%% 1.1 Set Up

% a)
N = 30;

% b)
k = 1:N;
a_denom = 2.^k;
a = 1./a_denom;

% c)
x = transpose(linspace(0,4*pi,150));

% d)
f = zeros(150,1);
f_prime = zeros(150,1);

figure(1)
stem(k,a)
title("Fourier Coefficients");
xlabel("Term Number"); ylabel("Coefficient");

%% 1.2 Compute Function

%fi = @(x) sum(a.*sin(k.*x));
%f = fi(x);
% I wasn't sure how to make element operations work, so I opted for a nested for loop
% instead

f = [];
for i = 1:150
    f_new = [];
    sum_in = [];
   for j = k
       in = a(j)*sin(j*x(i));
       sum_in = [sum_in, in];
   end
   f_new = sum(sum_in);
   f = [f, f_new];
end
 f = transpose(f);

 figure(2)
 plot(x,f)
 title("f (x)");
 xlabel("x");ylabel("y");

%% 1.3 Compute Derivative

f_prime = [];
for i = 1:150
    f_new = [];
    sum_in = [];
   for j = k
       in = a(j)*j*cos(j*x(i));
       sum_in = [sum_in, in];
   end
   f_new = sum(sum_in);
   f_prime = [f_prime, f_new];
end
 f_prime = transpose(f_prime);

 figure(3)
 plot(x,f, 'k')
 hold on
 plot(x,f_prime, 'b')
 title("f(x) and f'(x)");
 xlabel("x");ylabel("y");
 legend( "f (x)", "f ' (x)");