6.3 Taylor and Maclaurin Series

[STARTS AT 28:13]

Professor: Now, we saw previously that the Maclaurin polynomials for sin x are given by this form here, this generic form. So use the fifth Maclaurin polynomial for sin x to approximate sin of pi over 18, and then bound the error. Well, based off of that calculation, the fifth Maclaurin polynomial is given by x minus x cubed over 3 factorial plus x to the fifth over 5 factorial, which means, if we evaluate this at pi over 18, which is pi over 18 minus pi over 18 cubed over 3 factorial plus pi over 18 to the fifth over 5 factorial, that'll be our estimate, our approximation, so 0.173648. 

Now, to bound the error-- that part is simple enough-- to bound the error, we want to consider the remainder. Well ideally, we would want to find the remainder of this at pi over 18. However, the sixth Taylor polynomial is exactly the same as the fifth, so that doesn't actually help us any. So we're going to go to the next distinct polynomial, and that is the seventh. So we're going to actually find the remainder-- the sixth remainder there. 

So that is going to be f, a seventh derivative, evaluated at some c over 7 factorial x minus 0, so we'll have pi over 18 to the 7. Now, this doesn't seem like it helps, because we need to come up with a bound for the function. Well, if you recognize that the seventh derivative of sin is either going to be sin or cosine, which is always bounded by 1, then the absolute value of this remainder is going to be less than or equal to pi over 18 to the 7 over 7 factorial because that is positive there. That is our error bound. 

Now, that's a pretty good error bound. It's actually a fairly small number, I believe. Let's check that out and see what that is. Pi over 18 to the 7 divided by 7 factorial-- yeah, 9.78 times 10 to the negative 10. And that's a very small error. 

Now next question, for what values of x does that Taylor polynomial, that Maclaurin polynomial, approximate sin x to within 0.0001. Well, for any general x, we know that our fifth Taylor polynomial gives us this-- 1 over 7 factorial absolute value of 7-- or of x to the 7. We want to know when that is less than or equal to 0.0001. In fact, let's go and say less than. Well that-- by rearranging that, we get absolute value of x is less than 0.907. So as long as our x is-- or the absolute value of x is less than 0.907, then we're going to have a very, very close fit, that 0.0001. 

[ENDS AT 32:16]