# How do you estimate the quantity using Linear Approximation and find the error using a calculator of #1/(sqrt(95)) - 1/(sqrt(98))#?

Therefore the linear approximation will be

This means the percent error is

So our approximation is quite precise.

Hopefully this helps!

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To estimate the quantity using Linear Approximation, you first find the tangent line to the function at a point close to the value you're interested in. Then, you evaluate the function at that point and use the equation of the tangent line to estimate the value.

For ( f(x) = \frac{1}{\sqrt{x}} ), the linear approximation near ( x = a ) is given by:

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

Where ( f'(x) ) is the derivative of ( f(x) ).

In this case, we want to estimate ( f(x) ) near ( x = 100 ), so we'll choose ( a = 100 ).

First, we find ( f(100) ) and ( f'(100) ) by evaluating the function and its derivative at ( x = 100 ).

[ f(100) = \frac{1}{\sqrt{100}} = \frac{1}{10} ]

[ f'(x) = -\frac{1}{2x^\frac{3}{2}} ]

[ f'(100) = -\frac{1}{2 \times 100^\frac{3}{2}} = -\frac{1}{200} ]

So, the linear approximation near ( x = 100 ) is:

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

To estimate ( f(95) ) and ( f(98) ), plug in these values into the linear approximation equation:

[ L(95) = \frac{1}{10} - \frac{1}{200}(95-100) = \frac{1}{10} + \frac{1}{40} = \frac{3}{40} ]

[ L(98) = \frac{1}{10} - \frac{1}{200}(98-100) = \frac{1}{10} + \frac{1}{100} = \frac{11}{100} ]

Now, find the actual values of ( f(95) ) and ( f(98) ):

[ f(95) = \frac{1}{\sqrt{95}} \approx \frac{1}{9.7468} \approx 0.1025 ]

[ f(98) = \frac{1}{\sqrt{98}} \approx \frac{1}{9.8995} \approx 0.1005 ]

Finally, find the error for each approximation by subtracting the linear approximation from the actual value:

[ Error(95) = |0.1025 - \frac{3}{40}| \approx |0.1025 - 0.075| \approx 0.0275 ]

[ Error(98) = |0.1005 - \frac{11}{100}| \approx |0.1005 - 0.11| \approx 0.0095 ]

Thus, the error for ( f(95) ) is approximately ( 0.0275 ) and for ( f(98) ) is approximately ( 0.0095 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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