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In order to simplify the notation we will only

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Unformatted text preview: orem directly to e−x . The simple answer to this question is DE Math 128 330 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series that the method we outlined above results in much easier calculations. Because all of the derivatives of eu are just eu , it was rather easy to obtain an upper bound for the error in using linear approximation to estimate eu . This then gave us a very quick estimate of the 2 difference between 1 − x2 and e−x . If we wanted to apply Taylor’s Theorem directly to 2 e−x , viewing 1 − x2 as the third degree Taylor polynomial for more accuracy, we would 2 need to be able to determine how large the fourth derivative of e−x might be. This is not 2 2 2 2 an easy task since the fourth derivative of e−x is the function 12e−x − 48x2 e−x +16x4 e−x . The method we outlined avoids this difficulty. We will now see how Taylor’s Theorem can help in calculating various limits. In order to simplify the notation, we will only consider li...
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