Tous les niveauxKALKÜLÜS III: UYGULAMALAR / CALCULUS III: APPLICATIONS
Dersin adı nerden geliyor?
Eski zamanlarda güncelS hesaplamaları yaparken çakıl taşları kullanılırmış. Eski Yunanca’da çakıl taşının adı “Kalkülüs”. Dersimizin adı buradan geliyor. Kalker, kalsiyum,… gibi kelimeler de aynı kökten.
Ders Kısım II ile temel kavramlarla başlıyor ve Kısım III, uygulamalar olarak sürüyor. Neden?
Geleneksel kalkülüs kitaplarında önce türev kavram ve uygulamalarıyla, sonra da entegral kavram ve uygulamalarıyla sunuluyor. Geleneksel yaklaşımda, türevleri ve entegralleri öğrenip uygulamalara girince oldukça karışık hesaplamalar öğretiliyor. Öğrenciler de bu karışık hesaplamalar arasında kaybolup, hatta sıkıcı bulup ana kavramları geçiştiriyorlar. Entegral kısmına gelindiğinde bir yorgunluk oluşmuş durumda. Bunun sonucunda öğrencinin entegraldeki temel kavramları anlaması gecikiyor, hatta öğrenciler işi ezberlemeyle geçiştiriyor. Öğrenciler bir bakıma haklı: çünkü sınavlarda türev hesaplaması ve entegral hesaplaması soruluyor.
Türev çıkartma ve bölme işlemleriyle, entegral de çarpma ve toplamayla yapılıyor. Her ikisinde de küçük değerlerle limite gidilerek sonuca varılıyor. Kavramsal ve işlemsel olarak türev ve entegral birbirinin tamamlayıcısı. Her birisi de diğer işlemi anlamakta yararlı. Karmaşık hesaplamalara girince, konuların özü anlaşılamadan, öğrenciler de ezberlemeyle teknikleri öğrenip kaybolup gidiyor.
Bilgisayarların ve yazılımların çok gelişmiş olduğu dönemimizde, çok karmaşık entegrali hesaplamak pek büyük bir kazanç değil, bunları çeşitli tablolardan görmek mümkün. Yine günümüzdeki iş yaşamında uygulamalar bilgisayarlar yardımıyla sayısal yöntemlerle sağlanıyor. Tabii, sayısal hesaplamaları kurgulamak için temel konuları bilmek önemli, büyük ölçüde yeterli ve gerekli. Çağımızda, kalkülüs eğitiminin uygulamalarda bilgisayarlara hazırlamayı yok sayması beklenemez.
“Calculus”: Where does the name come from?
In ancient times, pebbles were used for daily calculations. In ancient Greek, "Calculus" means pebble, small regular stone pieces. This is where the name of our course “Calculus” comes from. Words like “calcareous”, “calcium”,... are also from the same root.
Part I, the preparation of calculus has the goal to recognize functions of one variable. Why?
Mathematics is a language. In verbal languages, we combine words with the rules of grammar to form observations and thoughts. Similarly, in mathematics the task of grammar is provided by non-contradictory assumptions (axioms/postulates). The task of words in the language of mathematics is provided by functions. While verbal languages require thousands of words, only dozens of "words", i. e. functions, are sufficient in mathematics. From this perspective, mathematics should be easier than verbal languages...
The aim of this PART I of the course is limited to introduce the basic functions. Not knowing well these functions would lead to failure in advancing in mathematics, particularly in calculus. Without success, the course turns into an unpleasant experience for students as well as for the instructors.
The course here starts with core concepts as Part II, and applications follow as Part III. Why?
In most traditional calculus books, the ordering is different: concepts of derivative and their applications are the starting phase. The second phase is made of the core concepts of integration and their applications. In this traditional approach, derivatives and integrals that are intimately connected appear as two different worlds. Moreover, the students are exposed to rather detailed and somewhat complex calculations with derivatives. When the course comes to integration, fatigue sets in, students would already have forgotten the basic concepts of derivatives trying to solve problems. As a result, the student's understanding of basic concepts of integration becomes a tedious work and understanding is replaced by memorization with the efforts being geared towards solving problems again. In a way, the students are right: in the exams they are asked questions about calculations of tricky derivatives and integrals.
The core concepts of derivative and integral are complementary. Derivative is defined by subtraction and division; integral is defined by multiplication and addition. In both, the conclusion is reached by going to the limit with small values. Conceptually and operationally, derivative and integral are the inverse of each other. Each of them is useful in understanding the other process, as well as complementing each other. When they are taught in the beginning, the students are not yet tired and understand more easily these two complementary core concepts.
Impact of the computers age
Computers and software are rather advanced now. It is not a big asset to calculate complex derivatives and integrals, as it is possible to see the results from various tables. Moreover, in professional life, solutions for applications can be obtained with numerical methods with the help of computers. Of course, it is important and necessary to know the basics of calculus for constructing numerical calculations. The teaching of calculus in modern times cannot neglect the preparation for applications using the computer.
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