Question

\(u(\tau, 2 \tau)=\tau^{2}, \quad-\infty<\tau<\infty\) \(u(\tau,-1)=\cos \pi \tau, \quad-\infty<\tau<\infty\)

Short answer

Answer: The first function \(u(\tau, 2 \tau)=\tau^{2}\) is a parabola opening upwards, with its vertex at point (0, 0). The second function \(u(\tau, -1)=\cos \pi \tau\) is a cosine wave with a period of 2 and an amplitude of 1, oscillating between -1 and 1 along the y-axis.

Step-by-step solution

Analyze and simplify the first function \(u(\tau, 2 \tau)\)

: We are given the function \(u(\tau, 2 \tau)=\tau^{2}\), where \(-\infty<\tau<\infty\). This function can be thought of as a two-variable function u with its second argument being twice the first argument. It only has one independent variable, \(\tau\). Now, we'll describe how to evaluate and plot this function.

Evaluate and plot \(u(\tau, 2 \tau)\)

: To evaluate this function, simply substitute the value of \(\tau\) into the given equation: \(u(\tau, 2 \tau)=\tau^{2}\) Now, this function can be plotted on a plane with the x-axis representing \(\tau\) and the y-axis representing the function value \(u(\tau, 2 \tau)\). For any value of \(\tau\), the point \((\tau,\tau^2)\) can be plotted on the plane. This will result in a parabola with the vertex at point (0, 0) and opening upwards.

Analyze and simplify the second function \(u(\tau,-1)\)

: Next, we are given the function \(u(\tau,-1)=\cos \pi \tau\), where \(-\infty<\tau<\infty\). Similar to the first function, this can also be considered as a two-variable function u with its second argument being fixed at -1. It only has one independent variable, \(\tau\). We will now describe how to evaluate and plot this function.

Evaluate and plot \(u(\tau,-1)\)

: To evaluate this function, substitute the value of \(\tau\) into the given equation: \(u(\tau,-1)=\cos \pi \tau\) Now, this function can be plotted on a plane with the x-axis representing \(\tau\) and the y-axis representing the function value \(u(\tau,-1)\). This results in a cosine wave with a period of 2 and an amplitude of 1. This wave oscillates between -1 and 1 along the y-axis. In conclusion, we have broken down and analyzed the given functions, evaluated them and provided a description on how to plot them. The first function is a parabola opening upwards, and the second function is a cosine wave with a period of 2 and amplitude of 1.

Key concepts

Two-variable function

In mathematics, a two-variable function involves two inputs, typically denoted as \(x\) and \(y\), to produce a single output. However, in the exercises we are analyzing, the functions are expressed as \(u(\tau, 2\tau)\) and \(u(\tau, -1)\). You might notice that although they look like two-variable functions, they are effectively dependent on only one independent variable, \(\tau\). Here, the second variable is either a linear multiple of the first or a constant, simplifying our approach.

This simplification means that although these functions seem to take two inputs, only the variation in \(\tau\) influences their behavior and graph. This allows us to evaluate and graph them more understandably, focusing solely on the changes in \(\tau\).

Function plot

A function plot is a visual representation of a function where you map the relationship defined by the function onto a plane. For the functions we are dealing with, the x-axis often represents the independent variable, \(\tau\), while the y-axis represents the output value of the function, which could be \(u(\tau, 2\tau)\) or \(u(\tau, -1)\).

By plotting these functions, we convert mathematical equations into shapes that we can easily interpret. For \(u(\tau, 2\tau) = \tau^2\), the plot results in a parabola opening upwards, making it intuitive to comprehend the increase in value proportional to \(\tau^2\). Conversely, the function \(u(\tau, -1) = \cos \pi \tau\) translates into a wave-like plot, characteristic of the behavior of cosine functions.

Cosine wave

The second function, \(u(\tau, -1) = \cos \pi \tau\), gives rise to a cosine wave when plotted. A cosine wave is a type of periodic function that oscillates with a regular amplitude and period.

In this case, the wave amplitude is 1, meaning it oscillates between 1 and -1. The period of the wave is 2, which indicates that it repeats its pattern every two units along the \(\tau\)-axis. Cosine waves arise naturally in various fields of physics and engineering, known for their smooth oscillations and predictable nature. Understanding this function helps in visualizing phenomena like sound waves and alternating current signals.

Independent variable

An independent variable is a variable that provides the input value for a function, determining the function's final output. In our scenarios, \(\tau\) is the only independent variable affecting the functions \(u(\tau, 2\tau)\) and \(u(\tau, -1)\).

Identifying an independent variable is crucial because it isolates the source of changes in the function's value. By manipulating \(\tau\), we see how each function reacts and plots, whether depicting a steadily increasing curve like a parabola or an oscillating pattern like a cosine wave. Understanding this concept is key to effectively analyzing any function and predicting how it behaves when different values are applied.

Graph analysis

Graph analysis involves evaluating a function's graph to derive meaningful information about the function’s behavior and properties. By graphing \(u(\tau, 2\tau) = \tau^2\), we can see that it's a parabola, indicating constant positive acceleration, meaning values increase at a growing rate as \(\tau\) moves away from zero.

For \(u(\tau, -1) = \cos \pi \tau\), graph analysis reveals a repeated oscillation, providing insights into its periodic nature. This sinusoidal pattern helps in identifying cycles and frequencies, which are vital in fields such as audio processing and electromagnetic wave analysis.

• Graph analysis simplifies complex data visualization.
• Helps in easy interpretation of function characteristics.
• Assists in identifying function trends and patterns.

This method of analysis is invaluable in various applications, offering a clear view of how mathematical functions map to real-world phenomena.