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Unformatted text preview: lus to evaluate integrals
involving this function. This is somewhat problematic since integrating functions similar
to f (x) is very often necessary in the statistical analysis of data, as well as many other
applications. As such we are left to use numerical methods to approximate these integrals.
In the next example, we will see how we can use Taylor polynomials and Taylor’s Theorem
to aid us with this approximation process.
Example. Estimate 0 .1 − x2
0 dx with an error of no more than 10−6 . We begin by using linear approximation to approximate the function h(u) = eu on the
interval [−0.01, 0]. (The reason we chose this interval will be clear very soon). Then, since
h (u) = eu , we get
T 1, 0(u) = e0 + e0 (u − 0) = 1 + u
and that h (c)
(u − 0)2 |
where −.01 ≤ u < c < 0. But h (u) = eu and since eu is positive and increasing,
| eu − (1 + u) |=| R1,0 (u) |=| 0 < e−0.01 < ec < e0 = 1.
Therefore, for any u ∈ [−0.01, 0], Taylor’s Theorem shows that
| eu − (1 + u) |=|
DE Math 128 h (c)
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