Example 4 Below
is the sketch of a function . Sketch the graph of the derivative of this
function, .
Solution
At first glance this seems to an all but impossible
task. However, if you have some basic
knowledge of the interpretations of the derivative you can get a sketch of
the derivative. It will not be a
perfect sketch for the most part, but you should be able to get most of the
basic features of the derivative in the sketch.
Let’s start off with the following sketch of the function
with a couple of additions.
Notice that at ,
,
and the tangent line to the function is
horizontal. This means that the slope
of the tangent line must be zero. Now,
we know that the slope of the tangent line at a particular point is also the
value of the derivative of the function at that point. Therefore, we now know that,
This is a good
starting point for us. It gives us a
few points on the graph of the derivative.
It also breaks the domain of the function up into regions where the
function is increasing and decreasing.
We know, from our discussions above, that if the function is
increasing at a point then the derivative must be positive at that
point. Likewise, we know that if the
function is decreasing at a point then the derivative must be negative at
that point.
We can now give
the following information about the derivative.
Remember that
we are giving the signs of the derivatives here and these are solely a
function of whether the function is increasing or decreasing. The sign of the function itself is
completely immaterial here and will not in any way effect the sign of the derivative.
This may still
seem like we don’t have enough information to get a sketch, but we can get a
little bit more information about the derivative from the graph of the
function. In the range we know that the derivative must be
negative, however we can also see that the derivative needs to be increasing
in this range. It is negative here
until we reach and at this point the derivative must be
zero. The only way for the derivative
to be negative to the left of and zero at is for the derivative to increase as we
increase x towards .
Now, in the
range we know that the derivative must be zero at
the endpoints and positive in between the two endpoints. Directly to the right of the derivative must also be increasing
(because it starts at zero and then goes positive therefore it must be increasing). So, the derivative in this range must start
out increasing and must eventually get back to zero at . So, at some point in this interval the
derivative must start decreasing before it reaches . Now, we have to be careful here because
this is just general behavior here at the two endpoints. We won’t know where the derivative goes
from increasing to decreasing and it may well change between increasing and decreasing
several times before we reach . All we can really say is that immediately
to the right of the derivative will be increasing and
immediately to the left of the derivative will be decreasing.
Next, for the ranges and we know the derivative will be zero at the
endpoints and negative in between.
Also, following the type of reasoning given above we can see in each
of these ranges that the derivative will be decreasing just to the right of
the left hand endpoint and increasing just to the left of the right hand
endpoint.
Finally, in the last region we know that the derivative is zero at and positive to the right of . Once again, following the reasoning above,
the derivative must also be increasing in this range.
Putting all of this material together (and always taking
the simplest choices for increasing and/or decreasing information) gives us
the following sketch for the derivative.
Note that this was done with the actual derivative and so
is in fact accurate. Any sketch you do
will probably not look quite the same.
The “humps” in each of the regions may be at different places and/or
different heights for example. Also,
note that we left off the vertical scale because given the information that
we’ve got at this point there was no real way to know this information.
That doesn’t mean however that we can’t get some ideas of
specific points on the derivative other than where we know the derivative to
be zero. To see this let’s check out
the following graph of the function (not the derivative, but the function).
At and we’ve sketched in a couple of tangent
lines. We can use the basic rise/run
slope concept to estimate the value of the derivative at these points.
Let’s start at . We’ve got two points on the line here. We can see that each seem to be about
onequarter of the way off the grid line.
So, taking that into account and the fact that we go through one
complete grid we can see that the slope of the tangent line, and hence the
derivative, is approximately 1.5.
At it looks like (with some heavy estimation)
that the second point is about 6.5 grids above the first point and so the
slope of the tangent line here, and hence the derivative, is approximately
6.5.
Here is the sketch of the derivative with the vertical
scale included and from this we can see that in fact our estimates are pretty
close to reality.
Note that this idea of estimating values of derivatives
can be a tricky process and does require a fair amount of (possible bad)
approximations so while it can be used, you need to be careful with it.
