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Chapter 4: Problem 20

Graph two periods of the given cotangent function. $$y=2 \cot 2 x$$

Short Answer

Expert verified
The graph of y=2cot(2x) consists of curves shooting up towards positive infinity at x=0, approaching zero at x=\(\frac{\pi}{4}\), descending towards negative infinity at x=\(\frac{\pi}{2}\), and repeating this pattern for the second period at x=\(\pi\) and x=\(\frac{3\pi}{2}\).

Step by step solution

01

Identify the Period and Interval

The period of the function cot(x) is \(\pi\), but here we have a frequency of 2 which affects the period. Hence, the period of the function y = 2cot(2x) is \(\frac{\pi}{2}\). So, two periods of the function are covered in the interval \([0, 2\pi]\).
02

Plot Key Points

Two periods of the function's cycle occurs every half-period, where the function changes from negative infinity to positive infinity, and back again. In one period of cot(x), the function goes from positive infinity at 0, crosses through 0 at \(\frac{\pi}{2}\), and then goes to negative infinity at \(\pi\). Given our period of \(\frac{\pi}{2}\), these events will happen at 0, \(\frac{\pi}{4}\), and \(\frac{\pi}{2}\) respectively, in just half of our period. The amplitude of the cotangent function is 2, which means it reaches a height of 2 (though the range is all real numbers for the cotangent function).
03

Sketch the Graph

At x=0, the graph should start from positive infinity. It should approach zero without actually reaching it at x=\(\frac{\pi}{4}\). After this point, it should decrease towards negative infinity by x=\(\frac{\pi}{2}\). This pattern will repeat for the second half of the period, starting again from positive infinity at x=\(\pi\) and going to negative infinity by x=\(\frac{3\pi}{2}\). By connecting the points smoothly, the graph of two periods of the function y=2cot(2x) can be drawn.

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Most popular questions from this chapter

Graph one period of each function. $$y=\left|2 \cos \frac{x}{2}\right|$$

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\cot 2 x$$

In Chapter \(5,\) we will prove the following identities: $$ \begin{aligned} \sin ^{2} x &=\frac{1}{2}-\frac{1}{2} \cos 2 x \\ \cos ^{2} x &=\frac{1}{2}+\frac{1}{2} \cos 2 x \end{aligned} $$ Use these identities to solve. Use the identity for \(\cos ^{2} x\) to graph one period of \(y=\cos ^{2} x\)

Use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. Convert each angle to \(D^{\circ} M^{\prime} S^{\prime \prime}\) form. Round your answer to the nearest second. $$30.42^{\circ}$$

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of h by adding or subtracting the corresponding \(y\) -coordinates on the graphs of \(f\) and \(g\) $$f(x)=-2 \sin x, g(x)=\sin 2 x, h(x)=(f+g)(x)$$
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