Question

Graph each damped vibration curve for \(0 \leq t \leq 2 \pi\). $$ d(t)=e^{-t / 4 \pi} \cos t $$

Short answer

Plot the damped vibration curve by combining the exponential decay and cosine functions over the interval from 0 to 2\(\tilde \text{π}\).

Step-by-step solution

Understand the Problem

The function we need to graph is a damped vibration curve given by \[d(t) = e^{-t / 4\tilde \text{π}} \text{ cos } t\]. The domain for this function is from \(0\) to \(2\tilde \text{π} \).

Identify the Components of the Function

The function consists of two parts: an exponential decay part \[e^{-t / 4\tilde \text{π}}\] and a cosine part \[\text{ cos } t\]. The exponential decay will decrease the amplitude of the cosine wave over time.

Plot the Exponential Decay Component

Plot the exponential decay function \[e^{-t/4\tilde \text{π}}\] over the interval \[0 \text{ to } 2\tilde \text{π}\]. This curve will start from 1 when \(t=0\) and gradually approach 0 as \(t\) increases.

Plot the Cosine Component

Plot the cosine function \[\text{ cos } t\] over the interval \[0 \text{ to } 2\tilde \text{π}\]. This curve will oscillate between -1 and 1 with a period of \[2\tilde \text{π}\].

Combine the Components

Multiply the values of the exponential decay function by the cosine function at each point to get the damped vibration curve. This will result in a cosine wave that decreases in amplitude over time.

Plot the Damped Vibration Curve

The final graph of \[d(t)=e^{-t / 4\tilde \text{π}} \text{ cos t}\] should show a cosine wave that starts at an amplitude of 1 and gradually decreases toward 0 as \(t\) approaches \[2\tilde \text{π}\].

Key concepts

Exponential Decay

Exponential decay is a fundamental concept that describes the process where a quantity decreases over time at a rate proportional to its current value. In this exercise, the exponential decay component is represented by \( e^{-t / 4 \tilde{\pi}} \). This function starts at 1 when \( t = 0 \) and gradually approaches 0 as \( t \) increases. This behavior is crucial because it modulates the amplitude of the cosine function, causing the wave to diminish over time instead of maintaining a constant amplitude.

Cosine Function

The cosine function, \( \text{cos} \, t \, \), is a well-known trigonometric function that oscillates between -1 and 1. It has a period of \( 2 \pi \), meaning it completes one full cycle within this interval. In our exercise, the cosine function will create the oscillatory part of the damped vibration curve. By itself, it would maintain consistent peaks and troughs, but when combined with the exponentially decaying function, its amplitude diminishes over time.

Graphing Functions

To graph a function properly, especially a composite one like our damped vibration curve, it’s important to understand each component and how they interact. First, graph the exponential decay \( e^{-t / 4 \pi} \) over the interval \( 0 \leq t \leq 2 \pi \, \), which will give you a curve that starts at 1 and approaches 0. Next, graph the cosine function \( \text{cos} \, t \) over the same interval, which will show its periodic nature oscillating between -1 and 1. When graphing the final damped vibration curve, multiply the values of the exponential decay function by the corresponding cosine values at each point. This will result in a dampening cosine wave.

Trigonometric Functions

Trigonometric functions like cosine are essential tools in mathematics, particularly in modeling periodic phenomena. The cosine function gives the oscillation, while the exponential decay controls the amplitude. When combined, these two functions produce a damped vibration curve. Understanding both these initial functions and their combined effect helps to effectively graph and interpret complex behaviors as shown in the exercise.