Expert: Alon Mandes - 5/13/2009

Question
Hi Alon,
I'm a physics student from india. I know basic calculus. No problems with that. I can differentiate, integrate, solve problems and all, and I'm preparing for the entrance exam of IIT(Indian Institute of Technology) currently.
Now the problem really is this. I read J.E. Thompson recently and it said" if y is a func. of x, then dy ( not necessarily a small change, just a change) will depend on the change in value of x i.e. dx and 'x itself' ". What i don't understand is that how do you logically reach the conclusion that a change in y will also depend on x. It can be logically understood that since the value of y depends on the value of x, so as the value of x changes, value of y will also change. So it can be logically understood that dy will depend on dx. But i fail to understand by plain logic that why should dy or change in the value of y should also depend on the value of x at that point.
I just somehow am not able to understand calculus the way i'd like it to. Though i do understand concepts such as limits and all but still i just don't know, what does calculus ultimately say. How can i understand every operation of calculus by pure logic or thought rather than by mathematical derivations and methods.
I do understand that differentiation means taking little bits of something and integration means summing up all these little tits and bits but how do we know how to sum up some particular sort of tits-bits? we certainly don't sit down and perform addition on every tit bit but yet we know there sum. How can we know that, or how do we get there logically without actually performing the addition?
I probably sound wierd, but it's just that i can't understand the way i'd like it to, probably like addition or substraction or even multiplication or normal algebra problems.
Please help..
Thanks
Shikhin


Answer
Hello Shikhin ,
I apologize for my late reply , had a rough 2 days of work , the
company is having a tied schedule for projects & I had 2 work extra
& was floated with work . Sorry !
As for calculus , you need to keep in mind that its a tool, a tool
to help you analyze theories , calculate entities , & to model
processes. The understanding of it essence is essential for using
it efficiently. It's more advanced than elementary math we studied
in High school. I will to the two issues you mentioned :
1. If y=x then dy=dx "The change in equals the change of x. But if
  y=f(x) then dy=f'(x)dx where f'(x) is a function of x. That
  means, the change of y not only depends on the change of x but
  it also depends on the position of x.
2. The integration is done according to an Algorithm of tit & bit.
  We don't know how the summation is performed , but we know the
  fashion of it's operation.
Math is very exciting subject, need practice & commitment. I wish
you success in your survey & I'm always here if you want help, or
need to ask anything .

Alon.