"How do I compute the Taylor series for cos(7x^2) at x=0?

[cathy]

1. Homework Statement [/b]

Determine the Taylor series for the function below at x=0 by computing P 5 (x)
f(x)=cos(7x^2)

Homework Equations

I used to taylor series for cosx and replaced it with 7x^2
so i used 1-49x^4/2! +2401x^8/4!... and so on.
That should be correct, my attempt below :(

The Attempt at a Solution

1-(49x^4/2)+(2401x^8/24)-(117649x^12/720)+7^8x^16/40320
I even tried it by adding one more
7^10(x^18)/10!
Can someone tell me where I went wrong? It's nothing with the formatting because entering it like this into my homework showed a preview and it showed up like it should have :( what did I do wrong? Please advise. Thanks in advance.

I know we're not supposed to upload pictures of the answers, but I uploaded mines. If someone would look at it and see it its correct? IT's attached in the thumbnailhttps://www.physicsforums.com/attachments/68644

 

[vela]

Probably too many terms. The problem asked you to find the fifth-degree Taylor polynomial, right?

 

[cathy]

i tried taking out one or two terms.still didnt work :/

 

[vela]

What's the highest power of $x$ that should appear (in principle)?

 

[cathy]

Shouldnt it be 20?
because that would be where n=5

 

[vela]

No. Suppose you didn't know about the Maclaurin series for cos x and just did the problem the hard way by calculating derivatives of f. How many derivatives would you have to take to calculate $P_5(x)$? Surely not 20.

 

[cathy]

5 derivatives

 

[cathy]

but i tried taking out one term, and that didnt work.

 

[vela]

5 derivatives

Right. So what would be the power of $x$ in the highest-order term?

 

[cathy]

would it be 5?

 

[vela]

Exactly. The problem asked for a fifth-degree polynomial, so the highest-power should be $x^5$, so throw out any terms with a higher power of $x$.

 

[cathy]

Oh! so, if it asks for a certain polynomial, the power can't be higher than what they're asking for? That P(5) refers to the power, and not the term?

 

[cathy]

Thank you:)

 

[vela]

Right. For $P_5(x)$, you can have up to 6 terms, but if some vanish, you'll have fewer.

 

[cathy]

If I had a function asking the same thing as above but the function was 4+15x+x^2sinx, how would I fin a taylor series for this? Would I have to expand out the x^2sinx? What would I do with the 4 and the 15x? I know if I expand out the x^2sinx, I would multiply them to each other, but where would the 4 and 15 x come into play?

Actually, how would i make expand the x^2*sin(x)?
I know that sinx x trend is x- x^3/3! + x^5/5!
How do I do the x^2? Since the derivatives are 2x and 2? I plug in 0?

 

[cathy]

Oh actually, that was silly. I got it.