%%% This code solves Ex1.1.17
syms x
format long 

%%% define function f
f = log(x^2+2)

%%% graph the function on interval [0, 2]
ezplot(f,[0,2])

%%%(a)
%%%% get the 3rd Taylor polynomial "tay_p3"  at expansion point 1.0
tay_p3 = taylor(f,4,1.0)

%%% (b)
%%% we define f - the 3rd Taylor polynomial
err = f - tay_p3

%%% Now use extreme value Thm on function "err"

der_f = diff(err, x)
root = fzero(char(der_f), [0, 1])
subs(err, x, root)

left = subs(err, x, 0.0)
right = subs(err, x, 1.0)

%%%% Answer: The Maximum error |f(x) - P3(x)| for 0<= x <= 1
%%%% is at the point 0.0. The max error = 0.026633657323934

%%% (c)
%%%% get the 3rd Maclaurin polynomial "tmac_p3" 
%%%% Note: Maclaurin polynomial is a special Taylor polynomial, which
%%%% expands at point 0.0
mac_p3 = taylor(f,4)

%%% (d)
%%% we define f - the 3rd Maclaurin polynomial
err_mac = f - mac_p3

%%% Now use extreme value Thm on function "err_mac"

der_f_mac = diff(err_mac, x)
root = fzero(char(der_f_mac), [0, 1])
subs(err_mac, x, root)

left_mac = subs(err_mac, x, 0.0)
right_mac = subs(err_mac, x, 1.0)

%%% Answer: The Max error is at point 1.0
%%% The max error = 0.094534891891836 (do not forget to take the abs. value)

%%% (e)
%%% Compute the relative error for Maclaurin polynomial approximation
err = subs(mac_p3, x, 1) - subs(f, x, 1)
relative_err = err/subs(f, x, 1)

%%% Compute the relative error for Taylot polynomial approximation
err_tay = subs(tay_p3, x, 0.0) - subs(f, x, 0.0)
relative_err_tay = err_tay/subs(f, x, 0)
%%%% Answer: Since the relative error of Taylor polynomial approximation is
%%%% smaller than that of the Maclaurin polynomial approximation. P3(0)
%%%% approximation is better.
%%%%