# How do you find the linear approximation of #(1.999)^4# ?

You can use the tangent line approximation to create a linear function that gives a really close answer.

The linear approximation we want (see my other answer) is

Bonus insight: The error depends on higher derivatives and can be predicted in advance! \ dansmath strikes again, approximately! /

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To find the linear approximation of ( (1.999)^4 ), we start by choosing a point close to ( 1.999 ) to use as the center of our approximation. Let's use ( x = 2 ) since it's close to ( 1.999 ). Then, we need to find the first derivative of the function ( f(x) = x^4 ) and evaluate it at ( x = 2 ), which gives us ( f'(x) = 4x^3 ), and ( f'(2) = 32 ).

Now, we use the linear approximation formula:

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

Substitute ( a = 2 ) and ( f'(a) = 32 ) into the formula:

[ L(x) = f(2) + 32(x - 2) ]

Now, find ( f(2) = 2^4 = 16 ):

[ L(x) = 16 + 32(x - 2) ]

This is the linear approximation of ( (1.999)^4 ).

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