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Chapter 4: Problem 98

For \(x>0,\) what effect does \(2^{-x}\) in \(y=2^{-x} \sin x\) have on the graph of \(y=\sin x ?\) What kind of behavior can be modeled by a function such as \(y=2^{-x} \sin x ?\)

Short Answer

Expert verified
The function \(y=2^{-x} \sin x\) starts like \(y=\sin x\) but the amplitude decreases due to the exponential decay factor, \(2^{-x}\). This behavior models natural phenomena with dampening over time such as a bouncing ball or a vibrating mechanical system.

Step by step solution

01

Understand the Basic Functions

Start with understanding the sine function \(y=\sin x\) and exponential function \(2^{-x}\). The sine function is periodic in nature, resulting in a wave-like graph, while an exponential function typically depicts growth or decay.
02

Understand the Modified Function

Move onto the function \(y=2^{-x} \sin x\). Here, \(2^{-x}\) is an exponential decay factor which decreases the amplitude of the wave function as \(x\) increases. Because of this factor, the output \(y\) gets closer to zero as \(x\) increases.
03

Effect on the Graph

The graph of \(y=2^{-x} \sin x\) would start like \(y=\sin x\) but because of the decay factor, the wave gradually flattens out as \(x\) increases. This results in dampened sine waves.
04

Behavior of the modelled function

This type of function, with a dampening factor applied to a periodic function, can model physical phenomena such as damped harmonic motion. An example of this could be the diminishing height of a bouncing ball or the decreasing amplitude of a vibrating mechanical system over time due to energy dissipation.

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Most popular questions from this chapter

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\tan \frac{x}{4}$$

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