# How do you use linear approximation about x=100 to estimate #1/sqrt(99.8)#?

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To use linear approximation about (x = 100) to estimate (1/\sqrt{99.8}), you first find the linear approximation of the function (f(x) = 1/\sqrt{x}) near (x = 100). The linear approximation is given by the formula:

[L(x) = f(a) + f'(a)(x - a)]

Where (a) is the point of approximation, (f(a)) is the value of the function at (x = a), and (f'(a)) is the derivative of the function evaluated at (x = a).

First, find the value of (f(100)) and (f'(100)):

[f(100) = \frac{1}{\sqrt{100}} = \frac{1}{10}] [f'(x) = -\frac{1}{2x^{3/2}}] [f'(100) = -\frac{1}{2(100)^{3/2}} = -\frac{1}{200}]

Now, plug these values into the linear approximation formula:

[L(x) = \frac{1}{10} - \frac{1}{200}(x - 100)]

Finally, plug in (x = 99.8) to get the estimate:

[L(99.8) = \frac{1}{10} - \frac{1}{200}(99.8 - 100)] [L(99.8) = \frac{1}{10} - \frac{1}{200}(-0.2)] [L(99.8) = \frac{1}{10} + \frac{1}{1000}] [L(99.8) = \frac{100 + 1}{1000}] [L(99.8) = \frac{101}{1000}]

So, the estimate of (1/\sqrt{99.8}) using linear approximation about (x = 100) is (101/1000).

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