Answer :
To address the problem, we need to analyze the function [tex]\(g(x) = 150{,}000 \csc\left(\frac{\pi}{36} x\right)\)[/tex] and understand its behavior over the specified interval from [tex]\(x = 0\)[/tex] to [tex]\(x = 42\)[/tex].
First, let's break the function down:
1. Function Definition:
- [tex]\(\csc(\theta) = \frac{1}{\sin(\theta)}\)[/tex]
- So, [tex]\(g(x) = 150{,}000 \cdot \frac{1}{\sin\left(\frac{\pi}{36} x\right)}\)[/tex]
2. Understanding the Behavior of [tex]\( \csc \left(\frac{\pi}{36} x\right) \)[/tex]:
- The sine function, [tex]\(\sin(\theta)\)[/tex], oscillates between -1 and 1.
- As [tex]\(\theta\)[/tex] approaches [tex]\(0\)[/tex] or multiples of [tex]\(\pi\)[/tex], [tex]\(\sin(\theta)\)[/tex] approaches 0, causing [tex]\(\csc(\theta) = \frac{1}{\sin(\theta)}\)[/tex] to approach infinity positively or negatively.
- Therefore, [tex]\(\csc(\theta)\)[/tex] has vertical asymptotes (goes to [tex]\(\pm\infty\)[/tex]) at values where [tex]\(\sin(\theta)\)[/tex] equals zero.
3. Identifying Vertical Asymptotes:
- We need to find when [tex]\(\sin\left(\frac{\pi}{36} x \right) = 0\)[/tex].
- [tex]\(\sin\left(\frac{\pi}{36} x \right) = 0\)[/tex] when [tex]\(\frac{\pi}{36} x = k\pi\)[/tex], where [tex]\(k\)[/tex] is any integer.
- Solving for [tex]\(x\)[/tex], we get [tex]\(x = 36k\)[/tex].
- Therefore, asymptotes occur at [tex]\(x = 0, 36, 72, \ldots\)[/tex].
4. Interval Consideration:
- The given interval is [tex]\([0, 42]\)[/tex], which includes [tex]\(x = 0\)[/tex] and [tex]\(x = 36\)[/tex], both being points where the sine function equals zero, thus [tex]\(\csc\)[/tex] has vertical asymptotes.
5. General Behavior Between Asymptotes:
- Between these vertical asymptotes (from [tex]\(x = 0\)[/tex] to [tex]\(x = 36\)[/tex]), the [tex]\(\csc\)[/tex] function will oscillate and have a range of very large positive and negative values.
- At [tex]\(x = 0\)[/tex] and [tex]\(x = 36\)[/tex], the function shoots to infinity ([tex]\(\infty\)[/tex] or [tex]\(-\infty\)[/tex]).
6. Graph Description:
- The graph of [tex]\(g(x)\)[/tex] will have vertical asymptotes at [tex]\(x = 0\)[/tex] and [tex]\(x = 36\)[/tex].
- It will begin from very large positive values as [tex]\(x\)[/tex] increases from just greater than [tex]\(0\)[/tex], decrease smoothly, cross through zero, continue decreasing to a very large negative value, then repeat this behavior toward [tex]\(x = 36\)[/tex].
- From [tex]\(x = 36\)[/tex] to [tex]\(x = 42\)[/tex], the graph will mirror the behavior seen in the first interval since it is repetitive due to the periodic nature of the sine function.
Given these detailed points, you should be able to visualize that the graph of [tex]\(g(x)\)[/tex] on the interval [tex]\([0, 42]\)[/tex] would display this oscillatory behavior with infinite peaks at the vertical asymptotes [tex]\(0\)[/tex] and [tex]\(36\)[/tex].
Therefore, it will have two main parts:
- One part on the interval [tex]\([0, 36]\)[/tex] where the function oscillates from very large positive values to negative and approaching infinity again.
- The second, smaller interval [tex]\([36, 42]\)[/tex] will exhibit a similar oscillation pattern as it heads towards the next potential vertical asymptote (not within our window).
The plot should show these characteristics over the specified interval, identifying the critical points and general nature of the periodic trend.
First, let's break the function down:
1. Function Definition:
- [tex]\(\csc(\theta) = \frac{1}{\sin(\theta)}\)[/tex]
- So, [tex]\(g(x) = 150{,}000 \cdot \frac{1}{\sin\left(\frac{\pi}{36} x\right)}\)[/tex]
2. Understanding the Behavior of [tex]\( \csc \left(\frac{\pi}{36} x\right) \)[/tex]:
- The sine function, [tex]\(\sin(\theta)\)[/tex], oscillates between -1 and 1.
- As [tex]\(\theta\)[/tex] approaches [tex]\(0\)[/tex] or multiples of [tex]\(\pi\)[/tex], [tex]\(\sin(\theta)\)[/tex] approaches 0, causing [tex]\(\csc(\theta) = \frac{1}{\sin(\theta)}\)[/tex] to approach infinity positively or negatively.
- Therefore, [tex]\(\csc(\theta)\)[/tex] has vertical asymptotes (goes to [tex]\(\pm\infty\)[/tex]) at values where [tex]\(\sin(\theta)\)[/tex] equals zero.
3. Identifying Vertical Asymptotes:
- We need to find when [tex]\(\sin\left(\frac{\pi}{36} x \right) = 0\)[/tex].
- [tex]\(\sin\left(\frac{\pi}{36} x \right) = 0\)[/tex] when [tex]\(\frac{\pi}{36} x = k\pi\)[/tex], where [tex]\(k\)[/tex] is any integer.
- Solving for [tex]\(x\)[/tex], we get [tex]\(x = 36k\)[/tex].
- Therefore, asymptotes occur at [tex]\(x = 0, 36, 72, \ldots\)[/tex].
4. Interval Consideration:
- The given interval is [tex]\([0, 42]\)[/tex], which includes [tex]\(x = 0\)[/tex] and [tex]\(x = 36\)[/tex], both being points where the sine function equals zero, thus [tex]\(\csc\)[/tex] has vertical asymptotes.
5. General Behavior Between Asymptotes:
- Between these vertical asymptotes (from [tex]\(x = 0\)[/tex] to [tex]\(x = 36\)[/tex]), the [tex]\(\csc\)[/tex] function will oscillate and have a range of very large positive and negative values.
- At [tex]\(x = 0\)[/tex] and [tex]\(x = 36\)[/tex], the function shoots to infinity ([tex]\(\infty\)[/tex] or [tex]\(-\infty\)[/tex]).
6. Graph Description:
- The graph of [tex]\(g(x)\)[/tex] will have vertical asymptotes at [tex]\(x = 0\)[/tex] and [tex]\(x = 36\)[/tex].
- It will begin from very large positive values as [tex]\(x\)[/tex] increases from just greater than [tex]\(0\)[/tex], decrease smoothly, cross through zero, continue decreasing to a very large negative value, then repeat this behavior toward [tex]\(x = 36\)[/tex].
- From [tex]\(x = 36\)[/tex] to [tex]\(x = 42\)[/tex], the graph will mirror the behavior seen in the first interval since it is repetitive due to the periodic nature of the sine function.
Given these detailed points, you should be able to visualize that the graph of [tex]\(g(x)\)[/tex] on the interval [tex]\([0, 42]\)[/tex] would display this oscillatory behavior with infinite peaks at the vertical asymptotes [tex]\(0\)[/tex] and [tex]\(36\)[/tex].
Therefore, it will have two main parts:
- One part on the interval [tex]\([0, 36]\)[/tex] where the function oscillates from very large positive values to negative and approaching infinity again.
- The second, smaller interval [tex]\([36, 42]\)[/tex] will exhibit a similar oscillation pattern as it heads towards the next potential vertical asymptote (not within our window).
The plot should show these characteristics over the specified interval, identifying the critical points and general nature of the periodic trend.
