Taylor Polynomial (can you help me?)

1. Mar 22, 2007

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!

2. Mar 22, 2007

Dick

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

3. Mar 22, 2007

kingwinner

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

4. Mar 22, 2007

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?

5. Mar 22, 2007

Dick

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

6. Mar 22, 2007

e(ho0n3

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?

7. Mar 22, 2007

kingwinner

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

8. Mar 22, 2007

ZioX

Look up the generalized leibniz product rule.

9. Mar 22, 2007

Dick

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