Suppose that we don’t have a formula for g(x) but we know that g(1) = −3 and g'(x) = √ x^2 + 8 for all x. Use a linear approximation to estimate g(0.9) and g(1.1)?

Linearization of f(x) at a
##L(x)=f(a)+f'(a)(x-a)##
Let us find the linearization ##L(x)## of ##g(x)## at ##1##.
Since ##g(1)=-3## and ##g'(1)=sqrt{(1)^2+8}=3##,
##L(x)=g(1)+g'(1)(x-1)=-3+3(x-1)=3x-6##.
So, we can approximate:
##g(0.9) approx L(0.9)=3(0.9)-6=-3.3##
##g(1.1) approx L(1.1)=3(1.1)-6=-2.7##
I hope that this was helpful.