Let's begin with a simple problem from geometry. We will consider all
rectangles whose perimeter is 100 cm and ask to find the one with the
largest area. Here is a demonstration which can give you a feel for
how the area changes as the rectangle changes.
If you think about it, this question might remind you of the first
homework assignment in which we studied the best size for a cell. The
crucial ingredient in our analysis was a consideration of how the
surface area and volume were related to one another. In an analogous
way, we are here relating the perimeter and area of a rectangle.
To approach this problem, let's set up some notation. First,
we'll consider a rectangle whose perimeter is 100 cm and call its
and its height
With this notation, the area of the rectangle becomes
Of course, not just any or will do:
we only want to consider rectangles with perimeter 100 cm which means
enables us to solve for as
rewrite the area as
Now that we have a function which measures the area of the
rectangle, we would like to find its maximum value. But before we jump
into this, let's remember that there are restrictions on the
value of : for instance, we want both
and to be positive which means we will only consider
values which satisfy
In this example, this observation will not turn out to be crucial, but
it is generally a good idea to consider any restrictions like this
imposed by the real world.
To find the maximum value, let's consider the derivative of
: it is
. That is, the
derivative is a simple linear function whose graph is a straight line
with slope -2 and y-intercept 50. In the demonstration below,
we graph both the original function (on the left) and the derivative
(on the right).
From the graph of the derivative, it is clear that when
which means that
is increasing. When
, we have
is decreasing. The
point at which the transition occurs is at
this is hence the maximum value.
Notice that when
, we also have that
and so the rectangle with the largest area is a
square. This is a rather common occurrence: nature likes to use
symmetry to optimize some property, in this case area.
Of course, the function
is a simple
quadratic and we could have read off its maximum value from the
graph. However, the analysis using the derivative that we have just
completed will help us tackle much more difficult situations.