Question

The Taylor polynomial approximation pn (r) for f(x) = sin(x) around x,-0 is given as follows: TL 2k 1)! Write a MATLAB function taylor sin.m to approximate the sine function. The function should have the following header: function [p] = taylor-sin(x, n) where x is the input vector, scalar n indicates the order of the Taylor polynomials, and output vector p has the values of the polynomial. Remember to give the function a description and call format. in your script, set x =-30.13]. Call function taylor-sin to get the Taylor polynomials with n = 0, 1, 2 and 3 and store them in variables t0, t1, t2, and t3, respectively. Make figure 4 (eg., put figure(4) in your script) to include two panels one on top of the other. The top panel includes 5 curves: the four Taylor polynomials and the function f versus x. The bottom panel includes 4 curves which show the corresponding absolute error of each polynomials. Use function semilogy in the bottom panel to plot the error in log scale. Remember to give axis labels, title and legend. Set p4 = 'See figure 4';

Answer 1

function p=taylor_sin(x,n)

fprintf('given value of n =%d\n',n); % Print value of n

for k=0:n % Loop to vary k from 0 ,1 ,2 ,3

for i=1:61 % loop to vary x from -3:0.1:3

  

terms(i) = (-1)^(k)*(x(i)^(2*k+1))/factorial(2*k+1); % Taylor's apx.

end

t(k+1) = sum(terms) % calculate sum of all terms

% E = abs((sin(x)-SINx)/sin(x));

figure(k+1)

plot(terms,x)

end

end

octave:3> taylor_sin(-3:0.1:3, 3)
given value of n =3
t =    1.7764e-15

3 2 L-2 3 3 2 0 2

4 3 2 0

4 3 2 0

2 -1 -2 -0.8 0.6 0.4 -0.2 0.2 0.4 0.6 0.8