Question

Rewrite the function \(y(t)\) in the form \(y(t)=R e^{a t} \cos (\beta t-\delta)\), where \(0 \leq \delta<2 \pi\). Use this representation to sketch a graph of the given function, on a domain sufficiently large to display its main features. $$ y(t)=e^{-2 t} \cos 2 t-e^{-2 t} \sin 2 t $$

Short answer

Question: Rewrite the given function \(y(t) = e^{-2t} \cos 2t - e^{-2t} \sin 2t\) in the form \(y(t) = R e^{at}\cos(\beta t - \delta)\) and sketch its graph, highlighting the main features such as the amplitude, frequency, and phase shift. Answer: After rewriting, the function is \(y(t) = e^{-2t}\cos(2t)\) with amplitude given by \(e^{-2t}\), frequency of \(2\), and no phase shift (\(\delta = 0\)). To sketch the graph, start at an amplitude of 1, have two oscillations within each time period of \(2\pi\), and let the amplitude decay exponentially as \(t\) increases. The graph will oscillate between positive and negative values, with the peaks getting closer to the horizontal axis as \(t\) increases.

Step-by-step solution

Rewrite the function using complex exponentials

We will use the property \(e^{i\theta} = \cos \theta + i\sin \theta\), which allows us to express the given function as a complex exponential. $$ y(t) = e^{-2t} \cos 2t - e^{-2t} \sin 2t = e^{-2t}( \cos 2t - i\sin 2t ) $$ Now, using the property \(e^{i\theta} = \cos \theta + i\sin \theta\) again, we can rewrite the expression inside the parentheses in terms of another complex exponential: $$ \cos 2t - i\sin 2t = e^{-i 2t} $$ Now the function becomes: $$ y(t) = e^{-2t}e^{-i2t} $$

Express the function in the desired form

Combine the exponentials and multiply them by a constant \(R\) that takes care of the initial amplitude: $$ y(t) = R e^{(-2 - i2)t} $$ We see that our function now has the form \(y(t) = R e^{at}\cos(\beta t - \delta)\) with \(a=-2\), \(\beta=2\), and \(\delta = 0\). So, $$ y(t) = R e^{-2t}\cos(2t) $$ Comparing this expression to our original function, we see that \(R=1\). Thus, our final rewritten function is: $$ y(t) = e^{-2t}\cos(2t) $$

Sketch the graph

To sketch the graph, first, identify the main features of the function: - Amplitude: The function amplitude varies as \(e^{-2t}\). This means the amplitude starts at 1 when \(t = 0\) and decays exponentially as \(t\) increases. - Frequency: The oscillatory part of the function is given by \(\cos(2t)\). The frequency is \(2\), which means the function completes two cycles in every time period of \(2\pi\). - Phase Shift: There is no phase shift as \(\delta = 0\). Based on these features, sketch the graph of the function \(y(t)\) starting at an amplitude of 1, with two oscillations within each time period of \(2\pi\), while the amplitude decays exponentially. The graph will oscillate between positive and negative values, with the peaks getting closer to the horizontal axis as \(t\) increases.

Key concepts

Complex Exponentials

Complex exponentials are a beautiful bridge between trigonometric functions and exponential functions. They are based on Euler's formula: \(e^{i\theta} = \cos \theta + i \sin \theta\). This formula tells us that we can express a cosine function and a sine function by using a complex exponential.
In the original problem, we used this idea to simplify the given function \(y(t) = e^{-2t} \cos 2t - e^{-2t} \sin 2t\). By combining the cosine and sine terms using their expressions in terms of complex exponentials, this results in a format of \(e^{at}\).

• Complex exponentials can simplify mathematical expressions significantly.
• They help in converting trigonometric identities into more manageable forms.
• This conversion makes it easier to analyze the behavior of a function especially in the context of differential equations.

Through the process above, our original function was transformed into the simpler exponential form \(y(t) = e^{-2t} e^{-i2t}\).

Amplitude and Frequency

When dealing with functions of the form \(R e^{at} \cos(\beta t - \delta)\), it's important to understand the role of amplitude and frequency. Amplitude describes the peak value of the oscillations, while frequency refers to how often these oscillations occur in a given time period.
In the rewritten function \(y(t) = e^{-2t} \cos(2t)\), let's explore these concepts further:

• Amplitude: Given by \(e^{-2t}\), the amplitude of this function decreases over time, due to the negative exponent in the exponential term.
• Frequency: Determined by the \(\cos(2t)\), the frequency is 2. This means there are two complete oscillations within each \(2\pi\) time period.

The amplitude decay is exponential, contributing to the damping of the oscillations. This means the waves get smaller as time progresses. However, the frequency remains constant at 2 cycles per \(2\pi\) interval, causing the oscillations' pace to remain steady despite reducing in size.

Graph Sketching

Graph sketching helps visualize the behavior of a function over time. For the function \(y(t) = e^{-2t} \cos(2t)\), sketching is especially useful to see the changes in amplitude and the frequency of oscillations.
To sketch the graph, follow these steps:

• Start with Amplitude: As \(t\) starts at 0, the amplitude is 1, because \(e^{-2 \cdot 0} = 1\). This is the highest point on the graph.
• Observe Decay: As \(t\) increases, note how the amplitude decreases exponentially, causing peaks in the graph to get lower.
• Track the Frequency: The graph will complete two oscillations within each \(2\pi\) interval because of the frequency \(\cos(2t)\).

Putting these observations together, the sketch should start at \(t = 0\) with an initial peak at 1, followed by oscillations which decrease over time. The graph will pass through the \(x\)-axis as it oscillates, gradually tapering off towards it as amplitude diminishes. This exponential decay reflects the damping effect from \(e^{-2t}\), illustrating the shrinking oscillations in sync with the persistent frequence of the cosine function.