Problem 101 - Calculus for Biology and Medicine

Question

Let $$ f(x)=4 \cos \left(\frac{x}{4}\right), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Step-by-Step Solution

Answer

Amplitude: 4, Period: \(8\pi\)

Step 1: Understand the Function Type

The function in question is a cosine function, which generally has the form \( a \cos(bx) \). The parameters \( a \) and \( b \) are critical in determining the function's amplitude and period.

Step 2: Identify the Parameters

For the given function \( f(x) = 4 \cos\left(\frac{x}{4}\right) \), we can identify \( a = 4 \) and \( b = \frac{1}{4} \). These parameters will allow us to find the amplitude and period.

Step 3: Calculate the Amplitude

The amplitude of a cosine function \( a \cos(bx) \) is the absolute value of \( a \). Therefore, the amplitude of \( f(x) \) is \( |4| = 4 \).

Step 4: Calculate the Period

The period of a general cosine function \( a \cos(bx) \) is calculated using the formula \( \frac{2\pi}{b} \). For this function, replacing \( b \) with \( \frac{1}{4} \), the period is \( \frac{2\pi}{\frac{1}{4}} = 8\pi \).

Key Concepts

Amplitude

Amplitude is a key concept when working with trigonometric functions like cosine. It refers to the maximum extent of the oscillation from the middle or equilibrium position. In simpler terms, amplitude measures how high and low the wave of the function goes.

To find the amplitude of a cosine function, you look at the coefficient in front of the cosine term, represented as \(a\) in the general form \(a \cos(bx)\). The absolute value of \(a\) gives you the amplitude. This ensures that amplitude is always a non-negative number as it is a measure of distance.

• For instance, with the function \(f(x) = 4 \cos\left(\frac{x}{4}\right)\), the coefficient \(a = 4\).
• The amplitude is thus \(|4| = 4\).

Understanding amplitude helps in knowing how much variation the function will have around its central axis. It's like looking at how tall the waves are in a sea.

Period

The period of a trigonometric function tells us how long it takes for the function to complete one full cycle of its pattern. Think of it like the time a pendulum takes to swing back and forth once.

For cosine and sine functions, the standard period is given by \(2\pi\). This represents a full rotation or cycle. However, when we have a function of the form \(a \cos(bx)\) or \(a \sin(bx)\), we need to adjust the period using the "\(b\)" value.

To compute the period of these functions, use the formula:

• \(\text{Period} = \frac{2\pi}{b}\)

Where \(b\) affects the frequency, and thus the period of the function.

• For \(f(x) = 4 \cos\left(\frac{x}{4}\right)\), \(b\) is \(\frac{1}{4}\).
• Applying the formula, the period is \(\frac{2\pi}{\frac{1}{4}} = 8\pi\).

Knowing the period allows you to predict how the function behaves over different x-values, indicating its regularity or frequency of oscillations.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often written as \(\cos(\theta)\), where \(\theta\) represents the angle. Cosine values range from -1 to 1, and it describes a wave-like pattern.

The general form of a cosine function is \(a \cos(bx)\), allowing modifications through parameters \(a\) and \(b\):

• \(a\) changes the amplitude, affecting the height of the peaks.
• \(b\) changes the period, affecting the stretch or compression of the wave.

A standard cosine function, like \(\cos(x)\), completes a cycle every \(2\pi\) units. Its graph is symmetric around the y-axis due to its even nature.

In the function \(f(x) = 4 \cos\left(\frac{x}{4}\right)\):

• The amplitude is 4, showing the maximum highs and lows as 4 units from the center line.
• The period is \(8\pi\), indicating the length over which each complete wave pattern repeats.

Understanding the cosine function and its components aids in analyzing various waveforms in multiple fields like physics, engineering, and economics.