Hello,
I am currently stuck on the following problem in a Calculus BC online course that I am taking while still in high school.
The first part : The Derivative Form of the Remainder for the Taylor series of f(x) will be denoted by Rsubk(x). Express Rsub5(6) in terms of the value c for the Taylor series for f(x) = sin x when sin x is expanded about a = 0.
The second part : Use the Derivative Form of the Remainder Rsubk(x) to determine the degree of the Taylor polynomial that approximates sin 6 to within 0.0005 of its actual value.
Any help whatsoever would be amazing.
I have been told that the series for f(x)=sin(x) at a=0 is equal to
tsub(2k+1)(x)= sigma (i=0 to k) of (((-1)^i)/(2i+1)!)*(x^(2i+1))
But where to go from here?
OK a bit of research tells us that the "Derivative form" is another name for
Lagrange's form for the remander.
So we have:
f(x) = f(a) + (x-a).f'(a)/1! + (x-a)^2.f"(a)/2! + (x-a)^3.f^{3}(a)/3!
........................ +...+ (x-a)^n.f^{n}(a)/n! + R_n
R_n = (x-a)^{n+1}.f^{n+1}(c)/(n+1)!, for some c between a and x.
(there is some confusion about if this remainder should have subscript n or n+1
at least in my mind, that is should n denote the highest power of x in the
un-ignored part of the expansion, or the number of terms in the expansion)
RonL