%% Lab 2
%
% This lab will be mostly focused constructing more and more complicated
% functions to handle tasks that need to be repeated.  
%
  
%% Evaluating the exponential function
%
% Consider the Taylor series expansion for $e^x$ about $x = 1$:
%
% $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}. $$
%
% The absolute error for partial sums is given by
%
% $$\left| e^x -  \sum_{n=1}^N \frac{x^n}{n!} \right| \leq \frac{e^{|x|} |x|^{N+1}}{N!}. $$
%


%% Exercise 1
% Write a function exp_frac(x) to compute the exponential function to a relative accuracy of
% 10e-15 for $0 \leq x \leq 1$.
%
% Create a plot of the absolute and relative error as $x$ increases.  Use
% the semilogy() command to help with this visualzation.

%% Exercise 2
% The exponential function has the important property that $e^{x/2} =
% (e^x)^2$.   Construct a function exp_series(x) to compute 
% the exponential function for any number $x$ using exp_frac(y) where $0 <
% y < 1$ and $x = y 2^m$ for some $m$. 
%
% Create a plot of the absolute and relative error as $x$ increases.  Use
% the semilogy() command to help with this visualzation.


%% Exercise 3
% Write a function bisection(f,a,b,TOL,Nmax) to find the root of a function
% f.

%% Exercise 4
% Write a function log_bisect(x) to compute the natural logarithm of $x \geq 1$
% using your exp_series() and bisection() functions.  What challenges do
% you encounter?
