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Chapter 10: Problem 49

Graph the function with the help of your calculator and discuss the given questions with your classmates. \(f(x)=e^{-0.1 x}(\cos (2 x)+\sin (2 x))\). Graph \(y=\pm e^{-0.1 x}\) on the same set of axes and describe the behavior of \(f\).

Short Answer

Expert verified
The function oscillates between the envelope functions, with amplitude diminishing as x increases due to exponential decay.

Step by step solution

01

Understand the Function

The given function is \( f(x) = e^{-0.1x} (\cos (2x) + \sin (2x)) \). This is the product of an exponential decay function and a trigonometric function. The exponential term \( e^{-0.1x} \) governs the amplitude of the oscillation, which decays as \( x \) increases.
02

Setup the Graphing Calculator

To graph the function using a calculator, input \( f(x) = e^{-0.1x} (\cos (2x) + \sin (2x)) \). Make sure to choose an appropriate window setting that captures the oscillations and decay. A typical setting might be \( x: [0, 20] \) and \( y: [-1, 1] \) in order to clearly observe the behavior.
03

Graph the Envelope Functions

Additionally, graph the envelope functions \( y = e^{-0.1x} \) and \( y = -e^{-0.1x} \). These functions represent the maximum and minimum bounds of \( f(x) \) over time as they decay towards zero along the y-axis.
04

Observe and Analyze the Behavior

Upon graphing, observe how \( f(x) \) oscillates between the envelope functions. The function starts with a larger amplitude at \( x = 0 \) and gradually diminishes due to the exponential term. As \( x \) increases, the amplitude gets smaller and eventually approaches zero. This behavior indicates that the function is oscillating with diminishing amplitude, eventually approaching a steady state at \( y = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Decay
Exponential decay is a mathematical concept where a quantity diminishes over time at a rate proportional to its current value. In the function \( f(x) = e^{-0.1x} (\cos (2x) + \sin (2x)) \), the exponential term \( e^{-0.1x} \) is responsible for the decay behavior. As \( x \) increases, \( e^{-0.1x} \) decreases, causing the overall function to shrink in amplitude. This means the peaks and valleys of the oscillating function will lessen over time.
  • The base of the exponential function is the natural number \( e \), approximately equal to 2.718.
  • The coefficient \(-0.1\) represents the rate of decay, indicating how fast the function decreases.
Exponential decay is crucial in many real-world applications like radioactive decay or cooling processes. It is a fundamental concept that helps in understanding the behavior of systems that lose energy or mass over time.
Envelope Curve
The envelope curve refers to a pair of bounding lines that frame an oscillating function. In our function \( f(x) = e^{-0.1x} (\cos (2x) + \sin (2x)) \), the envelope curves are represented by \( y = e^{-0.1x} \) and \( y = -e^{-0.1x} \). These curves define the maximum and minimum extents of the oscillations in the graph.
  • The upper envelope, \( y = e^{-0.1x} \), outlines the highest points the function can reach.
  • The lower envelope, \( y = -e^{-0.1x} \), defines the lowest points.
Graphing envelope curves provides a clear visualization of how the oscillations are constrained. While the function oscillates, the amplitude decreases as \( x \) increases due to the behavior of these decaying envelope curves. This concept helps understand not just how high or low a function can go, but also how its behavior changes over time.
Trigonometric Oscillation
Trigonometric oscillation describes the periodic behavior of functions involving sine and cosine. In the function \( f(x) = e^{-0.1x} (\cos (2x) + \sin (2x)) \), \( \cos(2x) + \sin(2x) \) creates the oscillating pattern.
  • The term \( \cos(2x) \) has a regular wave-like pattern with a period of \( \pi \), meaning it completes a cycle every \( \pi \) units.
  • Similarly, \( \sin(2x) \) also contributes to the oscillatory behavior with the same periodicity and phase shift.
When these functions are combined, they form a complex oscillating pattern with varying amplitudes, influenced by the exponential decay. Trigonometric oscillation is essential in modelling wave-like phenomena such as sound or light waves. It provides insight into how these repetitive patterns change, offering a predictable rhythm, even as the amplitude declines.
Graphing Calculator Techniques
Using a graphing calculator is a powerful way to visualize mathematical functions. Here are some steps to effectively graph a complex function like \( f(x) = e^{-0.1x} (\cos (2x) + \sin (2x)) \):
  • Set up your calculator by correctly entering the function. This includes using the exponential and trigonometric function buttons properly.
  • Choose an appropriate window setting. For this function, try \( x: [0, 20] \) and \( y: [-1, 1] \) to capture the primary behaviors within a focused range.
Additionally, graph the envelope functions \( y = e^{-0.1x} \) and \( y = -e^{-0.1x} \) on the same axes. This helps to compare and understand the constraints they impose on the oscillating pattern. Visualizing this way assists greatly in comprehending the interaction between exponential decay and trigonometric oscillation. A graphing calculator offers an interactive approach to explore the dynamics of various functions, making it an invaluable educational tool.

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