The graph of a differentiable function $f$ on the closed interval $[-3, 15]$ is shown in the figure above. The graph of $f$ has a horizontal tangent line at $x = 6$. Let $g(x) = 5 + \int_{6}^{x} f(t)\, dt$ for $-3 \leq x \leq 15$. (a) Find $g(6)$, $g^{\prime}(6)$, and $g^{\prime\prime}(6)$. (b) On what intervals is $g$ decreasing? Justify your answer. (c) On what intervals is the graph of $g$ concave down? Justify your answer. (d) Find a trapezoidal approximation of $\int_{-3}^{15} f(t)\, dt$ using six subintervals of length $\Delta t = 3$.
The graph of a differentiable function $f$ on the closed interval $[-3, 15]$ is shown in the figure above. The graph of $f$ has a horizontal tangent line at $x = 6$. Let $g(x) = 5 + \int_{6}^{x} f(t)\, dt$ for $-3 \leq x \leq 15$.\\
(a) Find $g(6)$, $g^{\prime}(6)$, and $g^{\prime\prime}(6)$.\\
(b) On what intervals is $g$ decreasing? Justify your answer.\\
(c) On what intervals is the graph of $g$ concave down? Justify your answer.\\
(d) Find a trapezoidal approximation of $\int_{-3}^{15} f(t)\, dt$ using six subintervals of length $\Delta t = 3$.