%% Lecture 5 fun
%
% Last week.  
%
% More throughts on Gauss, and practice with functions and 
%    variables in functions.
%
% This week.  Reprise of quiz 2, 
%   code traces.
%
% a) Taylor Series expansion, and linear interpolation.
% 
% b) Simulating time varying functions?
% 
% c) Functions with multiple outputs.

% In many physical systems, the quantity of interest is something that
% varies with time (such as the position of an object, the temperature of a
% surface).  Let's call this function "f".  The goal of good engineering
% is often to *predict* what value "f" will have in the future.
% Sometimes this can be done analytically, and classes in control theory,
% electrical and mechanical engineering often explore systems which can
% be mathematically solved.
%
% Computation tools are often used if the analytic version can't be
% solved.  Computation rests on "doing simple things over and over
% again".
%
% Today we will explore a simple physical phenomena, and see what all
% needs to be done to simulated it.
%
% So let us consider some function x(t).  To make this concrete, suppose
% we care about the x-position of a mass attached to a spring:

%|         |===|
%|wwwwwwwww|   |
%|         |===|
%____________________________
%
% ----x----

% x(t) will be the position x at time t.

% Now if we are going to update x, we need to understand how it changes.
%
% "Hooke's" law from physics, says that a spring, exerts a force F
% following the rule:
%
% F = -k * distension 
%
% where distension is how long the spring is extended.
% and "k" is how strong the spring is (

% F = m * a
% dv/dt = a;
% dx/dt = v
% x is the value we are interested in.

% Now, the key concept in simulation is "Taylor Series Expansion".

% Who's heard of this?  It really is that big of a deal.

% It says that *any* smooth function f(t) can be approximated near some
% time t as:

% x(t+dt) ~= x(t) + dx/dt * dt + d^2x/dt^2 * dt^2 + ....
%

% Then, the trick is to set dt small enough that dt^2 is very close to
% zero.

% so how do we apply this to our spring?



% How should we organize our program?
%  we'll need an initialization step?
%    what do we need to specify?
%  we'll need an update function
%    what do we need to update?
%  we'll need some visualization
%    what do we want to visualize?
%    Can we make the visualization pretty?