The derivative $f'(x)$ of a function $f(x)$ that is differentiable on the set of all real numbers is $$f'(x) = -x + e^{1 - x^{2}}$$ For a positive number $t$, let $g(t)$ be the area of the region enclosed by the tangent line to the curve $y = f(x)$ at the point $(t, f(t))$, the curve $y = f(x)$, and the $y$-axis. What is the value of $g(1) + g'(1)$? [4 points] (1) $\frac{1}{2}e + \frac{1}{2}$ (2) $\frac{1}{2}e + \frac{2}{3}$ (3) $\frac{1}{2}e + \frac{5}{6}$ (4) $\frac{2}{3}e + \frac{1}{2}$ (5) $\frac{2}{3}e + \frac{2}{3}$
The derivative $f'(x)$ of a function $f(x)$ that is differentiable on the set of all real numbers is
$$f'(x) = -x + e^{1 - x^{2}}$$
For a positive number $t$, let $g(t)$ be the area of the region enclosed by the tangent line to the curve $y = f(x)$ at the point $(t, f(t))$, the curve $y = f(x)$, and the $y$-axis. What is the value of $g(1) + g'(1)$? [4 points]\\
(1) $\frac{1}{2}e + \frac{1}{2}$\\
(2) $\frac{1}{2}e + \frac{2}{3}$\\
(3) $\frac{1}{2}e + \frac{5}{6}$\\
(4) $\frac{2}{3}e + \frac{1}{2}$\\
(5) $\frac{2}{3}e + \frac{2}{3}$