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The simple model for growth is
exponential growth, where
it is assumed that
is proportional to . That is,
Separating the variables and integrating, we have
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. The same formula is used for exponential decay, except the decay constant is negative.
What is usually called Newton's law of cooling is a simple model for
the change in temperature of an object that is in contact with an
environment at a different temperature. It says that the rate of
change of the temperature of an object is proportional to the
difference between the object's temperature and the temperature of the
environment. Mathematically, this can be expressed as the differential
where is the constant of proportionality and
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 .
Consider the functions defined by
To be able to get an inverse the function must be one-to-one. You can plot the
functions to get a hint as to whether they are invertible or not.
Both satisfy the vertical-line test but is not invertible since it does
not satisfy the horizontal-line test. Indeed is not one-to-one, for instance
. From the plot it seems that the function is one-to-one.
In order to determine its inverse we solve for x.
We observe that one of the solutions is not defined since the argument of the logarithm can only be positive. Thus:
Let's look at the plot along with the line to see if our functions seem to make sense.
Let's check that we have computed the right inverse. By definition the composot
ion of the functions should be the line since an inverse is the reflectio
n about this line.
We are having difficulty getting for the last composition. Think about what
issue the computer has in dealing with this simplification as you will come across this in the exercises.
- A 5 lb. roast having initial temperature of
is put into a pre-heated oven at
. How much longer will it take for the roast to be medium rare (a temperature of
if the roast is
after 75 minutes.
- For each function below, use the background example under the section titled Inverse Functions to help answer each of the following. Be sure to shrink your graphs.
- Plot each funtion over a range that gives a good representation of the function. Which functions are not invertible and why?
- Find the inverse of the invertible function(s).
- For the invertivle functions, plot the function and its inverse along with the line . Use a domain that shows that the functions are inverses.
- Show that you have the correct inverse by using the composite definition. (If you come across a simplifying problem type the command assume(x>0).)
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Dina J. Solitro-Rassias