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<center><b><font size=+2>Animations unifying the relationships among rates
of change and accumulations of power functions</font></b>
<br>&nbsp;
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$x^3 \leftrightarrow 3x^2$ Click here

$x^2 \leftrightarrow 2x$ Click here


The formulas illustrated are:<

$(n+1)^2=1+ 2 \sum_{j=1}^n j + 1 \sum_{j=1}^n 1$

$(n+1)^3=1+ 3 \sum_{j=1}^n j^2 + 3 \sum_{j=1}^n j + 1 \sum_{j=1}^n 1$

which may be rearranged to obtain the familiar summation formulas
used in integrating</b> $f(x)=x$ <b>and</b> $g(x)=x^2$ <b>directly:<

$\sum_{j=1}^n j = {n (n+1) \over 2}$

$\sum_{j=1}^n j^2 = {n(n+1)(2n+1) \over 6}$

These demonstrations show that the coefficients of $n$ and $1 \over n+1$
in the differentiation
and integration formulas share a common source: Pascal's triangle<

They foreshadow the deeper connection between rates of change and
geometric boundaries made systematic in the Stokes' theorems.

We ultimately wish to emphasize other aspects of the duality between
differentiation and integration in
a more unified and symmetrical fashion, exhibiting in parallel the
linearity rules for both differentiation
and integration, the product rule beside integration by parts, the
chain rule along with substitution, etc.

Along with summation formulas, discrete analogues of other calculus
rules, such as "summation by parts"
will also be highlighted.
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