Problem 62 - Calculus: One and Several Variables

Question

Determine the period. (The least positive number \(p\) for which \(f(x+p)=f(x)\) for all \(x\).) $$f(x)=\sin \frac{1}{2} x$$.

Step-by-Step Solution

Answer

The period of the function \(f(x) = \sin(0.5x)\) is \(4\pi\).

Step 1: Understanding the basic period of sine function

The basic sine function, \(f(x) = \sin(x)\), has a period of \(2\pi\). This is the least positive real number \(p\) for which \(\sin(x+p) = \sin(x)\) for every real number \(x\).

Step 2: Find the transformation factor

In the function \(f(x) = \sin(0.5x)\), 0.5 is the frequency of the sine function which affects the period of the function. It compresses the function horizontally by a factor of \(0.5\). Because of this, the period of this oscillation will be \(p = \frac{2\pi}{0.5}\).

Step 3: Calculate the period

Dividing \(2\pi\) by 0.5 will give us the period of the function. Therefore, \(p = \frac{2\pi}{0.5} = 4\pi\).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics focused on the relationships between the angles and sides of triangles, especially right-angled triangles. One of the key functions in trigonometry is the sine function, denoted as \(\sin(x)\), which is an oscillating function commonly used to model periodic phenomena. This function takes an angle as an input and outputs the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.

The sine function is also fundamental in the study of waves, such as sound waves, light waves, and tidal waves. Its graph is characterized by a smooth, continuous wave that repeats itself at regular intervals - a property known as periodicity.

Periodic Functions

A periodic function is one that repeats its values at regular intervals, called periods. The trigonometric functions, such as the sine function, are classic examples of periodic functions. The period is the smallest positive value for which the function repeats itself; mathematically, for any function \(f(x)\), if there's a number \(p\) such that \(f(x+p) = f(x)\) for all \(x\), then \(p\) is the period of the function.

Understanding the period is vital in analyzing the behavior of the function over time, predicting its future values, and finding its equivalence with other periodic functions. In real-world terms, it's like knowing the length of a day to predict sunrise and sunset.

Sine Wave Transformations

Sine wave transformations involve altering the basic sine wave \(\sin(x)\) to change its period, amplitude, phase shift, or vertical translation. Factors that affect the period include stretching or compressing the graph horizontally. If the sine function is written as \(f(x) = \sin(kx)\) where \(k\) is a constant, \(k\) influences the wave's frequency. When \(k > 1\), the wave is compressed, resulting in a shorter period. Conversely, if \(k < 1\), the wave is stretched, leading to a longer period.

Effect of Horizontal Stretches and Compressions

For the sine function \(f(x) = \sin(kx)\), the period is found by dividing the standard period \(2\pi\) by the absolute value of \(k\). As a result, transformations offer a powerful tool for modeling variations in periodic behavior, like adjusting the pitch in a musical note.

Function Period Calculation

To calculate the period of a transformed sine function like \(f(x) = \sin(bx)\), use the formula \(p = \frac{2\pi}{|b|}\), where \(b\) is a non-zero real number. In the case of the function \(f(x) = \sin(\frac{1}{2}x)\), the period is calculated by setting \(b = \frac{1}{2}\).

By substituting \(b\) into the formula, you can calculate the new period: \(p = \frac{2\pi}{|\frac{1}{2}|} = 4\pi\). This result tells you that the sine wave completes one full cycle over an interval of \(4\pi\), which is twice as long as the period of the basic sine function due to the fact that \(\frac{1}{2}\) is less than 1, indicating a horizontal stretch of the graph.