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 $\int _ { 3 } ^ { 5 } \left( 2 f ^ { \prime } ( x ) + 4 \right) dx$. (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 $\lim _ { x \rightarrow 1 } \frac { 10 ^ { x } - 3 f ^ { \prime } ( x ) } { f ( x ) - \arctan x }$.
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 $\int _ { 3 } ^ { 5 } \left( 2 f ^ { \prime } ( x ) + 4 \right) dx$.
(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 $\lim _ { x \rightarrow 1 } \frac { 10 ^ { x } - 3 f ^ { \prime } ( x ) } { f ( x ) - \arctan x }$.