Hey I need help with this problem guys!


How do I find, using Taylor's Theorem, the error of the taylor polynomial of f(x)=sqrt(x) of degree 2 to approximate sqrt(8)?

and, find a bound on the difference of sin(x) and x- x^3/6 + x^5/120 for x in [0,1]

 

Use the Taylor Remainder Theorem.

http://en.wikipedia.org/wiki/Taylor's_theorem
http://www.youtube.com/watch?v=yJWFAG1g4Jw

 

What about the second question? I dont think that has to do with it.

 

You need Weierstrass for the second part.

http://mathworld.wolfram.com/WeierstrassApproximationTheorem.html

 

I think that sin(x) and the polynomial you gave are equivalent for the first three terms. Since this is an alternating series, a bound should be the absolute value of the next term.

 

Well spotted papabravo!

Of course there are only three terms stated in the second part.

 

There might possibly be better bounds. For example a Chebyshev approximation has a much better absolute error bound for a given number of terms over the entire interval of approximation, which is typically [-1,1]