%% Annotated Matlab transcript -- 1/23
%% CS 323 - Doug DeCarlo

% operations on vectors...
>> a = 1:5
% scalar multiplication
>> 2*a
% addition of a scalar to each dimension
>> a+5
% * is for multiplying matrices, so this doesn't work here (error)
>> a*a
% use .* for termwise multiplication
>> a.*a
% same for exponentiation -- this squares a matrix (but gets an error)
>> a^2
% termwise exponentiation
>> a.^2
>> a.^3
% For matrix multiplication: this uses ' (transpose)
% so that this is just the inner/dot product
>> a * a'
% and the outer product
>> a' * a
% you can also apply functions to each term
>> exp(a)


% Now make a domain for graphing in [0,1] (points separated by 0.01)
>> x = 0:0.01:1;
% graph example: plot the line y = 2x+5
>> plot(x,2*x+5)

% Taylor series example for e^x around x=0, n=1
% plot e^x in red, and its linear taylor polynomial in green
% (notice how they intersect at x=0 and have the same tangent..)
>> plot(x,exp(x),'r', x,1+x,'g')
% the error
>> plot(x,exp(x)-(1+x))
% the upper bound on error from the taylor error
>> plot(x,x.^2/2.*exp(x))
% the actual error and the upper and lower bounds from the taylor error
>> plot(x,exp(x)-(1+x),'r',x,x.^2/2.*exp(x),'b',x,x.^2/2,'b')

% let's see how this error varies with the number of terms
>> n = 1:10;
>> plot(n,exp(1)./factorial(n+1))
% let's get a closer look for the example in lecture
>> n = 7:10;
>> plot(n,exp(1)./factorial(n+1))