Problem 68 - Contemporary Precalculus

Question

Provide further examples of functions with different graphs, whose graphs appear identical in certain viewing windows. Find a periodic function whose graph consists of "square waves." [Hint: Consider the sum $$ \left.\sin \pi t+\frac{1}{3} \sin 3 \pi t+\frac{1}{5} \sin 5 \pi t+\frac{1}{7} \sin 7 \pi t+\cdots\right] $$

Step-by-Step Solution

Answer

Question: Provide two examples of functions having different graphs but appearing identical when viewed in certain windows, and find a periodic function with a "square wave" graph. Answer: 1. Functions y = x^2 and y = x^2 + 5 appear identical in the window -2 < x < 2 and 0 < y < 10, even though they have different graphs. 2. Functions y = sin(x) and y = sin(x) + π look the same in the window 0 < x < 2π and -1 < y < 2, but they are different functions. 3. A periodic function with a "square wave" graph can be generated by summing an odd harmonic series of sine functions: f(t) = Σ(1/(2n+1))sin((2n+1)πt) for n = 0 to ∞.

Step 1: Example 1: Functions appearing identical in certain windows

Consider the functions \(y = x^2\) and \(y = x^2 + 5\). In the window -2 < x < 2 and 0 < y < 10, both functions appear to be the same parabolic shape. However, they are different functions with different graphs, as shown by the different equations.

Step 2: Example 2: Functions appearing identical in certain windows

Consider the functions \(y = \sin(x)\) and \(y = \sin(x) + \pi\). In the window \(0 < x < 2\pi\) and \(-1 < y < 2\), both functions have the same shape of one sine period. But they are indeed two different functions.

Step 3: A periodic function with "square wave" graph using summation of sine functions

Utilizing the hint, we can construct a square wave by summing an odd harmonic series of sine functions, as follows: \[f(t) = \sum_{n=0}^\infty \frac{1}{(2n+1)} \sin((2n+1)\pi t)\] This function represents a periodic "square wave" with a fundamental frequency of \(\pi\). The terms in the series, each with its own frequency, create a wave that approaches a square wave as the number of terms increases. For a small number of terms, the function resembles: \[f(t) \approx \sin(\pi t) + \frac{1}{3}\sin(3\pi t) + \frac{1}{5}\sin(5\pi t) + \frac{1}{7}\sin(7\pi t) + \cdots\] Adding more terms will generate a closer approximation of a square wave.

Key Concepts

Harmonic Series

Harmonic series are an essential component in constructing periodic functions with distinct wave shapes. In this context, they refer to an infinite series of sine functions that add together to form complex waves. These waves have frequencies that are integer multiples of a fundamental frequency, which is the smallest frequency in the series.

In the example of creating a square wave, we use an odd harmonic series. This series only includes sine functions with odd multiples of the fundamental frequency.

• The fundamental frequency is generally the lowest and most dominant frequency in the series.
• Odd multiples result in the specific shape of the square wave.
• The harmonics define the texture and complexity of the wave.

As more terms from the harmonic series are added, the wave becomes a more accurate representation of a square wave. This series helps illustrate how different harmonic components contribute to the overall shape of a complex wave.

Sine Functions

Sine functions serve as building blocks for various periodic functions seen in mathematics, such as waves. These functions are characterized by their smooth, oscillating wave shape that repeats at regular intervals.

In the creation of square waves from sine functions, each sine wave contributes a specific frequency to the overall function. Here are some essential features of sine functions in this context:

• The magnitude of each sine function diminishes as the frequency increases (e.g., \(\frac{1}{3}\sin(3\pi t)\)).
• The fundamental sine function sets the basic period, while higher frequencies add detail.
• The phase and amplitude affect the superposition and outcome of the wave shape.

By summing sine functions of alternating frequencies, especially odd frequencies, one can approximate the desired waveforms, like square waves, when plotting these mathematical functions.

Square Wave

A square wave is a type of periodic waveform that alternates between two levels in a symmetrical manner. This wave is characterized by its sharp transitions between the high and low states, resembling a square when plotted.

Creating a square wave using sine functions involves summing an odd harmonic series of sines:

• Odd harmonics ensure the step-like structure of the square wave.
• Each added harmonic refines the waveform, reducing the ripples or overshoots.
• The convergence to a true square wave becomes more apparent with more terms.

The beauty of the square wave lies in its purity; it is defined mathematically, allowing for precise repetitions without losing form. Although it's an abstraction, this concept finds utility in fields like signal processing, sound synthesis, and electronics.

Graphing Functions

Graphing functions allows us to visualize mathematical relationships and behaviors over a range of values. Through plotting points on a graph, one can observe how different functions behave, particularly periodic ones like sine and square waves.

In understanding the graphing of functions like those that form square waves, consider the following:

• It is crucial to choose the appropriate window to fully visualize waveforms, capturing their period correctly.
• Graphing helps illustrate the overlap of multiple sine waves contributing to the final shape.
• The contrast in appearance helps differentiate graphs that appear similar initially.

By carefully selecting and adjusting graph parameters, students can better interpret and compare different functional forms, furthering their comprehension of these mathematical concepts.