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Problem 2 Medium Difficulty

The graph of $ f $ is shown.
(a) Explain why the series
$ 1.6 - 0.8(x - 1) + 0.4(x - 1)^2 - 0.1(x - 1)^3 + \cdot \cdot \cdot $
is not the Taylor series of $ f $ centered at 1.
(b) Explain why the series
$ 2.8 + 0.5(x - 2) + 1.5(x - 2)^2 - 0.1(x - 2)^3 + \cdot \cdot \cdot $
is not the Taylor series of $ f $ centered at 2.

Answer

a. See details for answer.
b. See details for answer.

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Video Transcript

So for this problem, what we have is the Taylor series of F centered one. And what that's gonna look like is ffx equals the sum from an eagle zero to infinity of F to the 10th prime times one over and factorial times X minus one to the end. So that's gonna look like is F one plus F prime of one times X minus one um, plus F prime of 2/2 factorial and then times X minus one squared. So it's going to keep going on. If the series was a Taylor series of F than what we'd have is f prime of one equals a negative 08 However, since the function is increasing for X equals one, this is incorrect. So we know that the given series cannot be a Taylor series of F and then in and to be we want to show it centered it, too, so that this is going to look like is F of two plus F prime of two times X minus two plus F double prime of 2/2 times X minus two squared plus, and that's going to keep going on. But if this series was the Taylor series of F than what we'd end up having is that F F two equals 0.5. But that means that the function is increasing at X equals two. However, that's incorrect because the function actually reaches is it's extremely at X equals two, which would make a prime of two equal to zero, not 0.5. So that means that the series cannot be the proper Taylor series of F.

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