• Support PF! Buy your school textbooks, materials and every day products Here!

Taylor Series Remainder

  • #1

Homework Statement


What degree Taylor Polynomial around a = 0(MacLaurin) is needed to approximate cos(0.25) to 5 decimals of accuracy?


Homework Equations


taylor series....to complicated to type out here

remainder of nth degree taylor polynomial = |R(x)| <= M/(n+1)! * |x - a|^(n+1)
where a = 0 in this case
and
M >= |f^(n+1)(t)|



The Attempt at a Solution


I don't really get this question at all. I know that |R(0.25)| = 0.00001 <= M/(n+1)! * |x - a|^(n+1)
But how do I get M when |f^(n+1)(t)| is unknown? I don't even know what |f^(n+1)(t)| means!
 

Answers and Replies

  • #2
938
9

The Attempt at a Solution


I don't really get this question at all. I know that |R(0.25)| = 0.00001 <= M/(n+1)! * |x - a|^(n+1)
But how do I get M when |f^(n+1)(t)| is unknown? I don't even know what |f^(n+1)(t)| means!
f(n+1)(t) is the n+1:th derivative of f(t). So is your plan to find the lowest upper bound for Mn? It might be easier (and more likely to be correct too) if you just calculated enough terms from the series until you have the desired accuracy.
 
  • #3
But we aren't marked on that...it has to be through the remainder method.

Anyways, I know what f^(n+1)(t). I just don't know what to plug in for t. And after that, doesn't it just become a plug-and-check game for n until I get less than 0.00001?
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,833
955
All derivatives of cosine are [itex]\pm cosine[/itex] or [itex]\pm sine[/itex]. What is the largest possible value of a sine or cosine?
 

Related Threads on Taylor Series Remainder

  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
7
Views
4K
Replies
8
Views
882
Replies
1
Views
3K
Replies
1
Views
5K
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
2
Views
952
Replies
3
Views
4K
Top