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).(adsbygoogle = window.adsbygoogle || []).push({});

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:

R_{3}(x) = (f^{4}c)(x-0)^{4}/4!

f(x) = sin(x)

f^{1}(x) = cos(x)

f^{2}(x) = -sin(x)

f^{3}(x) = -cos(x)

f^{4}(x) = sin(x)

So |sin(c)| ≤ 1

|R_{3}(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?

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# Remainder Theorem and Error Help! Why are these 2 examples different?

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