So we had two examples in class, but I don't understand why they're different. And the professor is away today, which means I won't see him until the entire weekend has passed (a nightmare for students like me who obsess over a problem). 1. For which x is the approximation sin(x) ≈ x - (x^3)/6 correct within 1/100? The Taylor series for sin(x): x - (x^3)/3! + (x^5)/5! - (x^7)/7! + .... The answer was, For every x not equal to 0, this is an alternating series. Thus the error in sin(x) ≈ x - (x^3)/3! is no larger than |(x^5)/5!|. So |(x^5)/5!| < 1/100. 2. For which x is the approximation sin(x) = x - x^3/6 correct within 1/1000? My professor took the 3 order polynomial for the Taylor Series of sin(x) and used the Remainder Theorem: R3(x) = (f4c)(x-0)4/4! f(x) = sin(x) f1(x) = cos(x) f2(x) = -sin(x) f3(x) = -cos(x) f4(x) = sin(x) So |sin(c)| ≤ 1 |R3(x)| ≤ |x|4/4! Answer: |x|4/4! ≤ 1/1000 I don't understand why they used Remainder Theorem for the second one. Yes, I know the 1/100 and 1/1000 are different But why is the approach different?