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Chapter 4: Problem 75

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. $$f(x)=2^{-x / 4} \cos \pi x$$

Short Answer

Expert verified
As \(x\) increases without bound, the amplitude of the oscillations of the function \(f(x)=2^{-x / 4} \cos \pi x\) will decrease due to the damping factor and eventually approach zero.

Step by step solution

01

Graph the Function

Start by graphing the function \(f(x)=2^{-x / 4} \cos \pi x\). Make sure to choose a suitable range for \(x\), say for example from \(x=-10\) to \(x=10\).
02

Graph the Damping Factor

Next, proceed to graph the damping factor of the function which is \(2^{-x / 4}\). The damping factor is the part of the function that gradually reduces the amplitude of the oscillations over time. Also, graph this over the same \(x\) range as the function.
03

Analyze the Behavior

The last task is to observe and analyze the behavior of the function as \(x\) increases without bound. Observe that as \(x\) increases, the damping factor \(2^{-x / 4}\) causes the amplitude of the oscillations to decrease, and the oscillations of the function \(f(x)=2^{-x / 4} \cos \pi x\) to become smaller and smaller. Eventually, as \(x\) increases without bound, the oscillations should approach zero. This is a standard behavior of a damping function.

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