What is an approximate value of sqrt{199} cor | vedclass

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(D) Let a function $f(x) = \sqrt{x}$.We choose $x = 196$ and $\Delta x = 3$ because $196$ is the nearest perfect square to $199$.Using the differentia

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$14.1071$

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(D) Let a function $f(x) = \sqrt{x}$.We choose $x = 196$ and $\Delta x = 3$ because $196$ is the nearest perfect square to $199$.Using the differential formula $\Delta y \approx \frac{dy}{dx} 	imes \Delta x$:$\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$.At $x = 196$,$\frac{dy}{dx} = \frac{1}{2\sqrt{196}} = \frac{1}{2 	imes 14} = \frac{1}{28}$.Now,$\Delta y \approx \frac{1}{28} 	imes 3 = \frac{3}{28} \approx 0.10714$.Therefore,$\sqrt{199} = \sqrt{196} + \Delta y \approx 14 + 0.1071 = 14.1071$.

What is an approximate value of $\sqrt{199}$ corrected to $4$ decimal places?

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