MATH100 December 2010
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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?
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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!
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Hint
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We want to know the value of (or at least an upper bound of)
Taylor's remainder formula says that there is a such that
where M is an upper bound for the absolute value of the third derivative on .
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Solution 1
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Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies.
Taylor's remainder formula says that
where M is an upper bound for the absolute value of the third derivative on the interval . Computing the third derivative via the product rule gives
Notice that is increasing on and thus obtains its maximum at x = 1, i.e.
for .
The function is a parabola with roots at 0 and 1. Hence its minimum occurs at the midpoint of the roots, namely at . Since the value at the endpoints x = 0 and x = 1 is zero, the minimum value of the parabola maximizes its absolute value. Thus,
Since we want to bound the absolute value of the third derivative, we know that
- .
Hence, we have
as required.
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Solution 2
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Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies.
Proceed as in solution 1 to see that
We now need a bound on this function on . Notice that the third derivative is neither monotonically increasing nor is it monotonically decreasing on the interval. Hence we are not guaranteed to find the maximum value by just looking at the endpoints. In fact, at the endpoints we have , which is certainly not the maximum of the absolute value . Instead we do it the proper way: Take the derivative
and set it to zero
Recall that . Using the quadratic formula, we see that
only the positive such root is in . Let
- .
Then, our function obtains an extreme value at . Since the function is nonzero at this point, we know that is maximal at . Thus, we have that
completing the solution.
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MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Taylor series, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag
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