An example of a piecewise defined function
Imagine an icecream cone containing a single, perfectly spherical
scoop of ice cream. If the scoop is very small it will be
contained entirely within the cone. If it is somewhat larger the scoop will
rest against the sides of the cone and part of it will be within the
cone and part of it will extend beyond the cone. If the scoop it very
large it will rest on the outside lip of the cone and only a very
small part of the icecream ball will be within the cone itself.
Define a function of the radius R which describes the volume
in a ball of icecream of radius R which lies below the lip of
the icecream cone. You may find it useful to refer to
the formula
for the volume of the cap of height H
taken from a sphere of radius R.
Notice that defining the function requires
considering three three separate cases already mentioned.
To deal with this you should use
the piecewise command to define the function. (This is not
available MapleVR3. In this case use the if ...then ...elif
...else ...fi; command. An alternate is to use the
Heaviside fucntion).
To begin, assume that
that the angle formed at the tip of the cone is 
radians and
that the height of the cone is 1 unit.
Plot the function you have defined and ask yourself if it agreees with
common sense. Here you should use the discont=true option to the
plot command so that you do not falsely connect the function at the
point where the definition changes.
Use the graph to estimate the largest amunt of icecream whic will be
contained below the lip of the cone.
(Experiment with plotting the
function
with and without the discont=true option.)
To use this option
invoke the plot command on a function f as follows:
Next
solve the general icecream cone problem symbolically when the angle is 
radians.
Use plot3d to plot the volume as a function of both radius and
angle. Note that it is no lonegr possible to plot the function simply
as a function of R since the angle is not determined. What happens
if you do try to plot it?
Instructor
Juris Steprans
email address: steprans@mathstat.yorku.ca
Department of Mathematics and Statistics.
Ross 624 South, ext. 33952
York University
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