Step 1 :We are given that the function $g(x)$ has a value of $-3$ at $x=1$ and its derivative $g'(x)$ is given by $\sqrt{x^2 + 8}$.
Step 2 :We can use the linear approximation formula $f(a) + f'(a)(x-a)$ to estimate the values of $g(x)$ at $x=0.99$ and $x=1.01$.
Step 3 :For $x=0.99$, the linear approximation is $g(1) + g'(1)(0.99-1) = -3 + \sqrt{1^2 + 8}(0.99-1)$, which simplifies to approximately $-3.03$.
Step 4 :For $x=1.01$, the linear approximation is $g(1) + g'(1)(1.01-1) = -3 + \sqrt{1^2 + 8}(1.01-1)$, which simplifies to approximately $-2.97$.
Step 5 :Thus, the linear approximation estimates for $g(0.99)$ and $g(1.01)$ are approximately $-3.03$ and $-2.97$ respectively.