Calculus Law Of Exponential Change Calculus Law Of Exponential Change Law of Exponential Change means the exponential growth. It is the rate of growth whose values depends upon the current value of the function and growth rate will be negative. There is a model of exponential change and the name is Malthusian growth model. Now the formula of exponential change is Zt = z0 (1 + r)t Where z is a variable it should be positive or negative, r is the growth rate and t is the time interval which has values 0, 1, 2, 3……. Let us suppose that the growth rate of r = 4% = 0.04, it increases to another integer value with the second time to be 1.05 times which is larger than the 0.04. Now the basic formula of the exponential, Z (t) = p .qt/T It shows the any quantity which depends on the time.
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If t= (0) then z (0) = p Where p is a constant with initial value and q is the positive changing factor, z (t +T) = z(t).q When any value is changes that is increases or decreases at a rate proportional to the given value this is called the Law of Exponential Change. Mathematically we can write it as: dx/dt = cx, The above equation shows the rate of change or growth. So that x is increases or decreases exponentially. Now here is two case either x is increasing or decreasing, if c>0 then x is increasing otherwise x is decreasing. Now the example of the exponential change is population growth that size of population is a exponential growth, temperature is increases or decreases as per the environment condition. There is many more example of exponential growth. Some popular examples of the exponential growth could be related to the financial growth. The growth in the rate of the interest is also a example for this. The increase in the population, as the population of any country is say x at this time but after 10 years in future what will be the population of that country. Such problems are the good examples of the exponential growth or decay. Now, talking about the example of the exponential growth and decay and solution of those exponential growth decay we can take example of population.
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As discussed above, an important point about exponential growth is that even when it seems slow on the short run, it becomes impressively fast on the long run, with the initial quantity doubling at the doubling time, then doubling again and again. For instance, a population growth rate of 2% per year may seem small, but it actually implies doubling after 35 years, doubling again after another 35 years (i.e. becoming 4 times the initial population), etc. This implies that both the observed quantity and its time derivative will become several orders of magnitude larger than what was initially meant by the person who conceived the growth model. Because of this, some effects not initially taken into account will distort the growth law, usually moderating it as for instance in the logistic law. Exponential growth of a quantity placed in the real world (i.e. not in the abstract world of mathematics) is a model valid for a temporary period of time only.
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