MATH100 December 2014
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Question 11 (b)
Now let .Your friend uses a degree- Taylor polynomial about to approximate . Show that the absolute error in their approximation is no greater than .
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!
Recall the error in a degree- Taylor polynomial and bound it.
The error in a degree-2 Taylor polynomial is
for a between and .
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We need the third derivative of :
Then the absolute error is given by for between and . Hence
for between and . From the previous part we know this is maximised when and gives .
And since we have and so
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