Find the 3rd-order Maclaurin Polynomial (i.e. P3,o(u)) for the function f(u) = sin u, together with an upper bound on the magnitude of the associated error (as a function of u), if this is to be used as an approximation to f on the interval [0,2].
I did the question fine except for the upper bound error.
|Rn,o(u)| <= (k|u|^4)/(4!)
Where 0 <= sin(u) <= 2, and |sin(u)| <= k
The Attempt at a Solution
Okay, since 0 <= sin(u) <= 2 and |sin(u)| <= k, shouldn't k = sin(2)? The actual answer is k = 1, but if sin(u) is bounded between 0 and 2, it can't have a value greater than sin(2) so saying that it's max is 1 (at sin(pi/2) is an overestimation for this particular problem by my thought process.
Any enlightening comments as to why it's 1 and not sin(2)? Thanks!