The function $f$ is defined by $f ( x ) = \sqrt { 25 - x ^ { 2 } }$ for $- 5 \leq x \leq 5$.
(a) Find $f ^ { \prime } ( x )$.
(b) Write an equation for the line tangent to the graph of $f$ at $x = - 3$.
(c) Let $g$ be the function defined by $g ( x ) = \left\{ \begin{array} { l } f ( x ) \text { for } - 5 \leq x \leq - 3 \\ x + 7 \text { for } - 3 < x \leq 5 . \end{array} \right.$
Is $g$ continuous at $x = - 3$ ? Use the definition of continuity to explain your answer.
(d) Find the value of $\int _ { 0 } ^ { 5 } x \sqrt { 25 - x ^ { 2 } } d x$.