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
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.
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]