diff --git a/optimization/exercises/singVarCalcOptimization3.tex b/optimization/exercises/singVarCalcOptimization3.tex index a7843caaf..d7f26655c 100644 --- a/optimization/exercises/singVarCalcOptimization3.tex +++ b/optimization/exercises/singVarCalcOptimization3.tex @@ -34,11 +34,11 @@ \end{tikzpicture} \end{hint} \begin{hint} - The dimensions that will result in "least material" for the five sides are dimensions that result in "smallest surface area". - So, the quantity that we have to "minimize" is the surface area, $S$. + The dimensions that will result in ``least material'' for the five sides are dimensions that result in ``smallest surface area''. + So, the quantity that we have to ``minimize'' is the surface area, $S$. \end{hint} \begin{hint} - We have to "minimize" the surface area, $S$. + We have to ``minimize'' the surface area, $S$. $S=x^2 +4xy$ @@ -64,9 +64,8 @@ \begin{hint} So, we have to find the (global) minimum of $S$ on its domain, $(0,\answer{\infty})$. We have to find critical points of $S$. - Therefore, we have to compute $S'(x)$. - \end{hint} - \begin{hint} + Therefore, we have to compute $S'(x)$: + $S'(x)=2x-\frac{400}{\answer{x^2}}$. \end{hint} \begin{hint} @@ -76,9 +75,7 @@ $2x-\frac{400}{\answer{x^2}}=0$. It follows that $S$ has the only critical point $x=\answer{200^{\frac{1}{3}}}$. - \end{hint} - - \begin{hint} + Since $S'(x)=2x-\frac{400}{x^{2}}=\frac{2(x^{3}-200)}{x^{2}}$, it follows that