KALK L S III: UYGULAMALAR / CALCULUS III: APPLICATIONS

Dersin ad nerden geliyor? Eski zamanlarda g ncelS hesaplamalar yaparken ak l ta lar kullan l rm'
. Eski Yunanca'da ak l ta”n n ad ‘Kalk l s'. Dersimizin ad buradan geliyor. Kalker, kalsiyum,' gibi kelimeler de ayn k kten. Ders K s m II ile temel kavramlarla ba l yor ve K s m III, uygulamalar olarak s r yor. Neden? Geleneksel kalk l s kitaplar nda nce t rev kavram ve uygulamalar yla, sonra da entegral kavram ve uygulamalar yla sunuluyor. Geleneksel yakla”mda, t revleri ve entegralleri ”renip uygulamalara girince olduk a kar”k hesaplamalar ”retiliyor. ”renciler de bu kar”k hesaplamalar aras nda kaybolup, hatta s k c bulup ana kavramlar ge i tiriyorlar. Entegral k sm na gelindi inde bir yorgunluk olu mu durumda. Bunun sonucunda ”rencinin entegraldeki temel kavramlar anlamas gecikiyor, hatta ”renciler i i ezberlemeyle ge i tiriyor. ”renciler bir bak ma hakl : ”nk s navlarda t rev hesaplamas ve entegral hesaplamas soruluyor. T rev ”kartma ve b lme i lemleriyle, entegral de arpma ve toplamayla yap l yor. Her ikisinde de k”k de erlerle limite gidilerek sonuca var l yor. Kavramsal ve i lemsel olarak t rev ve entegral birbirinin tamamlay c s . Her birisi de di er i lemi anlamakta yararl . Karma”k hesaplamalara girince, konular n z anla”lamadan, ”renciler de ezberlemeyle teknikleri ”renip kaybolup gidiyor. Bilgisayarlar n ve yaz l mlar n ok geli mi oldu u d nemimizde, ok karma”k entegrali hesaplamak pek b y k bir kazan de il, bunlar e itli tablolardan g rmek m mk n. Yine g n m zdeki i ya am nda uygulamalar bilgisayarlar yard m yla say sal y ntemlerle sa lan yor. Tabii, say sal hesaplamalar kurgulamak i in temel konular bilmek nemli, b y k l”de yeterli ve gerekli. a”m zda, kalk l s e itiminin uygulamalarda bilgisayarlara haz rlamay yok saymas 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 …