# How do you estimate the quantity using the Linear Approximation and find the error using a calculator #(15.8)^(1/4)#?

The linear approximation (about

# (15.8)^(1/4) ~~ 1.975 #

This is within

Calculator Result Using a calculator we find:

Error Analysis The %age error is calculated using:

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To estimate the quantity using linear approximation, we use the formula:

[ f(x) \approx f(a) + f'(a)(x-a) ]

where ( f(x) ) is the function we want to approximate, ( f'(a) ) is the derivative of ( f(x) ) evaluated at ( x = a ), and ( a ) is the value near which we want to approximate.

In this case, we want to estimate ( (15.8)^{\frac{1}{4}} ). Let's choose ( a = 16 ) because it's a convenient value near ( 15.8 ). Now, we find the derivative of ( f(x) = x^{\frac{1}{4}} ) and evaluate it at ( x = 16 ).

[ f'(x) = \frac{1}{4}x^{-\frac{3}{4}} ] [ f'(16) = \frac{1}{4}(16)^{-\frac{3}{4}} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} ]

Now, we plug these values into the linear approximation formula:

[ f(15.8) \approx f(16) + f'(16)(15.8 - 16) ] [ (15.8)^{\frac{1}{4}} \approx (16)^{\frac{1}{4}} + \frac{1}{8}(15.8 - 16) ] [ \approx 2 + \frac{1}{8}(-0.2) ] [ \approx 2 - 0.025 ] [ \approx 1.975 ]

Now, to find the error, we calculate the actual value of ( (15.8)^{\frac{1}{4}} ) using a calculator:

[ (15.8)^{\frac{1}{4}} \approx 1.976 ]

The error is:

[ |1.975 - 1.976| = 0.001 ]

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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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