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

cathy
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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
 

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Probably too many terms. The problem asked you to find the fifth-degree Taylor polynomial, right?
 
i tried taking out one or two terms.still didnt work :/
 
What's the highest power of ##x## that should appear (in principle)?
 
Shouldnt it be 20?
because that would be where n=5
 
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.
 
5 derivatives
 
but i tried taking out one term, and that didnt work.
 
cathy said:
5 derivatives
Right. So what would be the power of ##x## in the highest-order term?
 
  • #10
would it be 5?
 
  • #11
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##.
 
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  • #12
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?
 
  • #13
Thank you:)
 
  • #14
Right. For ##P_5(x)##, you can have up to 6 terms, but if some vanish, you'll have fewer.
 
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  • #15
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?
 
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  • #16
Oh actually, that was silly. I got it.
 
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