What is the meaning of differentiation and integration in calculus?

[Amar.alchemy]

Hi Guys,

Actually I am searching for study material or any site which explains concepts of differentiation and Integration. What I mean is I don't want the techniques or different formulae to solve the problems. I want the meaning of differentiation and integration and why we use differentiation and integration. Kindly provide me any link or book which explains the conceptuality of calculus.

For ex: If we differentiate the equation Y= x^2 we get dy/dx = 2x. What exactly it means?? Similarly with integration.

Thanks in advance :-)

 

[quasar987]

Y is a map that assigns to each number x a number Y(x). For instance, it could be that x is the reading on a chronometer and Y is the distance traveled by a car. The concept of a map (aka function aka mapping aka application aka transformation) is the first that must be understood in order to understand the concept of the derivative of a function.

The second concept that must be understood on the way to understanding differentiation is that of the mean rate of change of a function between two points.

The third is that of the limit of a function at a point.

Finally, the derivative of a map Y at a point a is obtained by fusion of the 3 previous concepts:

1° Construct from Y a new function

$$\mbox{MRC}(Y;a,x):=\frac{Y(a)-Y(x)}{a-x}$$

which measures the mean rate of change (hence the letters "MRC") of Y between the points a and x (a being fixed and x being the variable).

2° Construct from MRC(Y;a,_) yet a third function $$Y'(a):=\lim_{x\rightarrow a}\mbox{MRC}(Y;a,x)$$. The map Y' is also written dY/dx and is called the "derivative" of Y.

If you have understood well the 3 previous concepts, you will agree that the number Y'(a) represents the "instantaneous rate of change" of Y at the point a. This is quite a difficult idea to grasp as can be expected from the complexity of the definition, but your understanding will grow as you go along and do examples.

For instance in the case where x is time and Y is the distance traveled by a car, Y'(a) is just the speed of the car at time a!I would expect that any calculus book explains the concepts I wrote in bold above and provide the reader with exercices to help him develop his understanding of these concepts.

 

[quasar987]

The concept of the integral

$$\int_a^bY(x)dx$$

of a map Y between two points a and b is much simpler: it is simply a measure of the signed area enclosed between the graph of Y and the x axis. (Where "signed area" means that the part of the erea where Y is <0 is substracted from the part where Y>0.)

The idea is that we know the formula for the area of a rectangle (width times height) so we approximate the signed area enclosed between the graph of Y and the x-axis by the area of a couple rectangles. See the pictures here: http://en.wikipedia.org/wiki/Integral.

Then we agree that as the number of rectangles increase, the sum of the signed areas of the rectangles approximates better and better the signed area enclosed between the graph of Y and the x-axis and so we define [itex]\int_a^bY(x)dx[/tex] as the sum of the signed areas of an infinite number of rectangles.

Although the integral is simpler conceptually, the rigourous construction of it is as involved, if not more, as that of the derivative.


Then there is the fundamental theorem of differential and integral calculus (FTC) which establises a link between the concept of derivative and that of integral. It says simply that

$$\int_a^bY'(x)dx=Y(b)-Y(a)$$

It is useful because derivatives are easy to calculate and integrals are hard to calculate. So for instance, if you're wondering how to integrate the function 2x*cos(x²), then having mastered the rules of differentiations, you will recognized after a bit of reflection that by the chain rule, this is just the derivative of sin(x²). So the FTC above allows you to write

$$\int_a^b2x\cos(x^2)dx=\int_a^b\frac{d(\sin(x^2))}{dx}dx=\sin(b^2)-\sin(a^2)$$.

Initially, one might not see the point of practicing using the rules of differentiation over and over like and automaton, but in fact, as a corollary of the preceeding little example, we see that it is very useful. For withouth that practice, I would not have realized that 2x*cos(x²)=d(sin(x²))/dx. The conceptual understanding of differentiation and integration allows you to translate a real word problem into a mathematical problem and a mastery of the rules of differentiation allow you to solve it.

 

[Amar.alchemy]

Thanks a lot dude...:-)

Yesterday i found below link in this foum:

http://www.freebookcentre.net/Mathematics/Calculus-Books-Download.html"


In this site i just checked the book "calculus made easy" by Thompson and it was amazing. Later i found that this is one of the classic book written in 1910. May be for advanced users the concepts of this book may look obvious but for begginers like me it is really very helpful. It is exactly just what i wanted

 

[quasar987]

This looks like a good book, but be warned that it is old and this is not how calculus is told nowadays (and for a good reason!). In Thompson's book, the "geometrical interpretation of differentiation" appears only in chapter 10 while in modern calc books, this is how differentiation is introduced. Also, the author says things like "let us divide this equation by dx". This is frowned upon nowadays.