and proposed limits as an alternative to infinitesimals;
however, it took another hundred and fifty years
for mathematicians to provide
a formal and rigorous definition of calculus.
Convergence
Newton knew something about the convergence
of the infinitesimally small
and the infinitely large.
Zeno really messed people up with his paradoxes.
It took genius to say,
“You can actually run to the finish line.”
Zeno said that motion and change are illusions,
but calculus, the mathematic of motion and change,
shows us why Zeno was wrong.
For his first contribution to mathematics
Newton generalized the binomial theorem
using infinite series.
Infinitesimals are arbitrarily small;
choose any small number;
we can choose a number that is smaller.
Infinities are arbitrarily large;
choose any large number;
we can choose a number that is larger.
However, the sum of an infinite series
isn’t necessarily infinite,
but can converge to a finite number.
Steps
Except for geniuses,
whom I have known,
people do learn this stuff
one step at a time.
Everything is related
to everything else,
pretty much, so you
can relate to it.
Should we take little steps
and risk boring you,
or big steps
and risk losing you?
The derivative of a function f(x) = y
is the slope of the tangent of the curve at the point (x,y).
The tangent trigonometric function,
defined as the ratio of the sine over the cosine,
represents the slope of the circle at the point
where the radius touches the circle.
Did I get that right?
The derivative of a function
f(x) = y
is the slope of the tangent of the curve at the point(x,y)
. The tangent trigonometric function, defined as the ratio of the sine over the cosine, represents the slope of the circle at the point where the radius touches the circle. Did I get that right?See also in The book of science:
Readings on wikipedia: