/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Sketch a graph of the given func... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 30

Sketch a graph of the given function. $$f(x)=e^{-x / 4} \sin x$$

Short Answer

Expert verified
The graph of the function starts as a standard sine wave but its amplitude reduces gradually as x increases, due to the function \(e^{-x / 4}\). This leads to the graph showing a diminishing sine wave as x approaches infinity.

Step by step solution

01

Understand the function

Given function is \(f(x)=e^{-x / 4} \sin x\) which is a product of an exponential function and a sine function. We can imagine the graph as a sine wave where the amplitude of the wave is scaled by the function \(e^{-x / 4}\). As x gets larger, the amplitude will get smaller due to the exponential part.
02

Plot the Key Points

To help sketch the function, we can plot a few key points. These are going to be the points which form the minimum and maximum value of the function for few periods. As x approaches infinity, the wave's amplitude will shrink towards 0.
03

Sketch the Graph

Drawing on the key points plotted, we can sketch the function. The graph would start with standard sine wave shape, but over the time (as x increases), the amplitude of our sine wave should decrease because of the effect of \(e^{-x / 4}\). This will result in a graph that gradually diminishes in its peaks and troughs as x approaches infinity.

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Key Concepts

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

Exponential Functions
Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. In the given exercise, the exponential function is represented by \(e^{-x/4}\). Here, **e** is the base of the natural logarithm, which is approximately equal to 2.718. This function grows or decays at an exponential rate as the value of x changes.

  • In the context of the problem, the negative exponent \(-x/4\) indicates that the function's value decreases exponentially as **x** increases.
  • This results in the amplitude of the sine wave shrinking over time, as **x** progresses.
  • The nature of exponential functions allows them to approach zero but never actually reach it, creating a curve that gets closer and closer to the x-axis.
Understanding this concept helps in visualizing how the exponential component influences the graph of the given function by scaling down the amplitude of the sine wave.
Trigonometric Functions
Trigonometric functions, such as sine, are periodic functions that repeat their values in regular intervals. In this exercise, the sine function \(\sin x\) is central to the composite function.

  • The sine function oscillates between -1 and 1, generating a wave-like graph.
  • Its period is \(2\pi\), meaning the wave pattern repeats every \(2\pi\) units along the **x-axis**.
  • This regular oscillation results in a smooth, repetitive pattern which is foundational to the behavior of the specified function.
When multiplied by the exponential function \(e^{-x/4}\), the sine function's amplitude decreases over time, giving the graph its diminishing appearance. This highlights the combined effect wherein the period remains constant, while the peaks and troughs lessen as **x** increases.
Amplitude Modulation
Amplitude modulation is the process of varying the amplitude of a wave, often in response to an additional influencing function. In this context, the sine wave \(\sin x\) has its amplitude modulated by the exponential decay function \(e^{-x/4}\).

  • The amplitude of the sine wave is controlled by the value of the exponential function, which itself decreases over time due to the negative exponent.
  • This results in progressively smaller wave peaks and troughs as x increases.
  • In essence, amplitude modulation combines the variations of both functions, visually representing how the sine wave's size adapts due to the decay factor.
Amplitude modulation is a concept prevalent in many real-world applications, such as radio transmission, where similar principles are used to encode information into signals. In mathematical graphs, it showcases how one function can dynamically influence another, creating complex and informative visuals.

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Most popular questions from this chapter

Rewrite the expression as a single logarithm. $$\ln 3-\ln 4$$

Adjust the graphing window to identify all vertical asympotes. $$f(x)=\frac{2 x}{x+4}$$

Refer to the hyperbolic functions. Show that \(\cosh ^{2} x-\sinh ^{2} x=1\) for all \(x\)

Iterations of functions are important in a variety of applications. To iterate \(f(x)\), start with an initial value \(x_{0}\) and compute \(x_{1}=f\left(x_{0}\right), x_{2}=f\left(x_{1}\right), x_{3}=f\left(x_{2}\right)\) and so on. For example, with \(f(x)=\cos x\) and \(x_{0}=1\), the iterates are \(x_{1}=\cos 1 \approx 0.54, x_{2}=\cos x_{1} \approx \cos 0.54 \approx 0.86\) \(x_{3} \approx \cos 0.86 \approx 0.65\) and so on. Keep computing iterates and show that they get closer and closer to \(0.739085 .\) Then pick your own \(x_{0}\) (any number you like) and show that the iterates with this new \(x_{0}\) also converge to 0.739085

Find all vertical asymptotes. $$f(x)=\frac{4 x}{x^{2}+4}$$
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