Problem 96 - Calculus for Biology and Medicine

Question

Find the amplitude and the period of \(f(x)\) : $$ f(x)=3 \cos 4 x, \quad x \in \mathbf{R} $$

Step-by-Step Solution

Answer

Amplitude is 3; Period is \(\frac{\pi}{2}\).

Step 1: Identifying the Amplitude

The amplitude of a cosine function in the form \( a \cos(bx) \) is given by the absolute value of \(a\). In this function \(f(x) = 3 \cos 4x\), the value of \(a\) is 3. Therefore, the amplitude is \(|3| = 3\).

Step 2: Determining the Period

The period of a cosine function in the form \( a \cos(bx) \) is calculated as \( \frac{2\pi}{b} \). Here, \(b = 4\), so the period is \( \frac{2\pi}{4} = \frac{\pi}{2}\).

Key Concepts

Understanding Amplitude in Trigonometric Functions

Amplitude is a crucial property of trigonometric functions such as sine and cosine. It's the height of the wave and essentially measures the extent of its oscillation from the center line or equilibrium position. In simple terms, amplitude is the maximum distance a wave can go up or down from its rest position.

The mathematical representation of a cosine function generally takes the form \( a \cos(bx) \). Here, \(a\) defines the amplitude. It's always a positive value because amplitude is about magnitude, which is inherently non-negative.

For the cosine function \( f(x) = 3 \cos 4x \), the amplitude is determined by taking the absolute value of the coefficient \(a\), which in this case is 3.

Thus,

• The amplitude of \( f(x) = 3 \cos 4x \) is 3.

This means that the wave moves 3 units above and 3 units below its central axis, defining the height of peaks and depth of troughs.

How the Period of a Cosine Function is Determined

The period of a trigonometric function refers to the distance over the x-axis it takes for the wave to repeat its shape. It's the interval after which the wave pattern begins to duplicate itself. Understanding the period is key to predicting wave behavior over time.

For a standard cosine function \( a \cos(bx) \), the period is calculated using the formula \( \frac{2\pi}{b} \). This formula involves dividing \(2\pi\), the natural period of a cosine function, by the coefficient \(b\) of \(x\).

In the example of \( f(x) = 3 \cos 4x \), the coefficient \(b\) is 4. Employing the period formula, we find:

• Period = \( \frac{2\pi}{4} = \frac{\pi}{2} \)

This means every \( \frac{\pi}{2} \) units along the x-axis, the wave completes one full cycle before starting over again.

The Basics of the Cosine Function

The cosine function is one of the fundamental trigonometric functions used to model wave-like patterns. Uniquely characterized by its symmetric shape and periodic nature, it plays a pivotal role in fields such as physics and engineering.

The standard cosine wave begins at its maximum value when \(x = 0\) because \(\cos(0) = 1\). As \(x\) increases, the wave cycles through its pattern of peaks and troughs, due to its periodic nature.

A cosine function in the general form of \( a \cos(bx) \) can be manipulated by changing \(a\) and \(b\) to adjust its amplitude and frequency respectively.

Consider \( f(x) = 3 \cos 4x \):

• Amplitude: Given by \(a = 3\), which sets the wave height.
• Frequency: Influenced by \(b = 4\), changing how quickly the cycles repeat.

Thus, options for scaling and shifting make the cosine function highly versatile in plotting various rhythmic phenomena, from sound waves to electronic signals.