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- 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.
- Plot the three approximations along with the exact function for x=[-2π 2π].
- Plot the relative true error for each of the approximations
- Calculate the value of sin(x) and the error for x=π and x=3π/2 for each of the approximations
- 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'])