Use a linear approximation (or differentials) to estimate the given number.

Question

(1.999)^4

Answer 1

f(x) = f(a) + (x-a) f'(a)

f(x) = x^4
df/dx = 4x^3

let a = 2
then f(a) = 2^4 = 16
f'(a) = 4*8 = 32

f(x) = 16 + (x-a)(32)
x-a = - .001
so
f(1.999) = 16 -.001(32) = 16 - .032
f(1.999) = 15.968

with calculator it is 15.968 also