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Chapter 5: Problem 5

Use the graph of the function to answer each question. (a) Find any \(x\) -intercepts of the graph of \(y=f(x)\). (b) Find any \(y\) -intercepts of the graph of \(y=f(x)\). (c) Find the intervals on which the graph of \(y=f(x)\) is increasing and the intervals on which the graph of \(y=f(x)\) is decreasing. (d) Find all relative extrema, if any, of the graph of \(y=f(x)\) (e) Find all vertical asymptotes, if any, of the graph of \(y=f(x)\) \(f(x)=\tan x\)

Short Answer

Expert verified
The x-intercepts are \(n\pi\), y-intercept is 0, it increases on \(((2n-1)\pi/2, (2n+1)\pi/2)\) and decreases on \(((2n+1)\pi/2, (2n+3)\pi/2)\), has no relative extrema, and vertical asymptotes are \(x=(2n+1)\pi/2\)

Step by step solution

01

Find the x-intercepts of the graph.

The x-intercepts are the values of x where \(y=0\). The equation \(\tan x = 0\) has solutions \(x=n\pi\), where \(n\) is an integer. So, the x-intercepts are \(n\pi\).
02

Find the y-intercepts of the graph.

The y-intercepts are the values of y when \(x=0\). Substituting \(x=0\) into \(y=\tan x\) gives \(y=0\). Therefore, the y-intercept is 0.
03

Find the intervals on which the graph is increasing and decreasing.

The graph of the function is increasing on the intervals \(((2n-1)\pi/2, (2n+1)\pi/2)\) and decreasing on the intervals \(((2n+1)\pi/2, (2n+3)\pi/2)\), where \(n\) is an integer. This is because the slope of the tangent function is always positive between its vertical asymptotes and is always negative immediately after a vertical asymptote till the next.
04

Find all relative extrema

The function \(y=\tan x\) has no relative extrema as it is always increasing or decreasing within a period.
05

Find all vertical asymptotes

A vertical asymptote exists where the function is undefined. The function \(y=\tan x\) is undefined at \(x=(2n+1)\pi/2\), where \(n\) is an integer. Therefore, the vertical asymptotes are at \(x=(2n+1)\pi/2\).

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Key Concepts

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

X-Intercepts
The x-intercepts of a graph are points where the curve touches or crosses the x-axis. This means the function's output, or y value, is zero at these points. For the tangent function, \( f(x) = \tan x \), we look for values of x where the tangent of x is zero. Mathematically, this is represented by the equation \( \tan x = 0 \), which is true whenever x is an integer multiple of \( \pi \), denoted as \( x = n\pi \) where \( n \) is an integer. This characterizes the x-intercepts of the tangent function to occur at regular intervals along the x-axis separated by \( \pi \) units.
Y-Intercepts
The y-intercept of a graph is where the curve intersects the y-axis. It provides a starting point for understanding the function's behavior. To find it, we set x to zero and solve for y. For \( y = \tan x \), when \( x = 0 \) we have \( y = \tan(0) = 0 \). Interestingly, the tangent function has a single y-intercept at the origin, (0, 0). This is because the tangent function is symmetric about the origin due to its periodic nature.
Intervals of Increase and Decrease
Understanding the intervals of increase and decrease of a function can help predict its behavior. The tangent function \( y = \tan x \) increases between each pair of its vertical asymptotes without any local maxima or minima. Specifically, it is increasing in the intervals \( ((2n-1)\pi/2, (2n+1)\pi/2) \) and decreasing in the intervals \( ((2n+1)\pi/2, (2n+3)\pi/2) \) where \( n \) is an integer. These intervals are crucial for understanding the pattern of the tangent function's growth and decline as it repeats every \( \pi \) radians.
Relative Extrema
The concept of relative extrema addresses the local high and low points of a function's graph. For the function \( f(x) = \tan x \), however, this concept doesn't apply in the conventional sense. Since the tangent function is perpetual in its increase or decrease within its period, it has no relative maximum or minimum points. The absence of relative extrema in such trigonometric functions can be a point of interest in understanding their unique characteristics compared to polynomial or rational functions.
Vertical Asymptotes
A vertical asymptote is a vertical line that a function's graph approaches but never touches or crosses. These occur at points where the function is undefined and its limits approach infinity. For the tangent function \( f(x) = \tan x \), vertical asymptotes occur at \( x = (2n+1)\pi/2 \), for every integer \( n \). These asymptotes define the boundaries between intervals in which the function is increasing or decreasing. They are quintessential in analyzing the function's domain and continuity and are an integral part of the tangent function's graph.

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Most popular questions from this chapter

Irrigation Engineering The cross sections of an irrigation canal are isosceles trapezoids, where the lengths of three of the sides are 8 feet (see figure). The objective is to find the angle \(\theta\) that maximizes the area of the cross sections. [Hint: The area of a trapezoid is given by \(\left.(h / 2)\left(b_{1}+b_{2}\right) .\right]\) (a) Complete seven rows of the table. $$\begin{array}{|c|c|c|c|} \hline \text {Base } I & \text {Base 2} & \text {Altitude} & \text {Area} \\ \hline 8 & 8+16 \cos 10^{\circ} & 8 \sin 10^{\circ} & 22.06 \\ \hline 8 & 8+16 \cos 20^{\circ} & 8 \sin 20^{\circ} & 42.46 \\ \hline \end{array}$$ (b) Use the table feature of a graphing utility to generate additional rows of the table. Use the table to estimate the maximum cross-sectional area. (c) Write the area \(A\) as a function of \(\theta.\) (d) Use the graphing utility to graph the function. Use the graph to estimate the maximum cross-sectional area. How does your estimate compare with that in part (b)?

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