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