Taylor Polynomial (can you help me?)

[kingwinner]

1) Let f(x) = (x^3) [cos(x^2)].
a) Find P_(4n+3) (x) (the 4n + 3-rd Taylor polynomial of f(x) )
b) Find f^(n) (0) for all natural numbers n. (the n-th derivative of f evaluated at 0)

I know the definition of Taylor polynomial but I am still unable to do this quesiton. I tried to find the first few terms but I can't see an obvious pattern. I have no problem using the definition to find the first few terms...but this is a weird question. Can someone nicely give me some hints on both parts?

Any help is greatly apprecaited!

 

[Dick]

Do you know a formula for the Taylor expansion of cos(x) around x=0? What does this imply for cos(x^2)?

 

[kingwinner]

Yes, I know that formula, I guess I can replace x by x^2 and get a formula for cos(x^2) too...

 

[kingwinner]

I also have another terrible quesiton...I am dying from it...

2) Find the 2006-th derivative evaluated at 0 and the 2007-th derivative evluated at 0 for f(x)=tan^(-1) x (inverse tan)

Any hints?

 

[Dick]

I also have another terrible quesiton...I am dying from it...

2) Find the 2006-th derivative evaluated at 0 and the 2007-th derivative evluated at 0 for f(x)=tan^(-1) x (inverse tan)

Any hints?

I'm not very clear on what you mean f(x) to be.

 

[e(ho0n3]

2) Find the 2006-th derivative evaluated at 0 and the 2007-th derivative evluated at 0 for f(x)=tan^(-1) x (inverse tan)
Any hints?

Find the first couple of derivates. Do you see a pattern? Can you determine a formula for the nth derivative of arctan from it? Can you prove your formula works for all n?

 

[kingwinner]

Find the first couple of derivates. Do you see a pattern? Can you determine a formula for the nth derivative of arctan from it? Can you prove your formula works for all n?

There isn't an obvious pattern...what can I do?

 

[ZioX]

Look up the generalized leibniz product rule.

 

[Dick]

You could also simply look up the series expansion of arctan.