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T4,0 = 1+x+ x5
x2 x3 x4
24 120 Observe that this time these polynomials are distinct. That is because ex , and hence all of
its derivatives, is never 0.
The diagram below shows the graphs of ex and its Taylor polynomials up to degree 5. 4.5.2 Taylor’s Theorem and
Errors in Approximations We have seen that using linear approximation and higher order Taylor polynomials enable
us to approximate potentially complicated functions with much simpler ones with surprising accuracy. However, up until now we have only had qualitative information on the
behaviour of the possible error. We know that the error in using Taylor polynomials to
approximate a function seems to depend on how close we are to the focal point We have
also seen that the error in linear approximation seems to depend on the potential size of the
second derivative and that the approximations seem to improve as we encode more local
DE Math 128 324 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series information. However, we do not have any precise mathematical statements to back-up
these claims. In this section, we will correct this deﬁciency by introducing an upgrade...
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