Suppose there is no known formula for $g(x)$, but we know that $g(1)=-3$ and $g^{\prime}(x)=\sqrt{x^{2}+8}$ for all $x$.
(a) Use a linear approximation to estimate $g(0.99)$ and $g(1.01)$.
\[
\begin{array}{l}
g(0.99) \approx \\
g(1.01) \approx
\end{array}
\]

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.