Problem 63 - Calculus Single Variable

Question

A function \(f\) is said to have period \(p\) if there is a smallest positive number \(p\) such that \(f(x+p)=f(x)\) for all \(x\) in the domain of \(f\). Find the period of the function defined by the given expression. \(\tan (2 x)+\sin (3 x)\)

Step-by-Step Solution

Answer

The period of the function \( \tan(2x) + \sin(3x) \) is \( 2\pi \).

Step 1: Identify individual periods

Identify the periods of the individual functions within the expression. The tangent function, \( \tan(2x) \), has a period of \( \frac{\pi}{2} \), because its regular period \( \pi \) is divided by its coefficient \( 2 \). The sine function, \( \sin(3x) \), has a period of \( \frac{2\pi}{3} \), because its regular period \( 2\pi \) is divided by its coefficient \( 3 \).

Step 2: Determine the least common multiple

To find the period of the combined function \( \tan(2x) + \sin(3x) \), compute the least common multiple (LCM) of the individual periods \( \frac{\pi}{2} \) and \( \frac{2\pi}{3} \).

Step 3: Convert to common terms

Express both periods \( \frac{\pi}{2} \) and \( \frac{2\pi}{3} \) in terms of the least common multiple. Convert them to have a common denominator if necessary. Here, \( \frac{\pi}{2} = \frac{3\pi}{6} \) and \( \frac{2\pi}{3} = \frac{4\pi}{6} \).

Step 4: Calculate least common multiple

Find the LCM of \( \frac{3\pi}{6} \) and \( \frac{4\pi}{6} \). The LCM combines the numerators over their common denominator: LCM of 3 and 4 is 12. So, the period is \( \frac{12\pi}{6} = 2\pi \).

Key Concepts

Tracing the Path of Trigonometric Functions

Trigonometric functions, like sine and tangent, tell us how angles relate to circles. They are periodic, meaning they repeat themselves over specific intervals. In our task, we have two trigonometric expressions: \( \tan(2x) \) and \( \sin(3x) \). Each has its own period because of the angles they pass through.
The tangent function, noted as \( \tan(x) \), typically has a period of \( \pi \). When we include a multiplier inside the tangent, like \( 2x \), it alters the period. Divide \( \pi \) by this multiplier (2) to find the new period. Thus, the period of \( \tan(2x) \) is \( \frac{\pi}{2} \).
The sine function, noted as \( \sin(x) \), begins with a period of \( 2\pi \). Similar to tangent, if we modify it to \( \sin(3x) \), the 3 changes its rate of repetition. Divide \( 2\pi \) by 3 to learn that \( \sin(3x) \) repeats every \( \frac{2\pi}{3} \). Understanding periods of trigonometric functions helps pave the way to grasp function behaviors and solve intricate problems.

Finding the Least Common Multiple: A Unifying Factor

The least common multiple (LCM) is all about bringing terms together. It helps in finding a shared period when combining functions, such as \( \tan(2x) \) and \( \sin(3x) \).

When expressions operate at different rates, their overall repetition period depends on both rates aligning. For example, we first express \( \frac{\pi}{2} \) and \( \frac{2\pi}{3} \) using a common language, or denominator, to compare them easily. So we obtain \( \frac{3\pi}{6} \) and \( \frac{4\pi}{6} \). Here, the common multipliers or factors unite to become 12, resulting in \( \frac{12\pi}{6} \). Thus, the overall period is influenced by looking at when cycles of the separate functions align, marked by the smallest shared multiple, 2\( \pi \).

Understanding LCM is important when dealing with any interconnected periods, especially in mathematics and harmonizing cycles.

Exploring Function Periods

The concept of a function period is akin to a repeating chorus in music. It lets you anticipate when the cycle of a function will repeat its sequence. With functions like \( \tan(2x) + \sin(3x) \), pinpointing the period helps make sense of the overall behavior.
Given individual periods, we seek the collective one by spotting the LCM. In our example, we combined the periods \( \frac{\pi}{2} \) and \( \frac{2\pi}{3} \) using the least common multiple to arrive at 2\( \pi \). Thus, \( \tan(2x) + \sin(3x) \) repeats its values every 2\( \pi \) units.
Knowing a function's period guides you to predict its pattern, anticipate peaks and troughs, and solve complex mathematical functions systematically. It's like having a roadmap that shows exactly where repeating paths will meet and shape ranges as expected.