00:01
In this question, we're going to estimate the error if p2 of x, which is 1 minus x squared over 2, is used to estimate the value of the cosine of x at x equals 0 .2.
00:14
So i'm going to start with my memorized maclaurin series for cosine x.
00:20
You may recall that the memorized series for cosine x is 1 minus x squared over 2 factorial plus x to the fourth over 4 factorial minus x to the sixth over 6 factorial plus dot dot dot.
00:39
And note that 2 factorial is the same thing as 2, so we can just replace the 2 factorial with a 2, which is what they have here.
00:49
So now, what does this tell us about the cosine of 0 .2? on the left, i've replaced x with 0 .2, so on the right, i should replace x with 0 .2 as well.
01:01
I'm getting 1 minus 0 .2 being squared over 2 plus 0 .2 being raised to the fourth over 4 factorial minus the quantity of 0 .2 raised to the sixth over 6 factorial plus dot dot dot.
01:23
And so we have here on the right -hand side a convergent alternating series.
01:28
And i am using the first two terms as my approximation for the cosine of 0 .2.
01:35
According to the alternating series error bound, the maximum error is given by the first omitted term from that series.
01:47
In this case, that's 0 .2 to the fourth over 4 factorial...