Problem 71 - Algebra and Trigonometry

Question

The functions in Exercises \(70-71\) model motion in which the amplitude decreases with time due to friction or other resistive forces. Graph each function in the given viewing rectangle. How many complete oscillations occur on the time interval \(0 \leq x \leq 10 ?\) $$ y=-6 e^{-0.09 x} \cos 2 \pi x ;[0,10,1] \text { by }[-6,6,1] $$

Step-by-Step Solution

Answer

Use a graphing calculator or tool to plot the function and determine the number of oscillations visually. The exact count may differ based on the analytic solution.

Step 1: Understand the Function Properties

The given function is a product of an exponential function and a cosine function, which characterizes a damping oscillatory motion. As \(x\) increases, the amplitude of the cosine wave decreases due to the multiplier \(-6 e^{-0.09 x}\).

Step 2: Draw the Graph

Plot the graph of the function \(y = -6e^{-0.09x} \cos 2\pi x \) over the interval \(0 \leq x \leq 10\). Use the viewing rectangle \([0,10,1]\) by \([-6,6,1]\) to observe the behavior of the function.

Step 3: Count the Oscillations

An oscillation is said to be complete when the wave starting from one point returns to the same point after going up and down or vice versa. Count the number of times this happens in the function from \(0 \leq x \leq 10\).

Key Concepts

Exponential Decay

When learning about damped oscillatory motion, it's crucial to understand the concept of exponential decay, which describes how the amplitude of oscillations decreases over time. In essence, exponential decay occurs when a quantity reduces at a rate proportionate to its current value, leading to a rapid decrease initially that slows down over time. This behavior is represented mathematically by the exponential function \( e^{-kx} \), where \( k \) is a positive constant, and \( x \) usually represents time.

Employing an exponential decay function as part of a damped oscillation equation, such as \( -6 e^{-0.09 x} \) in our example, implies that the oscillation's magnitude gets smaller and smaller as time \( x \) goes on. This mimics real-world phenomena such as a swinging pendulum that gradually comes to a stop due to air resistance or friction.

Cosine Function

Oscillations can be visualized by employing trigonometric functions, and the cosine function is a prime example in depicting wave-like patterns in mathematics. It's defined as \( \text{cos}(\theta) \) for an angle \( \theta \) and generates a wave that starts from a maximum value. Our function \( \text{cos} 2 \text{pi} x \) exhibits this repeating wave behavior, with \( 2 \text{pi} \) indicating that the wave cycles every unit of \( x \) since \( 2 \text{pi} \) radians is the equivalent of one complete rotation.

The cosine function is specifically selected in the modeling of oscillatory motion due to its smooth and regular wave pattern, which authentically captures the essence of many natural oscillatory processes, like the back-and-forth movement of a mass on a spring.

Graphing Functions

Graphing functions is a powerful tool for visualizing equations and understanding their behavior. For the function \( -6e^{-0.09x} \text{cos} 2\text{pi} x \), graphing it can expose the oscillation pattern and how it's affected by the exponential decay. To graph it accurately, you will need to mark the viewing rectangle, which frames the section of the coordinate plane you're interested in.

To illustrate the function, begin plotting points starting from \( x = 0 \) and observe how the function behaves as \( x \) increases. Due to the exponential decay component, expect the amplitude of the waves - their height from the central axis - to shrink as \( x \) progresses. Using the given viewing rectangle, you can observe the damped waves and how they remain within the bounds of \( -6 \) and \( 6 \) on the y-axis.

Oscillations

Oscillations refer to the repetitive variations, typically in time, of a physical quantity around a central value. In the context of our function, one complete oscillation can be seen when the cosine wave, affected by exponential decay, starts at a peak, drops to a trough, and returns back to the peak.

Counting the number of completed patterns between \( x = 0 \) and \( x = 10 \) on your graph will tell you how many complete oscillations occur. Each peak-to-peak or trough-to-trough segment represents one full cycle. As the exponential decay factor is at work, these oscillations are not uniform; they decrease in magnitude, depicting a damped motion. This portrayal is particularly helpful in fields like engineering and physics where understanding the behavior of oscillatory systems is crucial.