Approximate the function f(x)=\sin(x) using the corresponding Maclaurin polynomial: P_5(x), in a bound \epsilon(0,\delta). Determine a value of \delta>0, so that the rest R_5(x) verifies |R_5(x)|<0.0005 for all x\in{\epsilon(0,\delta)}

Well, the first thing that puzzles me a bit is that the order for P_5 and R_5 is the same. I assume that it is a mistake, and that the polynomial must be P_4, or the rest R_6. I have chosen to leave the degree of the polynomial as it was and turn up one grade to the rest.

So I have to find a delta for which any value within the environment (0,\delta) is less than the error 0.0005

So I did:

R_6(x)=|\displaystyle\frac{\sin(\alpha)x^6}{6!}|<0  .0005 0<\alpha<x

So I must find x:

R_6(x)=|\sin(\alpha)x^6|<0.36 0<\alpha<x

I must find x that satisfies the equation: \sin(x)x^6<0.36

I don't know how to go ahead.

Bye there.