MATH100 December 2011
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Question 04 (b)
Full-Solution Problem. Justify your answers and show all your work. Simplification of answers is not required unless explicitly stated.
Let be the second degree Taylor polynomial about for
Is larger than ? Please justify your answer.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
What can you say about the error between the Taylor polynomial and the actual function? That error is also called the remained.
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Lagrange's Remainder Formula states that:
Where is a number in
When is positive, then is an underestimate; if it is negative, then is an overestimate. For this problem,
Since and are both positive, we can conclude that is also positive since the factorial is positive as well. This means that:
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