Point Corresponding To $\theta = \frac { \pi } { 4 }$

question 67

Multiple Choice

$\text { Find } \frac { d y } { d x } \text { and } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { if possible, and find the slope and concavity (if possible) at the }$ point corresponding to $\theta = \frac { \pi } { 4 }$ .
$\begin{array} { l } x = 4 \cos \theta \\y = 4 \sin \theta\end{array}$

• A) \(\frac { d y } { d x } = \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 1 } { 4 \sin ^ { 3 } \theta }\); at \(\theta = \frac { \pi } { 4 }\) : slope 1 and concave down
• B) \(\frac { d y } { d x } = - \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 1 } { 4 \sin ^ { 3 } \theta }\); at \(\theta = \frac { \pi } { 4 }\) : slope \(- 1\) and concave down
• C) \(\frac { d y } { d x } = - \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 1 } { 4 \sin ^ { 3 } \theta }\); at \(\theta = \frac { \pi } { 4 }\) : slope \(- 1\) and concave up
• D) \(\frac { d y } { d x } = \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 1 } { 4 \sin ^ { 3 } \theta }\); at \(\theta = \frac { \pi } { 4 }\) : slope 1 and concave up
• E) \(\frac { d y } { d x } = - \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 1 } { 4 \sin ^ { 3 } \theta }\); at \(\theta = \frac { \pi } { 4 }\) : slope of \(- 1\) and concave down