Sentence:  In order to get a true bound for the range we may replace the Taylor series in (2) by the <WSD>mean value theorem</WSD>, which tells us that / :  f(\tilde x)=f(\check x) +f'(\zeta)(\tilde x - \check x) / for some \zeta on the line segment between \tilde x and \check x. 
Meaning of the word marked with <WSD> tags:  The theorem that for any real-valued function that is differentiable on an interval, there is a point in that interval where the derivative of the curve equals the slope of the straight line between the graphed function values at the interval's end points.