the catenary has equation y=(e^x+e^-x)/2. graph it and y=x^2 on the same set of coordinate axes. for what values of x are the y values within 1 of each other? within 0.5 of each other?

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anonymous · Answer

$$\begin{cases} y_1=\dfrac{e^x+e^{-x}}{2}\\ y_2=x^2 \end{cases}$$ The $y$ values will be within 1 of each other if $|y_1-y_2|\le1$: http://www.wolframalpha.com/input/?i=%7CCosh%5Bx%5D-x%5E2%7C%3C%3D1 and will be within 0.5 of each other if $|y_1-y_2|\le0.5$: http://www.wolframalpha.com/input/?i=%7CCosh%5Bx%5D-x%5E2%7C%3C%3D1%2F2