Where should we start? Obviously this problem expects you to know what a "MacLaurin series" is. Do you know that? It is simply the Taylor's series at x= 0. Do you know what that is? If not look up "MacLaurin series" or "Taylor's series" in the index of your book.

And, I strongly suspect that in the section where this problem is given, there is a formula for the error you get in using cutting a McLaurin or Taylor'spolynomialof degree n (that is, cutting off the McLaurin or Taylor's series at the nth power). It should look something like this:

where " " is an upper bound on the absolute value of the n+1 derivative of the function.

Since this problem asked about the "9th order MacLaurin series for sin(x)" for x between -7 and 7. Okay you need . What is the 10th derivative of sin(x)? What is its largest value between -7 and 7 (radians: is about 6.28 so this is from less than to larger than ).