Problem 5 - Calculus Single Variable

Question

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=3 \sin (4 x) $$

Step-by-Step Solution

Answer

Local maxima at \(x = \frac{\pi}{8} + \frac{k\pi}{2}\) and local minima at \(x = -\frac{\pi}{8} + \frac{k\pi}{2}\) for integer \(k\).

Step 1: Understand the Sinusoidal Function

The function given is a sinusoidal function of the form \(f(x) = a \sin(bx)\). In this case, \(a = 3\) and \(b = 4\), indicating that the amplitude of the function is \(3\) and the period is determined by \(\frac{2\pi}{b}\).

Step 2: Calculate the Period

The period of the function \(f(x) = 3\sin(4x)\) is calculated by the formula for the period of a sine function, \(\frac{2\pi}{b}\). Substituting \(b = 4\), we find the period is \(\frac{2\pi}{4} = \frac{\pi}{2}\). This means the function repeats every \(\frac{\pi}{2}\) units.

Step 3: Identify the Maximum and Minimum Points of Sine Function

The sine function \(\sin(x)\) reaches its maximum value of 1 and minimum value of -1. For \(f(x) = 3\sin(4x)\), the local maxima occur when \(\sin(4x) = 1\) and local minima occur when \(\sin(4x) = -1\).

Step 4: Calculate Points for Local Maxima

The maxima occur when the angle \(4x\) is equal to \(\frac{\pi}{2} + 2k\pi\) where \(k\) is an integer. Solving \(4x = \frac{\pi}{2} + 2k\pi\) gives \(x = \frac{\pi}{8} + \frac{k\pi}{2}\). These are the points where the function has local maxima.

Step 5: Calculate Points for Local Minima

The minima occur when the angle \(4x\) is equal to \(-\frac{\pi}{2} + 2k\pi\) where \(k\) is an integer. Solving \(4x = -\frac{\pi}{2} + 2k\pi\) gives \(x = -\frac{\pi}{8} + \frac{k\pi}{2}\). These are the points where the function has local minima.

Key Concepts

Sinusoidal Functions

Sinusoidal functions represent oscillating motions like waves. A common example is the function of type \( f(x) = a \sin(bx) \), where \( a \) and \( b \) are constants. In this context, the function resembles the regular sine function but is modified due to the coefficients \( a \) and \( b \). These functions exhibit characteristics of wave propagation, rising and falling in a consistent pattern.

In our exercise, the function \( f(x) = 3\sin(4x) \) is precisely such a sinusoidal function. The inclusion of coefficients affects two main features: amplitude and period, which will be explored in more detail. Sinusoidal functions consistently repeat their values over cycles, often described by their period, making them fundamental to understanding various scientific phenomena.

Amplitude

Amplitude refers to the "height" of a wave and represents the maximum displacement from the central axis of the function. In the typical sine function \( \sin(x) \), the amplitude is \(1\). However, when the function includes a coefficient \( a \) as in \( f(x) = a\sin(bx) \), the amplitude becomes \(|a|\).

• This coefficient \( a \) stretches or compresses the wave vertically.
• In our example \( f(x) = 3\sin(4x) \), the amplitude is \(3\), indicating that this function will reach up to 3 units above and below the horizontal axis.

Amplitudes are always positive, as they describe absolute distances from the mean line of oscillation, and usually have significant implications in fields such as physics and engineering.

Period

The period of a sinusoidal function determines how frequently the wave pattern repeats. For the basic sine function \( \sin(x) \), the period is \( 2\pi \). However, when a coefficient \( b \) modifies the input \( x \) in the function \( f(x) = a\sin(bx) \), the period changes.

• The period \( T \) can be calculated using the formula \( T = \frac{2\pi}{b} \).
• In our function \( f(x) = 3\sin(4x) \), the period is determined as \( \frac{2\pi}{4} = \frac{\pi}{2} \).
• This indicates that the wave pattern of the function repeats every \( \frac{\pi}{2} \) units.

Understanding periods is essential for predicting when a function will return to a specific value, making it valuable in the analysis of repetitive processes.

Local Extrema

Local extrema are the points in a function where it reaches local maximum or minimum values within an interval. In the context of sinusoidal functions, these extrema correlate with points of peaks and troughs.

• The local maxima occur where the function reaches its highest points on any given cycle, such as when \( \sin(x) = 1 \).
• Local minima occur at the lowest points, like when \( \sin(x) = -1 \).

For \( f(x) = 3\sin(4x) \):

• The points of local maxima occur where \( 4x = \frac{\pi}{2} + 2k\pi \), solving for \( x \) gives \( x = \frac{\pi}{8} + \frac{k\pi}{2} \).
• Local minima occur where \( 4x = -\frac{\pi}{2} + 2k\pi \), resulting in \( x = -\frac{\pi}{8} + \frac{k\pi}{2} \).

Recognizing these points helps identify periodic peaks and troughs, essential in harmonic analysis and signal processing.