Growth Rate of Functions

3 Key excerpts on "Growth Rate of Functions"

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  eBook - ePub

  An Elementary Approach to Design and Analysis of Algorithms

  • Lekh Raj Vermani, Shalini Vermani(Authors)
  • 2019(Publication Date)
  • WSPC (EUROPE)

    (Publisher)

  Chapter 2

  Growth of Functions

  Given a real-valued function f defined on the set of real numbers, by the growth of f we mean how rapidly or slowly it increases in comparison with another real-valued function when the real number x increases. So for the growth of functions, there need to be considered two functions. How rapidly or slowly the function f increases in comparison with another function g is expressed in terms of certain notations called asymptotic notations. In this chapter, we are concerned with this concept. Of course, asymptotic means that how the two functions behave only for large values of x.

  2.1.Introduction

  Let f be a function from the set R of real numbers to R. By the growth of the function f we mean to say how the function f(x) grows when the variable x increases or becomes large. In the analysis of algorithms we are generally concerned with the number of algebraic computations involved as the running time of an algorithm is proportional to this number. The number of computations depends on the size n (which is a positive integer) of the problem instance. Thus we come across functions f : N → N, where N denotes the set of all natural numbers (or non-negative integers). Also there may be more than one algorithm for solving a certain problem so that for a given problem we may get two functions, say f(n) and g(n), giving the performance of the two algorithms (for the same problem). Which one of the two algorithms may be preferred will normally depend on the comparative growth of the two functions f(n) and g(n). In this chapter, we study this problem of comparative study of functions. A function f(n) which grows more slowly than the function g(n) is called a superior function (to g(n)). In fact we are not concerned with the growth of two functions for all n but only for large n

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  Available until 5 Dec |Learn more

  Basic Mathematics for Economists

  • Mike Rosser, Piotr Lis(Authors)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  rate of growth is therefore

  d y

  d t

  y

  =

  r A

  e

  r t

  A

  e

  r t

  = r

  Even though r is the instantaneous rate of growth at any given moment in time, it must be expressed with reference to a time interval, which is usually a year in economic applications, e.g. 4.5% per annum.

  Example 14.5

  Owing to continuous improvements in technology and efficiency in production, an empirical study found that a factory’s output of product Q at any moment in time was determined by the function

  Q = 40

  e

  0.03 t

  where t is the number of years from the base year in the empirical study and Q is the output per year in tonnes. What is the annual growth rate of production?

  Solution

  When the accumulated amount from continuous growth is expressed by a function in the format y = Ae

  rt

  then the growth rate r can simply be read off from the function. Thus when

  Q = 40

  e

  0.03 t

  the rate of growth is

  r = 0.03 = 3 %

  Initial amounts

  What if you wished to find the initial amount A that would grow to a given final sum y after t time periods at continuous growth rate r? Given the continuous growth final sum formula

  y = A

  e

  r t

  then, by dividing both sides by e

  rt

  , we can derive the exponential growth initial sum formula

  A = y

  e

  − r t

  Example 14.6

  A parent wants to ensure that their child will have a fund of £35,000 to finance their study at university, which is expected to commence in 12 years’ time. They wish to do this by investing a lump sum now. How much will they need to invest if this investment can be expected to grow continuously at an annual rate of 5%?

  Solution

  Given values are: final amount y = 35,000, continuous growth rate r = 5% = 0.05, and time period t

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  eBook - ePub

  Essential Demographic Methods

  • Kenneth W. Wachter(Author)
  • 2014(Publication Date)
  • Harvard University Press

    (Publisher)

  R is the slope of the graph of the logarithm of population size over time. It is the proportional rate of change in population size.

  This definition gives us a measure of growth which is unchanging when the ratios of births and deaths to population size are unchanging, and only changes when the ratios change.

  The slope of any graph is equal to the “rise” over the “run”. The “rise” is the vertical increase corresponding to a given “run” along the horizontal axis. When we calculate the growth rate from time t = 0 to time t = T , the run is the length of the interval, T − 0 or T , and the rise is the increase in log(K (t )). The slope is the quotient:

  Using rules for differences of logarithms, we see that this expression is the same as an expression with a ratio of population sizes inside the logarithm:

  If we are measuring growth between two times t 1 and t 2 , we have another version of the same formula:

  Any one of these versions of the formula can be used to calculate a rate of population growth.

  For a contrast to present-day growth rates, we can calculate R for earlier periods, like the long period between the start of agriculture around 8000 B.C. and the birth of Christ. Guesses at populations for these epochs, 5 million and 250 million, are given later in the chapter, in Table 1.4 . We calculate the growth rate R to be

  This value is less than 1 per thousand per year. The units (millions of people) on 250 and 5 cancel when we take their ratio, while the units on 8,000—that is, years—in the denominator give us a measure per year .

  If we average the growth rates for two successive years—say, this year (t goes from 0 to 1) and next year (t goes from 1 to 2)—we end up with the overall growth rate for the 2-year period from t = 0 to t = 2:

  This property of our formula for growth is convenient. Knowledge of intermediate populations gives us no extra information when we seek the average growth rate between the start and the end of a long period.

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