Problem 23 - Algebra and Trigonometry

Question

In Problems 23-26, graph each damped vibration curve for \(0 \leq t \leq 2 \pi\). $$ d(t)=e^{-t / \pi} \cos (2 t) $$

Step-by-Step Solution

Answer

Graph the combined function \[d(t)=e^{-t / \pi} \cos (2 t)\] over \[0 \leq t \leq 2 \pi\] to display a decaying oscillation.

Step 1 - Identify the function

The function to graph is the damped vibration curve given by \[d(t)=e^{-t / \pi} \cos (2 t)\].

Step 2 - Understand the components

Recognize that the function is composed of two parts: 1. The exponential decay function \[e^{-t / \pi}\]2. The cosine function \[\cos (2 t)\]. These will affect the graph differently.

Step 3 - Determine the domain

The domain for the given problem is \[0 \leq t \leq 2 \pi \]. This means we only need to consider the values of the function between 0 and \[2 \pi \].

Step 4 - Calculate key points

Calculate d(t) at key points within the domain: \[t=0,\ \frac{\pi}{2},\ \pi,\ \frac{3\pi}{2},\ 2\pi\]. Evaluate d(t) at these points to sketch the curve accurately.

Step 5 - Plot the exponential decay

Plot the exponential decay part \[e^{-t / \pi}\] within \[0 \leq t \leq 2\pi\]. This curve will start at \[e^{0}=1\] at \(t=0\) and decay towards 0 as \(t\) approaches \(2\pi\).

Step 6 - Plot the cosine wave

Plot the cosine function \[\cos (2 t)\]. This wave oscillates with a period of \(\pi\), having peaks and troughs at regular intervals.

Step 7 - Multiply the functions together

Combine the two plots by multiplying them together. The resulting graph will show an oscillating cosine wave that gradually decays in amplitude due to the exponential function.

Step 8 - Draw the final graph

Sketch the final curve based on the combined function \[d(t)=e^{-t / \pi} \cos (2 t)\], ensuring it accurately reflects the decaying oscillations within the domain \[0 \leq t \leq 2 \pi\].

Key Concepts

Exponential Decay

The curve will show a steep decline initially and then slowly approach the x-axis. This exponential decay modulates the amplitude of the resulting wave, diminishing its height over time.

Cosine Function

Note that it completes one full cycle over a period of \(\pi\), which is half the period of the standard cosine function. This means that within 0 to 2\pi, the cosine wave will complete two full oscillations. Recognize the standard peaks at \(t = 0, \pi,\) and \(2\pi\), and troughs at \(t = \frac{\pi}{2}\) and \(\frac{3\pi}{2}\).

Function Graphing

Evaluate \(d(t)\) at \(t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},\) and \(2\pi\) to gather important values that help sketch the curve accurately. By combining the graphs of \(e^{-t / \pi}\) and \(\cos (2 t)\), multiply the values point by point to see how the cosine wave decreases in amplitude.

Amplitude Modulation

As you move along the x-axis through \(0 \leq t \leq 2\pi\), you will notice the peaks and troughs of the cosine wave becoming less pronounced, pointing gradually toward the x-axis. This shows that the damping effect reduces the height, or amplitude, of the oscillations as time progresses. This combination results in a damped vibration curve distinctively showing decaying oscillations.