deletions | additions       diff --git a/textbf_textit_Calculus_of_the__.tex b/textbf_textit_Calculus_of_the__.tex  index 99d90f7..773790a 100644  --- a/textbf_textit_Calculus_of_the__.tex  +++ b/textbf_textit_Calculus_of_the__.tex 
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\end{array}\right).  \end{align*}  Rearranging, this yields  $$  f\left(\frac{k}{b}\right)=4\left(1-\sqrt{1-\frac{k}{b}}\right)-2\frac{k}{b}-\frac{1}{2}\left(\frac{k}{b}\right)^{2}\mbox{ for all }0\leq\frac{k}{b}\leq1\mbox{.}  $$  $$  s^{*}\left(b,k\right)=\left(b-\frac{1}{2}\right)+\frac{\left(1-k\right)^{2}}{2\left(b-k\right)+\left(1-k\right)^{2}}\left(\frac{1}{2}-2b+k+\frac{1}{3}b^{2}f\left(\frac{k}{b}\right)\right),  $$  where we define  $$  f\left(\frac{k}{b}\right)=4\left(1-\sqrt{1-\frac{k}{b}}\right)-2\frac{k}{b}-\frac{1}{2}\left(\frac{k}{b}\right)^{2}\mbox{ for all }0\leq\frac{k}{b}\leq1\mbox{.}  $$  $\blacksquare$