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LIFE'S CURVE



                                                         N = n*e^(r*t)

   The equation yields the natural growth or decay of any quantity.  Here, the initial amount of quantity  n becomes N after time lapse of  t at the rate of growth r.  When we apply the equation for decay, the r becomes negative. we know Euler's constant e=2.71...
       The applet given below draws the curve for the equation, you can play with the rate r  and see it's effect on the curve. The initial quantity n = 1.  The curve gives the quantity N (y) as the time t (x) progresses.  The curve rapidly grows or decays and then it becomes slow.  But the curve tends to go to infinity.  To view the curve properly adjust the graph sheet in the applet.


                                       

   1.   When a capacitor is connected to a battery, it charges up.  When disconnected, it discharges.  The discharging voltage is given by the following relationship.
                                                              Vc = Vo*e^(-t/RC)
     here Vo is full voltage.  Vc is the decaying voltage as the time t progresses and -1/RC is the rate of fall of voltage.

    2.  The bulging earth's population can  also be calculated with this formula
                                                              .p1 = p*e^(r*t)
       p - initial population and world's rate of growth is said to be 1.68% (r = 0.0168) 
.
  3.   If you want your money to double every year in the bank, what should be the rate of interest?  It is not 100%.  But only 69.31% per annum ( r= 0.6931).  But on one condition the interest must be added to the principle very frequently.
                                           Amount = principle*e^(rate*no.of years)

      Hence this exponential relationship is applicable to rise and fall of current, radioactive decay, growth of a child, multiplication in a bacteria colony, cooling of cup of coffee and so on.certainly the curve plays vital role in our life.
     In this context, I want to mention another formula, no, a most beautiful formula in all of mathematics.

                                 e^(𝝿*i)+1 = 0

     The formula connects 5 most important constants in maths (e.pi,1,0,i).  It uses three operations(*,+,^).  It represents 4 branches of maths.  Arithmetic by 1 and 0:  Algebra by i; Geometry by pi; analysis by e. (i = root of -1)
     Regarding the formula, a Harvard's mathematician said "It is absolutely paradoxical; we cannot understand it and we do not know what it means.  But we have proved it and therefore it must be true."

   Science update:  Scientists have developed a compact low cost infrared laser microscope with 10 times more magnifying power than light based ones.         
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