By Sloughter D.

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**Extra resources for Difference equations to differential equations**

**Example text**

For example, the size of a certain population of owls will be a function of the number of years from some starting date. Example The area of a circle is a function of the radius of the circle. Example The distance of the earth from the sun is a function of the time of year. 1 The temperature at a certain fixed point in space is a function of time. In mathematical terminology, if y is a function of x, then we call x the independent variable and y the dependent variable. Also, the domain of this function is the set of permissible values for x and the range is the set of all values of y which correspond to some value of x.

And α > 1, called the uninhibited, or natural, growth model, although often accurate as a model of population growth over short periods of time, is usually too simplistic for predictions over long time spans. The inhibited growth model Suppose we wish to model the growth of a certain population which, without ecological constraints, would grow at a rate of 100β% per unit of time. 5) for n = 0, 1, 2, . .. However, suppose that, because of the limitation of resources, the population will begin to decline if it ever has more than M individuals.

5. 11. How many terms of the harmonic series are needed to obtain a partial sum larger than 5? How many terms are needed to obtain a partial sum larger than 10? 12. Plot the points (n, sn ), where sn is the nth partial sum of the harmonic series, for n = 1, 2, 3, . . , 1000. What does this show you about the rate of growth of the partial sums? 13. The first example of this section is a particular case of the more general problem of computing probabilities associated with the waiting time for some event to occur.