Science:Math Exam Resources/Courses/MATH100/December 2013/Question 10 (b)/Solution 1

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To apply the Lagrange Remainder Formula, we first need to find an such that , that is, .

Since is an even function, we choose the positive root without loss of generality.


By Taylor's Remainder theorem (as applied to Maclaurin polynomials),

where and denote the minimum and maximum values of on the interval , respectively.


Since we know that from part (a), we simply consider:


We now calculate the upper and lower bounds of the error. To do so, we substitute into the Lagrange Remainder Formula:

for some in in the interval . To find the error bounds, we maximize this function as a function of . But observe that

is monotonically decreasing on the interval (the denominator is increasing while the numerator is decreasing (note the minus sign), so overall is decreasing). Hence the minimum and maximum values of the error are attained at and , respectively.


Plugging this in we obtain the required lower and upper bound for the error: