Exponential Functions




Examples of exponential growth and decay……………………………. 2

·        unlimited population growth

·        serum blood drug levels

·        radioactive decay

·        atmospheric pressure

·        light absorption in seawater

·        compound interest growth

·        inflation rates

What do exponential functions look like?............................................... 6

What characteristics can be determined from exact exponential data?...... 7

Differential and difference models for exponential growth……………… 8    

Recognizing exponential growth from approximate data………………..10

Exponential Functions


Examples of exponential growth and decay

Unlimited Population Growth


A colony of bacteria living on a Petri dish under optimal conditions doubles in size every 20 minutes.  At noon on 24 March 1993, the dish was completely covered with bacteria.  At what times (to the nearest hour, minute and second) was the covered percentage of the dish

                   (a) 50 % ,

                   (b) 25 % ,

                   (c) 5 % ? [1]




At the start of 1988, the world population was about 5 billion.  For purposes of this exercise, which is largely fictional anyway, we assume the population was exactly 5 billion.  If the growth rate was 2% per year, what was the population at the start of 1989?  Suppose the 2% annual rate continued for the rest of the century.  What would the population be at the start of 2001?  Do you notice any pattern in you computation?  Still assuming the continued 2% rate, can you write a formula for the population any number of years after 1988?  What is the population half way through 1989?  Half way through the next year?  Explain your strategy for answering these questions.[2]


Exponential Functions


Examples of exponential growth and decay

Exponential Decay -- Serum Blood Drug Levels


The cardiac glycoside digoxin has a half life (in serum and in tissues) of approximately 48 hours.  If a patient has been on a maintenance dose of digoxin, then the serum level of the drug C at time t hours after it has been measured  to show  a level of 1.6 ng/ml is given by


if no further drug is administered. (Therapeutic serum levels in most individuals is 0.8-2.0 ng/ml when measured 8 hours after a dose is administered.)

          (a) After how many hours t will the serum level C of the drug reach

               0.8 ng/ml?


Digoxin is generally administered every 12 hours.  Peak serum levels are reached two hours after the drug is administered.  They then decrease (exponentially) until the next dose is administered.

          (b) Sketch a graph of serum level C as a function of time t assuming

               consistent dose administration every twelve hours.


The difference between the peak serum level (at two hours after a dose is administered) and the trough serum level (at twelve hours after a dose is administered) is the concentration that the patient is able to absorb from the dose administered.  If the patient’s serum level is consistently


where t=0 is when the serum level is measured, eight hours after a dose is administered,

          (c) what are the peak and trough levels for this patient?


Suppose this patient accidentally receives a double dose (OOPS!), and that the absorbtion is also doubled. 

          (d) What is the peak level in this event?

          (e) When does the patient get back down to a typical trough level?



Exponential Functions


Examples of exponential growth and decay

Exponential Decay – radioactive decay


In radioactive decay, we talk about half life for a substance.  This is a fixed time period during which the number of radioactive atoms present in a substance is reduced by half.


Carbon-14 Dating[3]

            Radiocarbon dating techniques, first developed by the American chemist Willard F. Libby and his associates at the University of Chicago in 1947, are frequently useful in deciphering time-related problems in archaeology, anthropology, oceanography, pedology, climatology, and recent geology. Through metabolic activity, the level of carbon-14 in a living organism remains in constant balance with the level in the atmosphere or some other portion of the earth's dynamic reservoir, such as the ocean. Upon the organism's death, carbon-14 begins to disintegrate at a known rate, and no further replacement of carbon from atmospheric carbon dioxide can take place. The rapid disintegration of carbon-14 generally limits the dating period to approximately 50,000 years, although the method is sometimes extended to 70,000 years. Uncertainty in measurement increases with the age of the sample.

            Although the method is suited to a variety of organic materials, accuracy depends on the half-life to be used, variations in levels of atmospheric carbon-14, and contamination. (The half-life of radiocarbon was redefined from 5570 ± 30 years to 5730 ± 40 years in 1962, so some dates determined earlier required adjustment; and due to radioactivity more recently introduced into the atmosphere, radiocarbon dates are calculated from ad 1950.) The radiocarbon time scale contains other uncertainties, as well, and errors as great as 2000 to 5000 years may occur. Postdepositional contamination, which is the most serious problem, may be caused by percolating groundwater, incorporation of older or younger carbon, and contamination in the field or laboratory.


Exponential Functions


Examples of exponential growth and decay

Exponential Decay – atmospheric pressure


Atmospheric pressure as a function of altitude is also represented as an exponential decay function.  For every 1000m increase of elevation, the atmospheric pressure is reduced by 11.5%.

Exponential Functions


Examples of exponential growth and decay

Exponential Decay – absorption of light in water


In lake and sea water, plant life can only exist in the top 10m or so, since daylight is gradually absorbed by the water.  The light intensity as a function of depth of water through which the light must pass is modeled as exponential decay.  (This is the Bouguer-Lambert law.)  The rate of absorption depends on the purity of the water and the wavelength of the light beam.


What function model would show that with an increase of depth by one meter, 75% of the light is absorbed?




Exponential Functions


What do exponential functions look like?


The adjectives and phrases often used to describe exponential growth are



                                      more rapid than x for any power of n...


This last description can be rather misleading without qualification. It can be difficult to distinguish graphs of exponential growth from graphs of polynomial growth when the scale is limited.


On the following page are graphs of five functions, y=x, y=x, y=x,y=x, and y=2-1, in various scales.  All graphs go through the points (0,0) and (1,1).  Identify which is the exponential function, and explain your choice.


Exponential Functions


What characteristics can be determined from exact exponential data?


Doubling time, half-life.

·        A cell population that doubles every hour.


·        A cell population that doubles every half hour.


·        The number of Carbon 14 atoms in dead organic material is reduced by half every 5730 years.


The percentage growth (or decay) over a fixed time period is always the same.

·        For every 1000m increase of elevation, the atmospheric pressure is reduced by 11.5%.




In the table below, identify doubling time, half life, percentage growth per unit time period as appropriate for each function.




Exponential Functions


Differential and Difference Models for Exponential Growth


Exponential growth of a value N means that from one unit time period to the next, the change in N  is a constant percentage of N.  This can be written as


The constant percentage change can be identified from data values as


(Note that with ,   has a constant value for data values measured at regular intervals.)


In the spreadsheet below, N˘, is used to denote ,[4]  the first difference for the N values; N˘˘ denotes the second differences, the difference of successive N˘ values.  This spreadsheet shows the formulas for examining the patterns of first and second differences, and of  for evenly spaced time units t.



The following page shows the above spreadsheet evaluated for the functions

N=2, N=t, N=t+ t, and N=e.  Note that for exponential functions, the N˘/N is constant, while for the polynomials, first or second or higher differences show special patterns.



In the tables, the first difference N˘ measures the change in N over a unit time period, thus giving an average rate of change during that time interval.  In calculus, the rate of change of a function is measured or calculated for an instant rather than for an interval.  Similar patterns appear in a sequence of values of first derivatives and first differences, of second derivatives and second differences, not value by value, but in the way the values change.  Thus the first derivative and first differences of a linear function are constant, the first derivative and first differences of a quadratic function are linear, etc.  And the first derivative and first differences of an exponential function are exponential.

Exponential Functions


Recognizing exponential growth from approximate data


Recognizing that numerical values of a function are exponential, as opposed to linear or polynomial or something else, is a difficult task.  Examining sample values of a quantity that is supposed to be constant, such as the slope, N˘, of a linear function, or the N˘/N for an exponential function, can provide useful information.


If the time intervals are not uniform in length, it is important to remember that the average rate of change over a time interval will be calculated as


Some examples follow.



Consider the growth of fruit flies in a favorable laboratory environment: unlimited food, unlimited space, and no predators. The data points are obtained by counting the flies at the same time each day for ten days.  Is growth of this fly colony exponential?[5]


      Day Number  t      0      1      2      3      4      5      6      7      8      9      10

Number of Flies  N    111  122  134  147   161  177  195  214  235  258  283


A chicken egg was incubated for three days at a temperature of 37°C. Subsequently during a 40 minute period, the temperature was reduced by t°C below 37°C and the number N of heartbeats per minute were measured[6]


          t°C     0.7     2.0     3.1     4.6     5.2     5.9     6.6     12.3     12.8

          N      154     133    110     94      83      82      75       38        36


Is the heart rate decreasing exponentially (as a function of degrees C below 37)?



The weight of a golden retriever puppy is recorded during her growth.  Is the puppy’s growth exponential?  Is the puppy’s growth exponential during the first seventy days?


AGE         WEIGHT

birth            1.48 kg                  100 days    13.64 kg

10 days       1.93 kg                  115 days    16.82 kg

20 days       2.50 kg                  150 days    24.55 kg

30 days       3.18 kg                  195 days    29.55 kg

40 days       4.09 kg                  230 days    31.82 kg

50 days       5.23 kg                  330 days    34.09 kg

60 days       6.82 kg                  435 days    35.00 kg

70 days       8.64 kg                 




The U.S. Census data from 1790 to 1900 is listed in the table below.  Does the growth of the U.S. population in this period appear to be exponential?


          Year     1790   1800   1810   1820   1830   1840   1850   1660   1870   1880   1890   1900

Pop’n(millions)    3.9    5.3      7.2      9.6     12.9    17.1    23.2    31.4    39.8    50.2    62.9    76.0



The following table shows the number of deaths per 100,000 women in the United States from stomach cancer and from lung cancer over the sixty year period from 1930-1990.  Are these death rates exponential with respect to time?[7]








[1] adapted from R. Fraga (ed.), Calculus Problems for a New Century

[2] adapted from Smith and Moore, The Calculus Reader, Chapter 2

[3] reprinted from Microsoft Encarta 1994

[4] In a more general setting later, N˘ denotes the rate of change per unit time period.  This is calculated as


In the special case of this spreadsheet, Dt=1.

[5] Data from Smith and Moore, Project CALC Instructor’s Guide

[6] Data from Dr. G. Wagner, Bern, Switz., in E. Batschelet, Introduction to Mathematics for Life Scientists, Chapter 10.

[7] Data adapted from  Gordon, Gordon, Fusaro, Siegel and Tucker, Functioning in the Real World: A Precalculus Experience, Chapter 3.