Carlson Hodgson
11/27/2023 · Middle School
\( q ( x ) = \frac { \csc x } { e ^ { x } } - e ^ { - x } , \quad a = \frac { \pi } { 6 } \)
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Step-by-step Solution
To analyze the function \( q(x) = \frac{\csc x}{e^x} - e^{-x} \) at the point \( a = \frac{\pi}{6} \), we can start by substituting \( x = \frac{\pi}{6} \) into the function.
First, we need to calculate \( \csc\left(\frac{\pi}{6}\right) \):
\[
\csc\left(\frac{\pi}{6}\right) = \frac{1}{\sin\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{2}} = 2.
\]
Now, we can substitute \( x = \frac{\pi}{6} \) into \( q(x) \):
\[
q\left(\frac{\pi}{6}\right) = \frac{\csc\left(\frac{\pi}{6}\right)}{e^{\frac{\pi}{6}}} - e^{-\frac{\pi}{6}}.
\]
Substituting \( \csc\left(\frac{\pi}{6}\right) = 2 \):
\[
q\left(\frac{\pi}{6}\right) = \frac{2}{e^{\frac{\pi}{6}}} - e^{-\frac{\pi}{6}}.
\]
Next, we can simplify \( e^{-\frac{\pi}{6}} \):
\[
e^{-\frac{\pi}{6}} = \frac{1}{e^{\frac{\pi}{6}}}.
\]
Thus, we can rewrite \( q\left(\frac{\pi}{6}\right) \):
\[
q\left(\frac{\pi}{6}\right) = \frac{2}{e^{\frac{\pi}{6}}} - \frac{1}{e^{\frac{\pi}{6}}} = \frac{2 - 1}{e^{\frac{\pi}{6}}} = \frac{1}{e^{\frac{\pi}{6}}}.
\]
Therefore, the value of \( q\left(\frac{\pi}{6}\right) \) is:
\[
q\left(\frac{\pi}{6}\right) = \frac{1}{e^{\frac{\pi}{6}}}.
\]
This is the evaluation of the function \( q(x) \) at the point \( a = \frac{\pi}{6} \). If you need further analysis, such as derivatives or limits, please let me know!
Quick Answer
Evaluate \( q(x) \) at \( x = \frac{\pi}{6} \), we get \( q\left(\frac{\pi}{6}\right) = \frac{1}{e^{\frac{\pi}{6}}} \).
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