# Matlab taylor series script

• MATLAB
I have been working on writing g a script file that will:

1. Calculate f(x)=5sin(3x) using the Taylor series with the number of terms n=2, 5, 50, without using the built-in sum function. 
2. Plot the three approximations along with the exact function for x=[-2π 2π]. 
3. Plot the relative true error for each of the approximations 
4. Calculate the value of sin(x) and the error for x=π and x=3π/2 for each of the approximations 
5. How many terms are necessary for an error E<.000001?

I have been able to get as far as the third part of the question, any advice, tips or pointers are greatly appreciated!

I pasted the script I have so far bellow:

clear, clc, close all
%Define the limits, the original function and the Taylor series.
syms x

a = -2*pi:2*pi;

g = (5*sin(3*x));

T_2 = taylor(g, 'Order', 2);

T_5 = taylor(g, 'Order', 5);

T_50 = taylor(g, 'Order', 50);

z = (5*sin(3*a));

%plot the original function and the three Taylor series.

fg=figure;
ax=axes;
ez1=plot(a,z, 'r--');
hold on
ez2=ezplot(char(T_2),[-2*pi, 2*pi]);
ez3=ezplot(char(T_5),[-2*pi, 2*pi]);
ez4=ezplot(char(T_50),[-2*pi, 2*pi]);

legend('5sin(3x)','T2','T5','T50')

set(ez2, 'color', [0 1 0])
set(ez3, 'color', [0 0 1])
set(ez4, 'color', [1 0 1])

title(ax,['Graph of 5sin(3x) and taylor expansions T2, T5 and T50'])

RUber
Homework Helper
for part 4, you should just evaluate the taylor series at the points mentioned and subtract that from the true value.
Once you have done that, you can set up a script to try different values for 'order' until you reach the accuracy needed.

NYK
kreil
Gold Member
3. "Relative true error" to me just sounds like you're plotting the absolute difference between the approximations and the true values.

4. Pretty self explanatory, I'm with RUber on this one ^^.

5. Use the remainder term to figure this out. See the following links:

http://en.wikipedia.org/wiki/Taylor's_theorem#Motivation

http://www.millersville.edu/~bikenaga/calculus/remainder-term/remainder-term.html [Broken]

Last edited by a moderator:
NYK
RUber
Homework Helper
If you wanted to do this without the built in Taylor functions, you could define the derivatives of g:
g(x)= 5sin(3x)
d(n,g(x))=5*3^n*sin(3x+n*pi/2)
and the taylor series is
(x-x_0)^n/n!*d(n,g(x_0)).
Often, the series is evaluated at x_0 = 0...which produces a pretty simple result for the sine function.

NYK
Thanks RUber, you are compltely correct, I read the problem statement a little more closely and releaized i wasnt do it correct.

So ive tried working with this:

clear;clc

n =[2 5 50]
do=linspace(-2*pi,2*pi,720);
for i =1:720

for k=1:1:50
ns=2*k+1
T(i)=T(i)+5*(-1)^k*(3*do(i))^(ns)/factorial(ns);
end
end

havent had any luck with it so far though