Unformatted text preview: s. It turns out that for the theorem to be of value, we really need to be able to
say something intelligent about how large | f (n+1) (c) | might be even without knowing c.
For an arbitrary function, this might be a diﬃcult task since higher order derivatives have
a habit of being very complicated. However, the good news is that for some of the most
important functions in mathematics, such as sin(x), cos x and ex , we can determine roughly
how large | f (n+1) (c) | might be and in so doing, to show that the estimates obtained for
these functions can be extremely accurate.
Use linear approximation to estimate sin(.01) and show that the error in using this approximation is less that 10−4 .
We know that f (0) = sin(0) = 0 and that f (0) = cos(0) = 1, so
L0 (x) = T1,0 (x) = x.
Therefore, the estimate we obtain for sin(.01) using linear approximation is
sin(.01) ∼ L0 (.01) = .01
Taylor’s theorem applies since sin(x) is always diﬀerentiable. Moreover, if f (x) = sin(x),
then f (x) = c...
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