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

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Aug 17, 2014

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

Let's put #f(x) = x^4,# we want #f(1.999)# so use x= 1.999 and the nearby point of tangency a = 2. We'll need #f'(x)=4x^3# too.

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

#f(x) ~~ f(a) + f'(a)(x-a)#

#f(1.999) ~~ f(2) + f'(2)(1.999-2)#

#~~ 2^4 + 4*2^3*(-0.001) = 16 - 0.032 = 15.968#

You can compare to the actual exact result of
#1.999^4 = 15.968023992001, #so we came pretty close!

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

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