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Chapter 4: Problem 74

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$f(x)=e^{-x} \cos x$$

Short Answer

Expert verified
As \(x\) increases, the function \(f(x)=e^{-x} \cos x\) depicts damped harmonic motion. The values oscillate above and below the x-axis with decreasing amplitude tending towards zero, demonstrating the behavior of a damping factor.

Step by step solution

01

Understanding the function

To use the graphing utility, first understand the function \(f(x)=e^{-x} \cos x\). It is a product of two simpler functions, an exponential function \(e^{-x}\) and a trigonometric function \(\cos x\). The function \(e^{-x}\) serves as a damping factor that reduces the amplitude of the regularly oscillating \(\cos x\) as \(x\) increases.
02

Graphing the function

Use a graphing utility to plot \(f(x)=e^{-x} \cos x\). In order to view the behavior of the damping factor along with the function, also plot \(y=e^{-x}\) in the same viewing window. This will allow a comparison between the oscillations of \(f(x)\) and its damping factor.
03

Interpreting the graph

From the graph, observe that as \(x\) increases, the value of \(f(x)\) tends to 0 from both positive and negative sides due to the damping factor \(e^{-x}\). This damping effect squishes the cosine function closer and closer to the x-axis as \(x\) increases, resulting in oscillations of decreasing amplitude. This is characteristic of a damped harmonic motion.

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