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Chapter 4: Problem 55

$$ y=\sin \frac{x}{2} \sin 2 x $$

Short Answer

Expert verified
The given function is \(y = \sin\frac{x}{2}\sin2x\), which is a product of sine functions with different arguments. The first part, \(\sin\frac{x}{2}\), stretches the sine function horizontally by a factor of \(2\) with a period of \(4\pi\) and amplitude \(1\). The second part, \(\sin2x\), compresses the sine function horizontally by a factor of \(2\) with a period of \(\pi\) and amplitude \(1\). Combining the two sine functions results in a more complex shape than a simple sine wave, but with periodic behavior.

Step by step solution

01

(Step 1: Identify the function type)

We are given a trigonometric function that involves a product of sine functions with different arguments, specifically, \(\sin\frac{x}{2}\) and \(\sin2x\). Trigonometric functions are used to model periodic phenomena and, in this case, we see a combination of sine functions.
02

(Step 2: Analyze the first part of the function)

Let's look at the first part of the function: \(\sin\frac{x}{2}\). The sine function is periodic with a period of \(2\pi\). When dividing the argument (\(x\)) by \(2\), we are stretching the sine function horizontally by a factor of \(2\). This means the period of this component will become \(4\pi\). The amplitude, or the maximum height of the sine function, will remain the same, which is \(1\).
03

(Step 3: Analyze the second part of the function)

Now, let's analyze the second part of the function: \(\sin2x\). In this case, the argument (\(x\)) is multiplied by \(2\), meaning that the sine function will be compressed horizontally by a factor of \(2\). This results in the period being reduced to \(\pi\). The amplitude remains the same, which is \(1\).
04

(Step 4: Combine the two trigonometric functions)

We have analyzed both parts of the function, and now we can combine them: \(y = \sin\frac{x}{2}\sin2x\). The final function is obtained by multiplying the two sine functions together. This means that the value of \(y\) will depend on the combination of the periodicities of each individual sine function. The final function will have a more complex shape than a simple sine wave, but still exhibit periodic behavior.

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Key Concepts

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

Sine Function
The sine function is one of the fundamental trigonometric functions, often noted as \(\sin x\). It oscillates smoothly, following the form of a wave, and plays a critical role in modeling periodic phenomena like sound waves or tides. This function repeats its values in a regular pattern and is known for its wave-like behavior.
  • Standard Range: The sine of an angle varies between \(-1\) and \(1\).
  • Wavelength: Called the period, represents when the function completes one full cycle.
  • Sine in components: The sine function is often modified (as in our example) by adjusting its argument, which changes its frequency, or through amplitude, which changes height.
When we modify the function like \(\sin \frac{x}{2}\), the essentials of sine's behavior stay, but the specifics like the wave's frequency adjust.
Periodic Functions
Periodic functions are those that repeat their values in regular intervals over the domain. In mathematics, they are critical when modeling anything with a repetitive pattern, such as cycles in nature or signals.
  • Nature: Periodic functions can smooth, sawtooth, or step in form; sine is smooth.
  • Repetition Interval: This is called the period, signifying the horizontal length of one complete cycle.
Our example function utilizes periodic nature, combining two sine functions \(\sin \frac{x}{2}\) and \(\sin 2x\), which individually repeat over different intervals but combined present rich symmetries and variations in periodicity.
Function Analysis
Function analysis involves examining the properties of functions to understand their behavior. It's a methodical approach that helps us break down functions into simpler components to study more comprehensively.
  • Steps: Identifying function type, behavior, and how operations like addition or multiplication affect it.
  • Graphical Understanding: Often analyzed through plotting on graphs to visualize changes.
In our process, analyzing \(\sin \frac{x}{2}\) and \(\sin 2x\) showed how modifying input values alters frequency but not amplitude, crucial for predicting final behavior in combination.
Amplitude
Amplitude refers to the peak value of a function, representing the maximum extent of its repeated oscillation.
  • For Sine: Standard sine (\(\sin x\)) has an amplitude of \(1\), representing the distance from middle to peak.
  • Modifications: Changes in amplitude show greater or lesser swings from the centerline of the graph.
In both \(\sin\frac{x}{2}\) and \(\sin2x\), the amplitude remains\(1\), meaning maximum heights remain the same, even when other properties like period change. This constancy in amplitude aids in the predictability of sine functions' behavior.
Period of a Function
The period of a function is the interval over which a function completes one full cycle before repeating.
  • For \(\sin x\): The classic period is \(2\pi\)
  • For Modifications: Dividing the angle by a number stretches the wave, increasing the period. Multiplying does the opposite, compressing it and decreasing the period.
In \(\sin \frac{x}{2}\), the period stretches to \(4\pi\) due to division by \(2\). In contrast, for \(\sin 2x\), it's compressed to \(\pi\), demonstrating how such manipulations affect the function's cyclical behavior.

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Most popular questions from this chapter

$$ y=x^{\ln x} $$

$$ \text { Given } \left.f(x)=\sqrt{x} \text { , find } f^{\prime}(0) \& f^{\prime}(1) \text { by first principles. \\{Ans. does not exist, } \frac{1}{2}\right\\} $$

$$ \left.y=\cos ^{-1} \frac{x^{2 n}-1}{x^{2 n}+1} \text { \\{ Ans. }-\frac{2 n x^{n-1}}{x^{2 n}+1} \text { if } n \text { is even and }-\frac{2 n x^{n}}{|x|\left(x^{2 n}+1\right)}, \text { if } n \text { is odd. }\right\\} $$

$$ \text { Given } \left.f(x)=e^{x} \text { , find } f^{\prime}(0), f^{\prime}(1) \& f^{\prime}(-1) \text { by first principles.

$$ \text { Given } f(x)=x^{a} \text { , show that } f^{\prime}(0) \text { does not exist if } 0
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