Calculus Solutions, Important Examples, Formulas and Videos

Calculus Important Questions - Page 6
Calculus Video Lecture - Page 7

Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain change. First, we give an intuitive idea of derivative (without act actually defining it). Then we give a naive definition of limit and study some algebra of limits. Then we come back to a definition of derivative and study some algebra of derivatives. We also obtain derivatives of certain standard function.

Limits: The above discussion clearly points towards the fact that we need to understand limiting process in greater clarity. We study a few illustrative examples to gain some familiarity with the concept of limit . Consider the function f(x ) = x^{2} bserve that as x takes values very close to 0, the value of f ( x ) also moves towards 0.

\lim_{x\rightarrow 0}f(x)

In genral as x\rightarrow a,f(x)\rightarrow l , then l is called limit of the function f(x) which is symbolically written as \lim_{x\rightarrow o}f(x)=l .

Consider the following function g(x) = |x| , x \neq . observer that g(0) is not defined. Computing the value of g(x) for the value of x very near to 0 we see that the value of g (x) moves towards zero .

So, \lim_{x\rightarrow 0}g(x)=0 .

This intuitively clear from the graph of y = |x | for x ≠ 0

Note : we say \lim_{x\rightarrow a}f(x) is the expected value of f at x = a given the values of f near x to the left of a. This value is called the left hand limit of f at a.

We say \lim_{x\rightarrow a}f(x) is the expected value of f at x = a given the values of f near x to the left of a. This value is called the right hand limit of of f(x) at a if the right and left limit coincide , we call common value as the limit of f(x) at x = a and denote it by \lim_{x\rightarrow a}f(x)

 

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