Problem 95 - Calculus for Biology and Medicine

Question

Find the amplitude and the period of \(f(x)\) : $$ f(x)=2 \sin \left(\frac{x}{2}\right), \quad x \in \mathbf{R} $$

Step-by-Step Solution

Answer

The amplitude is 2, and the period is \(4\pi\).

Step 1: Identify the Standard Form

The given function is \( f(x) = 2 \sin \left(\frac{x}{2}\right) \). The standard form of a sine function is \( a \sin(bx + c) + d \). Here, \( a = 2 \) and \( b = \frac{1}{2} \).

Step 2: Find the Amplitude

The amplitude of a sine function \( a \sin(bx) \) is the absolute value of \( a \). Therefore, the amplitude of \( f(x) = 2 \sin \left(\frac{x}{2}\right) \) is \( |2| = 2 \).

Step 3: Determine the Period

The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \). For \( f(x) = 2 \sin \left(\frac{x}{2}\right) \), the value of \( b \) is \( \frac{1}{2} \). Thus, the period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).

Key Concepts

Amplitude

The amplitude of a trigonometric function measures the maximum distance it extends from its central axis, or median line, to its highest or lowest points. For sine functions, this is straightforward. If you have a function of the form \(a \sin(bx)\), its amplitude is the absolute value of \(a\).

In the function \(f(x) = 2 \sin \left(\frac{x}{2}\right)\), the constant \(a\) in front of the sine function is 2. Therefore, our amplitude is \(|2|\). This means that the wave of this function reaches up to 2 units above and below the central axis, resulting in a vertical stretch of the graph. This characteristic does not change with variations in the rest of the sine function, so long as the constant \(a\) remains the same.

Period

The period of a sine function describes how often the wave completes a full cycle of its pattern. It's like the length of one complete wave. Calculating the period helps understand how tightly or widely the waves are spaced over the horizontal axis.

For sine functions, the period is determined using the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) inside the sine function. In the function \(f(x) = 2 \sin \left(\frac{x}{2}\right)\), \(b\) is \(\frac{1}{2}\). So, applying the formula, the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).

This indicates that every \(4\pi\) units along the x-axis, the wave pattern repeats itself. Knowing the period is crucial in graphing sine functions accurately, as it dictates the spacing of the wave's cycles.

Sine Function

The sine function is one of the essential trigonometric functions used in mathematics, particularly for modeling periodic behaviors. The standard form of a sine function is \(a \sin(bx + c) + d\). Here's what each part represents:

• \(a\): Amplitude, which determines the height of the wave.
• \(b\): Affects the period, reflecting how many cycles fit into a specific length.
• \(c\): Controls horizontal phase shifts, moving the wave right or left.
• \(d\): Adjusts the vertical position of the waveform.

In our function, \(f(x) = 2 \sin \left(\frac{x}{2}\right)\), we only have \(a\) and \(b\) to consider, making it a simpler form. Here, the sine function starts from the origin and continues to repeat its wave-like pattern indefinitely across the x-axis. It creates a smooth oscillation between a peak and a trough, which doesn't have vertical shifts since \(d = 0\) and no horizontal shift since \(c = 0\). This cyclical nature is what makes sine functions instrumental in describing oscillations and waves.