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underlie everything else in calculus, one of
which is so central that it is usually called the
Fundamental Theorem.
As it turns out, Lego bricks are a perfect
medium to do a very basic geometry-first
demonstration of this theorem. We think Newton
would have approved. If you want to follow along
at home, you’ll need a base plate and some 2×2
square bricks.
After that, we’ll try a more accurate way of
looking at a smooth curve, instead of a blocky one
made of columns of Lego bricks. We’ll give you
some downloadable models to 3D print or to print
on paper to play along there, too.
The Curved Wall
Let’s make a wall of red Lego bricks using
columns. For the first column, leave a space
with “zero bricks.” Then, place 1 brick in the next
column, then 4, 9, 16, and finally 25, as shown in
Figure
A
. The height of each column in the red
wall contains bricks corresponding to the square
of its position. We alternated colors in the tallest
one, just to help you count them.
Now, let’s take some blue bricks, and look
at the differences from one red column to the
next (Figure
B
). As we go along this curve, the
differences get bigger, too.
Next, let’s move those differences to make
a wall of their own, putting each blue brick in-
between the columns of red ones (Figure
C
).
That signifies that the blue brick is the difference
between those two red columns (including the red
“column” of zero bricks). There are 1, 3, 5, 7 and 9
bricks in the blue wall.
This blue wall made up of all the differences of
the original wall has some interesting properties.
It is climbing up by two bricks per column. In
algebra you may have learned about the slope
of a wall (sometimes expressed as rise/run). A
straight line (like the one made by the blue wall)
will have a slope that is some constant number.
In this case, the blue wall has a slope of 2; it rises
two bricks for every column.
Remember that the blue wall represents how
the red curve is changing. In calculus, we call this
more-general version of a slope the derivative of
the original curve. From here on out we will call
the blue wall of differences the “derivative wall.”
Rich Cameron
(There are technicalities we are passing over
here, because our wall is stair-stepping rather
than being a smooth curve, but we’ll ignore all
that for now.)
Now, about those interesting properties of the
blue derivative wall. Suppose we added up the
bricks in that wall. The number of bricks in the
first blue column is 1. That’s the same as the
number of bricks in the first nonzero column of
the original (the difference between 0 and 1). If we
now add up the first two columns of the derivative
wall, we get a column the same height as the
second (nonzero) column in the original wall. This
is true all the way to the end (Figure
D
) where we
get a blue totaled-up column equal to the tallest
red column.
You can prove to yourself that this will always
be true. If you start off with the first value in the
original curve and keep adding the differences
between subsequent points as you go along, the
running total of blue bricks is the same as the
number of red bricks at that point in the original
A
C D
B
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