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Very quick Taylor Approximation Question

122
0
1. Homework Statement
Let f(x) = sin x
a) find p_6 (taylor polynomial 6th degree) for f at x = 0
b) How accurate is this on the interval [-1,1]



2. Homework Equations



3. The Attempt at a Solution

I got p_6 = x + (x^3)/6 + (x^5)/120, which was correct as per the solution manual. My issue is with part b.

What's the procedure that one takes to estimate the accuracy of a taylor approximation within a given interval?

Thank you all for your help
M
 
32,849
4,568
One of your signs is wrong in your polynomial. The Maclaurin series for sin(x) (which is a Taylor series evaluated at 0) is an alternating series. Do you know a formula for estimating the error when you truncate an alternating series? There's also a formula for a bound on the error in a Taylor series.
 
122
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My mistake, the correct formula is:

p_6(x) = x - (x^3)/6 + (x^5)/120


I understand that, in order to find the error, we must f(x) - p_6(x) = R_6(x). Where R_6(x) represents the error. What I don't understand is where the interval [-1,1] come into play.

Thanks!

M
 
32,849
4,568
The error expression, R_6(x) is a function of x. Since x is in the interval [-1, 1], then R_6(x) has a maximum value somewhere on that interval.
 

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