The purpose of this lab is to use Maple to study applications of exponential and logarithmic functions. These are used to model many types of growth and decay, for example bacterial growth and radiaoctive decay. This lab also describes applications of exponential and logarithmic functions for heating and cooling and to medicine dosage

Separating the variables and integrating (see section 4.4 of the text),
we have

so that

In the case of exponential growth, we can drop the absolute value signs around , because will always be a positive quantity. Solving for , we obtain

which we may write in the form , where is an arbitrary positive constant.

where is a constant. This is the same equation as in exponential growth, except that replaces . The solution is

where is a positive constant. Physically, is the amount of material present at .

Radioactivity is often expressed in terms of an element's half-life.
For example, the half-life of carbon-14 is 5730 years. This statement means
that for any given sample of
, after 5730 years, half of it
will have undergone decay.
So, if the half-life is of an element Z is years, it must be
that
, so that and
.

where is the constant of proportionality and is the temperature of the environment. Using a technique called separation of variables it isn't hard to derive the solution

where is the temperature of the object at .

A problem facing physicians is the fact that for most drugs, there is a concentration, , below which the drug is ineffective and a concentration, , above which the drug is dangerous. Thus the physician would like the have the concentration satisfy

This means that the initial dose must not produce a concentration larger than and that another dose will have to be administered before the concentration reaches .

> f := x -> exp(-2*x);

> simplify(ln(3)+ln(9));

> ln(exp(x));

> simplify(ln(exp(x)),assume=real);

The

`assume=real`

is needed in the command above, because Maple
usually works with complex variables.
> solve(exp(-3*x)=0.5,x);

> plot(log[10](x),x=0..100);

Sometimes you need to use experimental data to determine the value of
constants in models. For example, suppose that for a particular drug,
the following data
were obtained. Just after the drug is injected, the concentration is
1.5 mg/ml (milligrams per milliliter). After four hours the
concentration has dropped to 0.25 mg/ml. From this data we can
determine values of and as follows. The value of is the
initial concentration, so we have

To find the value of we need to solve the equation

which we get by plugging in and using the data . Maple commands for solving for and defining and plotting the function are shown below.

> k1 := solve(0.25=1.5*exp(-4*k),k);

> C1 := t -> 1.5*exp(-k1*t);

> plot(C1(t),t=0..6);

- Exponential growth can be used to model the growth of certain
kind of investments. Suppose that the value of an investment
satisfies the differential equation

where is the interest rate. If and you start with an investment of $15,000 dollars, how many years does it take for the value to double? How many years does it take to quadruple? Is there an easy way to answer the second part of this question? Explain. - A Pizzeria uses insulated bags to keep delivery pizzas warm. Use
Newton's law of cooling to model the temperature of a pizza. Assume
that
,
and
. Plot the
temperature for
. Does this value of seem
reasonable, based on your experience with delivery pizza? If not,
try to come up with a more reasonable value.
- Suppose that for a certain drug, the following results were
obtained. Immediately after the drug was administered, the
concentration was 6 mg/ml. Five hours later, the concentration had
dropped to 1.2 mg/ml. Determine the value of for this drug.
- Suppose that for the drug in the previous exercise, the maximum
safe level is
and the minimum effective level is
. What is the maximum possible time between doses
for this drug? (Hint - the initial dose should give an initial
concentration of .)

2001-02-08