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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 arguement 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 pot of soup taken from the stovetop at
and a jug of apple juice taken from a refrigerator at
are put on a dinner table. The temperature in the dining room is held at
. After 10 minutes, the soup temperature is
and the juice temperature is
- How much longer will it take the soup to lose its taste quality, i.e., to cool down to the temperature of
? What is the juice temperature at this moment?
- Plot the graphs (temperature) versus (time) for both products. From the graphs, make an estimate when the soup should be eaten if it is supposed to be best at
- Copy every command from the background under the section titled Inverse Functions and paste them into Maple and execute each command. Be sure to shrink your graphs.
- Given the functions
(Hint: Use Maple's surd function when entering fractional exponents.)
- Plot the function over the interval
and plot the function over the same interval, but on a separate graph. Which function is not invertible and why?
- Find the inverse of the invertible function.
- Plot the function and its inverse along with the line on the domain of
- 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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