Asymptotics lecture
(for Maxim's class)

behaviour at infinity
$1/x, 1/x^2, 10^{-x}$: decay
$x, x^2, 1700x+x^2, 10^x$: growth (asymptotics kill lower order terms)

EG of algorithms, like searching in ordered list, or sorting
(guess number 1-100: 7 guesses ($\sim \log_2 n$); merge sort:
 clever: only touch each element 1 time in each merge, and each
 element goes through $\log_2 n$ merges, hence $n\log_2 n$)

behaviour at a finite point
$x,x^n$: zero of order $n$
  (visually, a simple crossing vs. a tangency vs. a -very- tangent point)

o/O notation:
o(1) just means ``goes to zero''
o(x) means ``goes to zero -faster- than x''

asymptotic is -a- notion of ``approximated''

defn of continuity and differentiability in these terms:
continuous: asymptotic to a -constant- (to within o(1))
differentiable: asymptotic to a -linear- (to within o(h))
  dissect equation: make it -sing-!
  [f(h) - (f(0)+ah)]/h  -- a is exactly the derivative!

Incidentally, this is the real reason behind why
``if a function is differentiable at a point, it's continuous''
...and so forth for higher differentiable.

within o(h) is -exactly- what we -mean- by ``tangent''

Give EG of |x| -in these terms-.

Show that constant value is unique, and that derivative is unique,
b/c of difference of polynomials.

Note that this definition has a -problem-:
it's non-constructive.
Of course, we know how to find it: take the derivative!

This polynomial is called the ``Taylor polynomial'':
it's the best asymptotic approximation of a given degree.

Taylor's theorem is that
$f(0)+f'(0)x+f''(0)x^2/2!+...$
is the Taylor polynomial.

Remember, the key point is that it's -asymptotic-.
In physics and other applications, this is why you look at
-small perturbations-.
EG of spring: for -small- movement it behaves like derivative.

In reality, you gotta watch out: it may be subject to large
perturbations.
If it's a spring, it breaks.
If it's an investing strategy, you go broke.
