Derek's Technical Notes: Fast calculate square root

Calculate sqrt(37) and tell me estimate of the error.

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I did in this way:

$\begin{align*} \sqrt{37}&=6+\Delta\\ &=\sqrt{(6+\Delta)^2}=\sqrt{(36+12\Delta+\Delta^2)}\\ &\approx \sqrt{36+12\Delta}\\ \Rightarrow\quad & 12\Delta=1\quad\Rightarrow\quad \Delta\approx 0.08\\ \textrm{Since}\quad &\sqrt{36+12\Delta+\Delta^2}\leq\sqrt{36+12\Delta}\,+\,\Delta\\ \textrm{and}\quad & \sqrt{36+12\Delta}<\sqrt{37} \end{align*}$

Therefore, I say the answer is 6.08, and the error is within 0.08. In other word, sqrt{37}=6.09~6.16. Indeed, calculator shows the answer is 6.083.

Derek's Technical Notes

Fast calculate square root

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