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Chapter 4: Problem 65

Without drawing a graph, describe the behavior of the basic tangent curve.

Short Answer

Expert verified
The basic tangent function has a domain of all real numbers excluding \(x = \frac{(2n+1)\pi}{2}\), a range of all real numbers, and a periodicity of \( \pi \). It is odd symmetric and its curve crosses the x-axis at \(x = n\pi\) and has vertical asymptotes at \(x = \frac{(2n+1)\pi}{2}\).

Step by step solution

01

Understanding the Domain of the Tangent function

To describe the behavior of a curve, start by defining its domain. For the tangent function, the set of all real numbers excluding the Values where the function is undefined can serve as the domain. The tangent function is undefined at values \(x = \frac{(2n+1)\pi}{2}\) where \(n\) is an integer. These correspond to points where the cosine function equals zero.
02

Establishing the Range of the Tangent function

In this step, proceed to define the range of the function. Unlike the domain, the range of the tangent function is all real numbers. That is, for any value in the range, a corresponding value in the domain exists.
03

Defining the Periodicity and Symmetry

The function tangent has a periodicity of \( \pi \), indicating that it repeats its values every \( \pi \) intervals. Additionally, it is odd symmetric - this means that for every point (x, y) on the tangent graph, the point (-x, -y) is also on the graph.
04

Evaluate the Behavior at Special Points

Lastly, study the behavior of the tangent curve at its special points. At \(x = \frac{(2n+1)\pi}{2}\) (where the function is undefined), the function approaches positive or negative infinity. This indicates that the function has vertical asymptotes at these points. Also, the function crosses the x-axis at \(x = n\pi\), where n is an integer.

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Most popular questions from this chapter

The following figure shows the depth of water at the end of a boat dock. The depth is 6 feet at low tide and 12 feet at high tide. On a certain day, low tide occurs at 6 A.M. and high tide at noon. If \(y\) represents the depth of the water \(x\) hours after midnight, use a cosine function of the form \(y=A \cos B x+D\) to model the water's depth.

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=4 \cos \left(2 x-\frac{\pi}{6}\right) \text { and } y=4 \sec \left(2 x-\frac{\pi}{6}\right)$$

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 a decimal in degrees. Round your answer to two decimal places. $$30^{\circ} 15^{\prime} 10^{\prime \prime}$$

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Solve: \(x^{2}+4 x+6=0\) (Section \(2.1,\) Example 5 )
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