Dersin ad谋 nerden geliyor?
Eski zamanlarda g眉ncelS hesaplamalar谋 yaparken 莽ak谋l ta艧lar谋 kullan谋l谋rm谋艧. Eski Yunanca鈥檇a 莽ak谋l ta艧谋n谋n ad谋 鈥淜alk眉l眉s鈥�. Dersimizin ad谋 buradan geliyor. Kalker, kalsiyum,鈥� gibi kelimeler de ayn谋 k枚kten. Ders K谋s谋m II ile temel kavramlarla ba艧l谋yor ve K谋s谋m III, uygulamalar olarak s眉r眉yor. Neden? Geleneksel kalk眉l眉s kitaplar谋nda 枚nce t眉rev kavram ve uygulamalar谋yla, sonra da entegral kavram ve uygulamalar谋yla sunuluyor. Geleneksel yakla艧谋mda, t眉revleri ve entegralleri 枚臒renip uygulamalara girince olduk莽a kar谋艧谋k hesaplamalar 枚臒retiliyor. 脰臒renciler de bu kar谋艧谋k hesaplamalar aras谋nda kaybolup, hatta s谋k谋c谋 bulup ana kavramlar谋 ge莽i艧tiriyorlar. Entegral k谋sm谋na gelindi臒inde bir yorgunluk olu艧mu艧 durumda. Bunun sonucunda 枚臒rencinin entegraldeki temel kavramlar谋 anlamas谋 gecikiyor, hatta 枚臒renciler i艧i ezberlemeyle ge莽i艧tiriyor. 脰臒renciler bir bak谋ma hakl谋: 莽眉nk眉 s谋navlarda t眉rev hesaplamas谋 ve entegral hesaplamas谋 soruluyor. T眉rev 莽谋kartma ve b枚lme i艧lemleriyle, entegral de 莽arpma ve toplamayla yap谋l谋yor. Her ikisinde de k眉莽眉k de臒erlerle limite gidilerek sonuca var谋l谋yor. Kavramsal ve i艧lemsel olarak t眉rev ve entegral birbirinin tamamlay谋c谋s谋. Her birisi de di臒er i艧lemi anlamakta yararl谋. Karma艧谋k hesaplamalara girince, konular谋n 枚z眉 anla艧谋lamadan, 枚臒renciler de ezberlemeyle teknikleri 枚臒renip kaybolup gidiyor. Bilgisayarlar谋n ve yaz谋l谋mlar谋n 莽ok geli艧mi艧 oldu臒u d枚nemimizde, 莽ok karma艧谋k entegrali hesaplamak pek b眉y眉k bir kazan莽 de臒il, bunlar谋 莽e艧itli tablolardan g枚rmek m眉mk眉n. Yine g眉n眉m眉zdeki i艧 ya艧am谋nda uygulamalar bilgisayarlar yard谋m谋yla say谋sal y枚ntemlerle sa臒lan谋yor. Tabii, say谋sal hesaplamalar谋 kurgulamak i莽in temel konular谋 bilmek 枚nemli, b眉y眉k 枚l莽眉de yeterli ve gerekli. 脟a臒谋m谋zda, kalk眉l眉s e臒itiminin uygulamalarda bilgisayarlara haz谋rlamay谋 yok saymas谋 beklenemez. 鈥淐alculus鈥�: 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 鈥淐alculus鈥� comes from. Words like 鈥渃alcareous鈥�, 鈥渃alcium鈥�,... 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.
