Taylor Polynomial (can you help me?)

  • Thread starter kingwinner
  • Start date
  • #1
1,270
0
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!
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,260
619
Do you know a formula for the Taylor expansion of cos(x) around x=0? What does this imply for cos(x^2)?
 
  • #3
1,270
0
Yes, I know that formula, I guess I can replace x by x^2 and get a formula for cos(x^2) too...
 
  • #4
1,270
0
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
Dick
Science Advisor
Homework Helper
26,260
619
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.
 
  • #6
1,357
0
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?
 
  • #7
1,270
0
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?
 
  • #8
370
0
Look up the generalized leibniz product rule.
 
  • #9
Dick
Science Advisor
Homework Helper
26,260
619
You could also simply look up the series expansion of arctan.
 

Related Threads on Taylor Polynomial (can you help me?)

Replies
3
Views
2K
Replies
8
Views
2K
  • Last Post
Replies
6
Views
4K
  • Last Post
Replies
1
Views
802
  • Last Post
Replies
2
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
3
Views
857
  • Last Post
Replies
1
Views
735
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
9
Views
790
Top