The continuous function $f$ is defined on the closed interval $-6 \leq x \leq 5$. The figure above shows a portion of the graph of $f$, consisting of two line segments and a quarter of a circle centered at the point $(5, 3)$. It is known that the point $(3, 3 - \sqrt{5})$ is on the graph of $f$. (a) If $\int_{-6}^{5} f(x)\, dx = 7$, find the value of $\int_{-6}^{-2} f(x)\, dx$. Show the work that leads to your answer. (b) Evaluate $\int_{3}^{5} \left(2f'(x) + 4\right) dx$. (c) The function $g$ is given by $g(x) = \int_{-2}^{x} f(t)\, dt$. Find the absolute maximum value of $g$ on the interval $-2 \leq x \leq 5$. Justify your answer. (d) Find $\lim_{x \to 1} \dfrac{10^x - 3f'(x)}{f(x) - \arctan x}$.
The continuous function $f$ is defined on the closed interval $-6 \leq x \leq 5$. The figure above shows a portion of the graph of $f$, consisting of two line segments and a quarter of a circle centered at the point $(5, 3)$. It is known that the point $(3, 3 - \sqrt{5})$ is on the graph of $f$.
(a) If $\int_{-6}^{5} f(x)\, dx = 7$, find the value of $\int_{-6}^{-2} f(x)\, dx$. Show the work that leads to your answer.
(b) Evaluate $\int_{3}^{5} \left(2f'(x) + 4\right) dx$.
(c) The function $g$ is given by $g(x) = \int_{-2}^{x} f(t)\, dt$. Find the absolute maximum value of $g$ on the interval $-2 \leq x \leq 5$. Justify your answer.
(d) Find $\lim_{x \to 1} \dfrac{10^x - 3f'(x)}{f(x) - \arctan x}$.