Symbolic Math Toolbox

Taylor Series

The statements

• syms x
  f = 1/(5+4*cos(x))
  T = taylor(f,8)
  

return

• T =
  1/9+2/81*x^2+5/1458*x^4+49/131220*x^6
  

which is all the terms up to, but not including, order eight in the Taylor series for f(x):

Technically, T is a Maclaurin series, since its basepoint is a = 0.

The command

• pretty(T)
  

prints T in a format resembling typeset mathematics:

•             2           4     49    6
  1/9 + 2/81 x  + 5/1458 x  + ------ x
                             131220
  

These commands

• syms x
  g = exp(x*sin(x))
  t = taylor(g,12,2);
  

generate the first 12 nonzero terms of the Taylor series for g about x = 2.

Next, plot these functions together to see how well this Taylor approximation compares to the actual function g:

• xd = 1:0.05:3; yd = subs(g,x,xd);
  ezplot(t, [1,3]); hold on;
  plot(xd, yd, 'r-.')
  title('Taylor approximation vs. actual function');
  legend('Taylor','Function')
  
  
  

Special thanks to Professor Gunnar Bäckstrøm of UMEA in Sweden for this example.

Symbolic Summation

Calculus Example