Question

Find the local linearization of g(x)=sqrt 4x near x=2
use local linearization to estimate g(2.05)
is your estimate too high or too low?

Answer 1

g(x) = sqrt 4x = 2 sqrtx
Local linearization near x = 2:
gl(x)= g(x=2) + [dg/dx@x=2])*(x-2)g
g(x=2) = 2 sqrt2
dg/dx = 2(1/2)/sqrtx = 1/sqrtx
dg/dx @ x=2 = 1/sqrt2
gl(x) = 2sqrt2 + (1/sqrt2)(x-2)
is the linearization about x = 2
For x = 2.05,
g(2.05) = 2.86356
The linearized approximation is
gl(2) + (0.05)/1.41421
= 2.8284 + 0.0353 = 2.86376