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# Find the linear approximation of the function f(x) = \sqrt{1-x} at a = 0 and use it to approximate the numbers \sqrt{0.9} and \sqrt{0.99}. (Round your answers to four decimal places.) L(x) = \sqrt{0.9} \approx \sqrt{0.99} \approx

Differential equations
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asked 2021-05-03
Find the linear approximation of the function $$f(x) = \sqrt{1-x}$$ at a = 0 and use it to approximate the numbers $$\sqrt{0.9}$$ and $$\sqrt{0.99}$$. (Round your answers to four decimal places.)
L(x) =
$$\sqrt{0.9} \approx$$
$$\sqrt{0.99} \approx$$

## Expert Answers (1)

2021-05-04

$$f(x) = \sqrt{1 - x}$$
f(0) = 1
pt (0,1)
$$f'(x) = \frac{1}{2} \cdot (1-x)^{-\frac{1}{2}} \cdot -1$$
$$f'(x) = (-\frac{1}{2}) \cdot (x-1)^{-\frac{1}{2}}$$
$$T(x) = 1 - \frac{1}{2} \cdot (x)$$
To estimate 0.9 we need to put in 0.1 and 0.01$$T(0.1) = 1 - \frac{1}{2} \cdot 0.1 = 1 - 0.05 = 0.95$$ $$T(0.01) = 1 - \frac{1}{2} \cdot 0.01 = 0.995$$

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