1 Exponential increase

  1. Look back at Section 6.2 to review the definitions of a n exponential function and t h e exponential function.
  2. List examples in this Workbook of contexts in which exponential functions are appropriate.
  1. A n exponential function has the form y = a x where a > 0 . T h e exponential function has the form y = e x where e = 2 . 718282 . . . . . .
  2. It is stated that exponential functions are useful when modelling the shape of a hanging chain or rope under the effect of gravity or for modelling exponential growth or decay.

We will look at a specific example of the exponential function used to model a population increase.


Given that

P = 12 e 0.1 t ( 0 t 25 )

where P is the number in the population of a city in millions at time t in years answer these questions.

  1. What does this model imply about P when t = 0?
  2. What is the stated upper limit of validity of the model?
  3. What does the model imply about values of P over time?
  4. What does the model predict for P when t = 10 ? Comment on this.
  5. What does the model predict for P when t = 25 ? Comment on this.
  1. At t = 0, P = 12 which represents the initial population of 12 million. (Recall that e 0 = 1 .)
  2. The time interval during which the model is valid is stated as ( 0 t 25 ) so the model is intended to apply for 25 years.
  3. This is exponential growth so P will increase from 12 million at an accelerating rate.
  4. P ( 10 ) = 12 e 1 33 million. This is getting very large for a city but might be attainable in 10 years and just about sustainable.
  5. P ( 25 ) = 12 e 2.5 146 million. This is unrealistic for a city.

Note that exponential population growth of the form P = P 0 e k t means that as t becomes large and positive, P becomes very large. Normally such a population model would be used to predict values of P for t > 0 , where t = 0 represents the present or some fixed time when the population is known. In Figure 6, values of P are shown for t < 0 . These correspond to extrapolation of the model into the past. Note that as t becomes increasingly negative, P becomes very small but is never zero or negative because e k t is positive for all values of t . The parameter k is called the instantaneous fractional growth rate .

Figure 6 :

{ The function $P = 12e^{0.01t}$}

For the model P = 12 e k t we see that k = 0.1 is unrealistic, and more realistic values would be k = 0.01 or k = 0.02 . These would be similar but k =0.02 implies a faster growth for t > 0 than k = 0.01 . This is clear in the graphs for k = 0.01 and k = 0.02 in Figure 7. The functions are plotted up to 200 years to emphasize the increasing difference as t increases.

Figure 7 :

{ Comparison of the functions $P = 12 e^{0.01t}$ and $P = 12e^{0.02t}$}

The exponential function may be used in models for other types of growth as well as population growth. A general form may be written

y = a e b x a > 0 , b > 0 , c x d

where a represents the value of y at x = 0 . The value a is the intercept on the y -axis of a graphical representation of the function. The value b controls the rate of growth and c and d represent limits on x .

In the general form, a and b represent the parameters of the exponential function which can be selected to fit any given modelling situation where an exponential function is appropriate.