clear
format compact

%% 
% Let's approximate exp(x) over the interval [-1,1]. We can sample it at,
% say, 40 points, and find the best-fitting straight line to that data. 
t = linspace(-1, 1, 40)';
y = exp(t);
plot(t, y, '.')
V = [t.^0 t];
c = V\y
p = @(t) c(1) + c(2)*t;
hold on
fplot(p, [-1 1])
title('Least-squares fit to data from exp(x)')
xlabel('x')
ylabel('y')
grid on
legend('y = exp(x)', 'y = c1 + c2*x', 'location', 'best')

%%
% There's nothing special about 40 points. By choosing more we get closer
% to the ``true'' exp(x).
t = linspace(-1, 1, 200)';
y = exp(t);
V = [t.^0 t];
c = V\y

%%
t = linspace(-1, 1, 1000)';
y = exp(t);
V = [t.^0 t];
c = V\y

%%
% It's quite plausible that the coefficients of the best-fit line are
% approaching a limit as the number of nodes goes to infinity.