27: You should get

The Lagrange estimate of the error is

Determine the maximum in the formula.

Alternative: An experienced problem solver would get scared by the above result, since he/she knows that worse is to come. The first idea, a = 0, is beginning to look better and better. Granted, the function f is not defined there, but it does have a convergent limit at 0. This means that the definition of this function can be extended also to 0 in such a way that we obtain a continuous function on the whole real line. Remarkably, also f ′ that we obtained above has a convergent limit at 0, so it also can be continuously extended to include 0 in its domain, and one can check that this extended function still works as the derivative of f (that is, it also works at 0). Even more remarkably, the same is also true for all derivatives that we calculated, including the ugly long ones, and miraculously convergence at 0 is always obtained just by using one l'Hospital's rule.

Thus it is possible to find the Taylor polynomial with a = 0, just instead of directly finding derivtives one has to work with limits of the above expressions. Do it.

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