2074HW4_1

function [Taylor_series,Taylor_error,Taylor_order]=Chapra_4p19(x,a,tol)
%% Taylor series for Chapra 4.19 -- Taylor series for Planet Coordinates
%
%% Input
%  x   = values on interval over which approximation is evaluated (vector)
%  a   = base point, x_i=a, used in the Taylor series expansion (scalar)
%  tol = maximum error tolerance that approximation should not exceed
%        on the interval (scalar)
%
%% Output
%  Taylor_series = Values of f(x) for Taylor series
%                  that meets 0.015 maximum absolute error (vector)
%  Taylor_error  = True absolute error values for each value in vector x
%                  that meet 0.015 maximum absolute error (vector)
%  Taylor_order  = Lowest-order Taylor series (integer)
%                  that meets tol=0.015 maximum absolute error (scalar)

%% Write your code here.
f = @(x) x-1-sin(x)/2;
y = f(x); %True value
T = 0*f(x)+ f(a); %Taylor approximation
%^^^Creates vector with as many entries as x, all filled to whatever value
%f(a) is

n = 0; %Order of the approximation
error = 1;

while error > tol
    n = n + 1;
    if n == 1
        deriv = 1 - cos(a)/2;
    elseif (mod(n,2) == 0)
        deriv = (-1^(0.5*n+1))*sin(a)/2;
    else
        deriv = (-1^(0.5*(n-1)+1))*cos(a)/2;
    end

    T = T + (deriv*(x-a).^n)/factorial(n);
    error = max(abs(y - T));
end

Taylor_series = T;
Taylor_error = error;
Taylor_order = n;

end