Problem 19 - University Calculus: Early Transcendentals

Question

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1. $$y=3 \sin x, \quad 0

Step-by-Step Solution

Answer

Function has absolute maximum at \(x=\frac{\pi}{2}\) and absolute minimum at \(x=\frac{3\pi}{2}\) on \((0, 2\pi)\).

Step 1: Analyze the Function

The function given is \( y = 3 \sin x \) and must be analyzed over the domain \( 0 < x < 2\pi \). This is a trigonometric function, where the amplitude is 3, which means the graph oscillates between \(-3\) and \(3\).

Step 2: Identify Critical Points

Find the derivative of the function: \( y' = 3 \cos x \). Set \( y' = 0 \) to find critical points: \( 3 \cos x = 0 \). This gives \( \cos x = 0 \) which occurs at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) within the domain.

Step 3: Analyze Endpoints

While the domain specifies \( 0 < x < 2\pi \), meaning the endpoints \( x = 0 \) and \( x = 2\pi \) are not included, we examine values infinitesimally close to these points if necessary. However, the sin function behavior right at these boundary conditions should suggest trends in extremal behavior.

Step 4: Evaluate Function at Critical Points

Calculate \( y = 3 \sin x \) at the critical points. At \( x = \frac{\pi}{2} \), \( y = 3 \), and at \( x = \frac{3\pi}{2} \), \( y = -3 \). These suggest local maxima and minima given that the function's range altogether lies within \(-3\) and \(3\) by the sinusoidal nature and amplitude.

Step 5: Conclusion on Extremes

Within the open interval, the highest value is \(3\) and lowest is \(-3\) at the critical points \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) respectively. Thus, these are the local and absolute extremes of the function within the given domain.

Step 6: Validate with Theorem 1

Theorem 1 generally refers to the Extreme Value Theorem, which is applicable in closed intervals. Since the interval is open \((0, 2\pi)\), the function does not technically satisfy conditions for the theorem but showcases local extremes at critical points. This aligns because the sin function is continuous and differentiable.

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational in mathematics, known for their periodic nature. In this exercise, we're dealing with the sine function, specifically expressed as \( y = 3 \sin x \). Here are a few crucial elements of this function:

• Amplitude: The amplitude, which is represented by the coefficient 3, indicates the height of the peaks and the depth of the valleys in the graph. For this function, it means the sine wave oscillates between -3 and 3.
• Period: The period of \( \sin x \) is \( 2\pi \), meaning the wave completes one full cycle over this interval.
• Domain: We're interested in the interval \( 0 < x < 2\pi \), which covers one full cycle of the sine wave.

Trigonometric functions are immensely important in modeling real-world phenomena, such as sound waves, tides, and circular motion.

Critical Points

In calculus, critical points are where the derivative of the function equals zero or does not exist. For the function \( y = 3 \sin x \), the derivative is \( y' = 3 \cos x \).

• Critical points occur where \( 3 \cos x = 0 \). This simplifies to \( \cos x = 0 \), which happens at specific values within the given domain.
• In this problem, the critical points within \( 0 < x < 2\pi \) are found at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

At these points, the function can have potential local maxima, minima, or points of inflection. It's crucial to analyze the behavior of the function at these points, using derivative analysis.

Extreme Values

Extreme values refer to the highest and lowest values of a function. For \( y = 3 \sin x \), these extremes are derived from the critical points found at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

• At \( x = \frac{\pi}{2} \), the function reaches a local maximum, \( y = 3 \).
• At \( x = \frac{3\pi}{2} \), the function hits a local minimum, \( y = -3 \).

Since the interval is an open one \((0, 2\pi)\), these local extremes also represent the absolute extremes within the domain. Hence, the highest value of the function is 3, and the lowest is -3 within the specified range.

Derivative Analysis

Derivative analysis involves using derivatives to understand the behavior of a function. For \( y = 3 \sin x \), the derivative is \( y' = 3 \cos x \), providing crucial insights into the nature of the function.

• Zero derivative: When \( y' = 0 \), critical points are found, indicating where the function could have extremes.
• Sign of derivative: The sign of \( y' \) changes in the intervals between critical points.
  • If \( y' > 0 \), the function is increasing.
  • If \( y' < 0 \), the function is decreasing.

By examining these intervals, you gain a deeper understanding of how the function behaves over its domain. The derivative doesn't just help find critical points, but informs if these points are maximums or minimums.