Problem 59 - Calculus: One and Several Variables

Question

Determine the period. (The least positive number \(p\) for which \(f(x+p)=f(x)\) for all \(x\).) $$f(x)=\sin \pi x$$.

Step-by-Step Solution

Answer

The period of the function \(f(x) = \sin \pi x\) is 2.

Step 1: Identify and present the function

The function to evaluate is \(f(x) = \sin \pi x\). This is a sine function where \(\pi\) is the coefficient of \(x\).

Step 2: Determine the period of the standard function

The standard sine function \(f(x) = \sin x\) has a period of \(2\pi\), which means it repeats every \(2\pi\) units.

Step 3: Determine the period of given function

If the standard sine function is altered by multiplying \(x\) by a factor, the period changes. The period \(P\) of \(f(x) = \sin(Bx)\), where \(B\) is the coefficient of \(x\), is given by \(P = 2\pi / |B|\). In this case, \(B\) is \(\pi\), so the period is \(P = 2\pi / |\pi| = 2\).

Key Concepts

Sine Function

The sine function, denoted as \( \sin(x) \), is one of the fundamental functions in trigonometry that corresponds to an angle's sine value in a right triangle. Its graph is a smooth, wave-like pattern that repeats at regular intervals, creating what we call a "periodic" behavior.

• The sine function outputs values between -1 and 1, which means it smoothly oscillates up and down between these bounds.
• In trigonometry, it is used to model cycles and oscillations such as sound waves, light waves, and tides.

The function \( f(x) = \sin(x) \) is known as the standard sine function. It is defined at all angles and is continuously differentiable, making it a handy tool for describing smooth periodic motion. When modified by multiplying its input by constants, like in \( f(x) = \sin(\pi x) \), the function's graph is stretched or compressed horizontally, altering its period while maintaining its distinctive wave shape.

Function Period

The function period is the length over which a function's pattern repeats itself. For a sine function like \( f(x) = \sin(Bx) \), the period determines the interval required for the function to complete one full cycle. To find the period of \( \sin(Bx) \), use the formula:\[P = \frac{2\pi}{|B|}\]where \( B \) is the coefficient of \( x \). This coefficient essentially scales the period of the standard sine function, \( \sin(x) \), which has a period of \( 2\pi \).

• If \( B = 1 \), the period remains at \( 2\pi \).
• If \( B \) is altered, the function repeats more or less frequently, depending on the value of \( B \).

For example, in \( f(x) = \sin(\pi x) \), the coefficient \( B \) is \( \pi \), which results in a period of \( 2 \) as calculated by:\[P = \frac{2\pi}{\pi} = 2\]This means that the function \( f(x) = \sin(\pi x) \) completes one cycle in an interval of 2 units along the x-axis.

Trigonometric Functions

Trigonometric functions are a set of functions that relate the angles of a triangle to the lengths of its sides. They are widely recognized for their role in describing periodic phenomena. The primary trigonometric functions include sine, cosine, and tangent, each contributing uniquely to analyzing cyclic patterns.

• Sine (\( \sin \)): Models the y-coordinate of a point on the unit circle as it moves around, smoothly oscillating between -1 and 1.
• Cosine (\( \cos \)): Represents the x-coordinate of the said point and also oscillates smoothly like the sine function but starts from 1.
• Tangent (\( \tan \)): Models the ratio of sine to cosine and has a different periodic behavior characterized by vertical asymptotes.

These functions are crucial for describing waves and oscillations, making them integral in fields such as physics, engineering, and even economics, where they model reoccurring trends. Understanding the periods of such functions, like we examined in \( \sin(\pi x) \), is essential for precise predictions and modeling of periodic events.