Taylor Polynomial (can you help me?)

In summary: There isn't an obvious pattern...what can I do?Look up the generalized leibniz product rule.You could also simply look up the series expansion of arctan.
  • #1
kingwinner
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!
 
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  • #2
Do you know a formula for the Taylor expansion of cos(x) around x=0? What does this imply for cos(x^2)?
 
  • #3
Yes, I know that formula, I guess I can replace x by x^2 and get a formula for cos(x^2) too...
 
  • #4
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
kingwinner said:
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
kingwinner said:
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
e(ho0n3 said:
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
Look up the generalized leibniz product rule.
 
  • #9
You could also simply look up the series expansion of arctan.
 

1. What is a Taylor Polynomial?

A Taylor Polynomial is a mathematical function that approximates a more complicated function by using a series of simpler polynomial functions. It is named after the mathematician Brook Taylor.

2. How is a Taylor Polynomial calculated?

A Taylor Polynomial is calculated using derivatives of the original function evaluated at a specific point. The more derivatives used, the more accurate the approximation will be.

3. What is the purpose of a Taylor Polynomial?

The purpose of a Taylor Polynomial is to provide a simpler function that can be used to approximate more complicated functions. This can be useful in solving mathematical problems or in making predictions about real-world phenomena.

4. What is the difference between a Taylor Polynomial and a Taylor Series?

A Taylor Polynomial is a finite series of polynomial terms, while a Taylor Series is an infinite series of polynomial terms. A Taylor Series is a more accurate representation of a function, but it requires an infinite number of terms to be completely accurate.

5. How do I know if a Taylor Polynomial is a good approximation?

The accuracy of a Taylor Polynomial can be determined by comparing it to the original function. The closer the values of the two functions are, the better the approximation. Additionally, the accuracy can be improved by using more terms in the polynomial.

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