Problem 33 - Thomas Calculus

Question

In Exercises \(21-36,\) find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$ g(x)=\csc x, \quad \frac{\pi}{3} \leq x \leq \frac{2 \pi}{3} $$

Step-by-Step Solution

Answer

The absolute extrema are both ≈1.155 at \(x = \frac{\pi}{3}\) and \(x = \frac{2\pi}{3}\).

Step 1: Understand the Function and the Interval

The function given is the cosecant function, \( g(x) = \csc x \). We need to evaluate it over the interval \( \frac{\pi}{3} \leq x \leq \frac{2\pi}{3} \). The cosecant function is the reciprocal of the sine function, \( \csc x = \frac{1}{\sin x} \).

Step 2: Determine Important Points

Evaluate \( g(x) = \csc x \) at the endpoints of the interval and any critical points within the interval. The endpoints are \( x = \frac{\pi}{3} \) and \( x = \frac{2\pi}{3} \). Any \( x \) in this interval where \( \sin x \) is either undefined or a critical point (where the derivative \( \csc x \) is zero or undefined) must also be checked.

Step 3: Calculate the Cosecant at the Endpoints

Calculate \( g(x) \) at the endpoints: - \( g\left(\frac{\pi}{3}\right) = \csc\left(\frac{\pi}{3}\right) = \frac{1}{\sin\left(\frac{\pi}{3}\right)} = \frac{2}{\sqrt{3}} \approx 1.155 \) - \( g\left(\frac{2\pi}{3}\right) = \csc\left(\frac{2\pi}{3}\right) = \frac{1}{\sin\left(\frac{2\pi}{3}\right)} = \frac{2}{\sqrt{3}} \approx 1.155 \)

Step 4: Evaluate Critical Points

Find \( \csc x \) at critical points between \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \). Since the first derivative involves calculating where \( \sin x = 0 \), but \( \sin x eq 0 \) within this interval, there are no points where the derivative is zero. Also, \( \csc x \) does not become undefined within the interval because \( \sin x \) is not zero.

Step 5: Identify Extremes

Because there are no critical points within the open interval that affect the extrema and the function is smooth, the extrema occur at the endpoints. Therefore, the absolute maximum and minimum values are both \( \approx 1.155 \) at both ends \( x = \frac{\pi}{3} \) and \( x = \frac{2\pi}{3} \).

Step 6: Graph the Function

Graph the function \( g(x) = \csc x \) over the interval \( \frac{\pi}{3} \leq x \leq \frac{2\pi}{3} \). The values at the endpoints are plotted and there are no other extrema within the interval. The graph will show symmetry about the midpoint since \( \csc x \) reflects about the origin for the symmetric sine values over this interval.

Step 7: Conclusion with Coordinates

The absolute minimum and maximum values of \( g(x) = \csc x \) on the interval \( \frac{\pi}{3} \leq x \leq \frac{2\pi}{3} \) are both \( \frac{2}{\sqrt{3}} \approx 1.155 \), occurring at the points \( (\frac{\pi}{3}, \frac{2}{\sqrt{3}}) \) and \( (\frac{2\pi}{3}, \frac{2}{\sqrt{3}}) \).

Key Concepts

Cosecant Function

The Cosecant Function, denoted as \( \csc x \), is a fascinating trigonometric function derived as the reciprocal of the Sine Function. It is defined as \( \csc x = \frac{1}{\sin x} \). You can think of it, quite simply, as turning the sine function upside down.

Since it is reciprocal of sine, wherever sine reaches its maximum or minimum, cosecant will flip these points to extremes. This transformation leads to several important properties:

• The body of the function experiences vertical asymptotes wherever the sine of an angle becomes zero, making the function undefined at those points.
• Cosecant function forms a series of upward and downward curves, often taking on either very large positive or negative values.

Understanding the behavior of \( \csc x \) is crucial in solving problems involving finding extrema or evaluating its function over specific intervals.

Critical Points

Critical Points are a central concept in calculus used to find where a function's derivative is zero or undefined. They indicate where possible maxima, minima, or points of inflection occur.

In the context of the Cosecant Function, for instance, these critical points can tell us where to find the absolute extrema. To find critical points for \( \csc x \):

• We first need to check where the derivative of \( \csc x \), which involves the derivative of its reciprocal \( \sin x \), becomes zero.
• Since \( \sin x eq 0 \) within the interval \( \left[\frac{\pi}{3}, \frac{2\pi}{3}\right] \), there aren't any critical points within this interval.
• Critical points lie where the function's rate of change flatlines or becomes undefined, but here the strength of \( \csc x \) behaves smoothly between the given endpoints.

This analysis illustrates that the trigonometric oscillation does not yield new extremum inside this specific interval, placing importance on the interval's edges.

Interval Analysis

Interval Analysis involves examining the behavior of a function over a given range. This process is essential in determining characteristics like continuity, differentiability, and extrema of the function.

For the function \( g(x) = \csc x \) over the interval \( \left[\frac{\pi}{3}, \frac{2\pi}{3}\right] \):

• We calculate the function values at the endpoints: \( \csc\left(\frac{\pi}{3}\right) \) and \( \csc\left(\frac{2\pi}{3}\right) \).
• Both values result in \( \frac{2}{\sqrt{3}} \), affirming that the endpoint values are equal, thus standing as the extrema.
• From the analysis, the function \( \csc x \) remains continuous and smooth within this range.

The investigation assures that no new extrema ever emerge apart from those at the endpoints, making Interval Analysis fundamentally useful.

Sine Function Reciprocal

The Sine Function Reciprocal plays an essential role in trigonometry and calculus. Understanding this relationship gives deeper insight into functions like the Cosecant Function. Recall:

The sine function \( \sin x \) varies between -1 and 1. When flipped to become its reciprocal:

• The range expands dramatically, with \( \csc x \) taking on values from \(-\infty \) to -1 and from 1 to \( \, \infty \).
• This provides insight into the increased sensitivity of \( \csc x \) to changes in \( \sin x \).
• Where \( \sin x = 0 \), the reciprocal process means \( \csc x \) becomes undefined, leading to vertical asymptotes in the graph.

This conceptual inversion adds complexity and intriguing behavior to the behavior of \( \csc x \), crucial for exploring absolute extrema in given intervals like this exercise shows.