How do i graph a certain amount of elements in a taylor series.

Here is my question: For all of these on the same figure , plot:

• The Taylor Series expansion using the first 2 terms as a DASHED red line.

• The Taylor Series expansion using the first 4 terms as a DOTTED cyan line.

• The Taylor Series expansion using the first 6 terms as a DASHED blue line.

How do i do this?

This graph has the following limits:

xlim([-4*pi,4*pi])

ylim([-2,2])

[David Hill]

What is the function?

[Nathen Eberhardt]

what do you mean?

The Taylor Series is approximation of the sine function

[David Hill]

There are Taylor Series for different functions.

[per isakson]

See what Wiki says: Taylor series

[Nathen Eberhardt]

Sorry, Im not all too familiar with the taylor series.

This is how my teacher described it for the problem:

The Taylor Series approximation of the sine function is T = x − x^3/3! + x^5/5! − x^7/7! + x^9/9! − x^11/11! + ...

Answer 1

[Aaron Pedersen]

As this is homework you may have to write your own taylor series expansion code but:

%from https://www.mathworks.com/help/symbolic/sym.taylor.html
%TS
syms x %use x as your independent variable
f = sin(x); %define f(x)
%use built in taylor function,define independent variable, define n order expansion
T2 = taylor(f, x, 'Order', 2);
T4 = taylor(f, x, 'Order', 4);
T6 = taylor(f, x, 'Order', 6);

%plotting
figure(1);
hold on;grid on;

%labeling the bottom 
xticks([-4*pi,-3*pi,-2*pi,-pi,0,pi,2*pi,3*pi,4*pi])
xticklabels({'-4\pi','-3\pi','-2\pi','-\pi','0','\pi','2\pi','3\pi','4\pi'})

%setting limits
xlim([-4*pi,4*pi]);
ylim([-2,2]);

%assing plots to variable names for easy legend creation
%use fplot so we dont need an xaxis variable
p1 = fplot(T2,"r--");l1 = "two terms"; %red dashed
p2 = fplot(T4,"c:");l2 = "four terms"; %cyan dotted
p3 = fplot(T6,"b--");l3 = "six terms"; %blue dashed
legend([p1,p2,p3],[l1,l2,l3],'location','northwest');
hold off

[T2,T4,T6] being your taylor series representations.

"help plot" in the command window will give you more information on plotting styles.

hope this helped.

[Nathen Eberhardt]

Thank you this will!!

The Taylor Series approximation of the sine function is T = x − x^3/3! + x^5/5! − x^7/7! + x^9/9! − x^11/11! + ...

So how exactly would I code to do the first two terms or four terms of this function?

[Aaron Pedersen]

I have edited my answer to show you how I would do it, however depending on your course they may not allow the use of the internal matlab functions. I recommend looking at other psudocode, from a quick google (this looks useful) there are a few examples online of different automated methods of doing this. David Hill has answered with another method which works but is less automated. If I had to wirte my own function I would look at the patterns in the Taylor Series aproximation of your function and use a for loop to construct the values based off of its sigma notation defining x as a linspace as David did.

Answer 2

[David Hill]

x=linspace(-pi,pi,100)';%Taylor Series is only good from -pi to pi
t=[x,-x.^3/factorial(3),x.^5/factorial(5),-x.^7/factorial(7),x.^9/factorial(9),x.^11/factorial(11)];
plot(x,sum(t(:,1:2),2),x,sum(t(:,1:4),2),x,sum(t,2));

[Nathen Eberhardt]

Thank You!!