAFAAAAAAACAP8D AAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgAC ABcAAABfCAAAGgMAAAEARgIBAAAAAP//AQBGAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAA AwA2AAAACgBPcGVuU3ltYm9sAADiAAAA1gEAAP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/ AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAALcJAAAa AwAAAQAAACgATQAAAAAA//8BACgAjAABAAAAAACLAAEAAgAAAB8AigABAEQAAAADAD4AAAAS AE5pbWJ1cyBSb21hbiBObzkgTAAAAAAAAKYBAAAAAAMAAAAFAAAAAAACAP8DAAAAAAAAAAAA /wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAAAhCgAA GgMAAAEAYtMAAAAAAP//AQBiAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBP cGVuU3ltYm9sAADiAAAA1gEAAP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAACoLAAAaAwAAAQAAACkA TQAAAAAA//8BACkAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1i b2wAAAAAAACmAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEA hwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAAeQsAABoDAAABAAAAEiJPAQAAAAD/ /wEAEiKMAAEAAAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5v OSBMAAAAAAAApgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAOsMAAAaAwAAAQBGAgEAAAAA//8B AEYAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAOIAAADW AQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/ ////AIYAAQAEAAAAAAAAAHIAAgAaAAAAKQ4AABoDAAABAAAAKABNAAAAAAD//wEAKACMAAEA AAAAAIsAAQACAAAAHwCKAAEARAAAAAMAPgAAABIATmltYnVzIFJvbWFuIE5vOSBMAAAAAAAA pgEAAAAAAwAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA /////wCGAAEABAAAAAAAAAByAAIAFwAAAK0OAAAaAwAAAQBh0wAAAAAA//8BAGEAjAABAAAA AACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAOIAAADWAQAA//8AAAAA BQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAE AAAAAAAAAHIAAgAaAAAAmw8AABoDAAABAAAAKQBNAAAAAAD//wEAKQCMAAEAAAAAAJUAAQAE AAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA== .