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Chapter 4: Problem 76

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. $$h(x)=2^{-x^{2} / 4} \sin x$$

Short Answer

Expert verified
The given function \(h(x)=2^{-x^{2}/4} \sin x\) behaves like a damped sine wave. As \(x\) tends to infinity, the function approaches 0 due to the damping factor, but it continues to oscillate between positive and negative values with decreasing amplitude.

Step by step solution

01

Graph the basic sine function

First, graph the parent function, \(\sin x\). The sine function oscillates between -1 and 1 for all values of \(x\).
02

Graph the damping factor

Next, graph the damping factor \(2^{-x^{2}/4}\). This function decreases rapidly as \(x\) increases, approaching zero but always staying positive. It also symmetrically mirrors to the negative \(x\) values.
03

Graph the given function

Overlay the graph of the function \(h(x)=2^{-x^{2}/4} \sin x\). The function exhibits a pattern of oscillation similar to the basic sine wave. However, its amplitude decreases as \(x\) increases, due to the damping factor.
04

Analyze the behavior of the function as \(x\) tends to infinity

As \(x\) tends to infinity, the amplitude of oscillations tends to 0 due to the exponential damping factor. Hence, this function approaches 0 as \(x\) tends to infinity, but keeps oscillating between the negative and positive values with decreasing amplitude.

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