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EASY CALCULUS



Take the function,

 y = x^2

 x        y=x^2
-3        9
-2        4
-1        1
0         0
1         1
2         4
3         9 

  If we draw the graph, we get a parabola.  Now we know how the function behaves. 







   I want to know if 'x' changes by 1[unity], What is a change in 'y' at any value of x.  That is, what is the rate of change of function at any point in the curve.  Is there any standard formula?  The question is easy to answer, if the function is linear or the graph is straight line.  The answer would be slope [gradient] of line and the slope will be the same at every point in the line. but Here we have parabolic relationship (refer figures).  

   Velocity is the rate of change of distance.Inflation is the rate of change of prices.Hence it is a very important quantity.

      Here, calculus come to our help.  It tells and proves that  if y=x^2,  dy/dx = 2x;  that is, the rate of change of function y= x^2 at any point is x is 2x.  Let us numerically verify it

 Let x changes from 2 to 3;  unit change
hence y = x^2 changes from 4 to 9
Net change in y or rate of change of y =  9-4 = 5
average x value = [2+3]/2 = 2.5
From calculus, rate of change of y = 2x = 2*2.5 = 5
So it tallies.

 Another example:
Let x changes from 10 to 11
Hence y changes from 100 to 121
Net change in y for unit change in x  = 121-100=21
average x = [10+11]/2= 10.5
rate of change is= 2x=2*10.5 = 21
So it tallies again.
  
 So the formula dy/dx = 2x works for any value for x (refer slopes in parabola graph).
Finding the rate of change of a function is called differentiation in calculus.  We have so many formula like this

 examples:
y=sin x;  dy/dx=cos x

 y=x^3;   dy/dx=3x^2 

 y=e^x;   dy/dx=e^x   [no mistake]

 Integration is the reverse of differentiation.

 Integration of 2x=x^2
integration of cos x=sin x

Differentiation and integration are a two sides of a coin and they constitutes calculus.  It is the central blank of mathematics and it plays major roles in physics, engineering and economics.  
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