**Exponential
Functions**

Contents

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

explosive!

unbounded!

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, D*t*=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.