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

Apply the concept of margin of error to interpret survey results and make informed conclusions.

Understand the implications of sample size on the reliability and precision of confidence intervals.

Identify the effects of increasing confidence levels on the width of confidence intervals.

Recognize the importance of using confidence intervals for estimating population parameters from sample data.

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Definitions:

True Value

The actual or real value of a parameter which is estimated in a study.

Sample Size

The number of individual observations or data points collected in a sample from a larger population.

ANOVA

A statistical method, short for Analysis of Variance, that assesses potential differences in means among multiple groups.

Assumptions

Assumptions are accepted truths or beliefs that are taken for granted without proof, used as a basis for reasoning, discussions, or theoretical modeling.