However if a function has a graph that is always

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Unformatted text preview: x2 ) | dx ≤ 0 0.1 ≤ 0 x4 dx 2 x 5 0.1 = | 5(2) 0 (0.1)5 = 10 = 10−6 DE Math 128 329 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series 0 .1 2 This tells us that if we approximate the integral 0 e−x dx by the much simpler integral 0.1 (1 − x2 ) dx, the estimate will have an error of no more than 10−6 which is exactly what 0 we wanted. Therefore, we get that 0.1 0 .1 2 e−x dx ∼ = (1 − x2 ) dx 0 0 x 3 0 .1 )| 30 0.001 = 0.1 − 3 = (x − with an error of no more than 10−6 . There are two more important observations we can make with respect to this example. The first observation is that for every u, 1 + u ≤ eu . This is clear from the diagram below, but it also follows from the fact that y = 1 + u is the tangent line to the graph of y = eu at u = 0 and the graph of eu is always concave up. However, if a function has a graph that is always concave up, then every tangent line sits below the function. Since 1 + u ≤ eu for every u, given x ∈ [0.0.1], when we let u = −x2 we get 2 1 + (−...
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This document was uploaded on 03/23/2013.

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