Let $f$ be the function defined by $f ( x ) = e ^ { x } \cos x$. (a) Find the average rate of change of $f$ on the interval $0 \leq x \leq \pi$. (b) What is the slope of the line tangent to the graph of $f$ at $x = \frac { 3 \pi } { 2 }$ ? (c) Find the absolute minimum value of $f$ on the interval $0 \leq x \leq 2 \pi$. Justify your answer. (d) Let $g$ be a differentiable function such that $g \left( \frac { \pi } { 2 } \right) = 0$. The graph of $g ^ { \prime }$, the derivative of $g$, is shown below. Find the value of $\lim _ { x \rightarrow \pi / 2 } \frac { f ( x ) } { g ( x ) }$ or state that it does not exist. Justify your answer.
Let $f$ be the function defined by $f ( x ) = e ^ { x } \cos x$.
(a) Find the average rate of change of $f$ on the interval $0 \leq x \leq \pi$.
(b) What is the slope of the line tangent to the graph of $f$ at $x = \frac { 3 \pi } { 2 }$ ?
(c) Find the absolute minimum value of $f$ on the interval $0 \leq x \leq 2 \pi$. Justify your answer.
(d) Let $g$ be a differentiable function such that $g \left( \frac { \pi } { 2 } \right) = 0$. The graph of $g ^ { \prime }$, the derivative of $g$, is shown below. Find the value of $\lim _ { x \rightarrow \pi / 2 } \frac { f ( x ) } { g ( x ) }$ or state that it does not exist. Justify your answer.