Optimal Degree for Approximating Cosine with Taylor Series

[hadroneater]

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!

 

[clamtrox]

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 M_n? 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.

 

[hadroneater]

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?

 

[HallsofIvy]

All derivatives of cosine are $\pm cosine$ or $\pm sine$. What is the largest possible value of a sine or cosine?