/*! 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 105 Use the fact that \(\sec x=\frac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 105

Use the fact that \(\sec x=\frac{1}{\cos x}\) to explain why the maximum domain of \(y=\sec x\) consists of all real numbers except odd integer multiples of \(\pi / 2\).

Short Answer

Expert verified
The domain of \( y = \sec x \) is all real numbers except odd multiples of \( \frac{\pi}{2} \).

Step by step solution

01

Understand the reciprocal relationship

The function \( y = \sec x \) is defined as the reciprocal of the cosine function, \( \sec x = \frac{1}{\cos x} \). This implies that wherever \( \cos x \) is zero, \( \sec x \) will be undefined.
02

Identify where \(\cos x\) is zero

The cosine function, \( \cos x \), equals zero at odd multiples of \( \frac{\pi}{2} \). This can be written as \( x = \frac{(2k+1)\pi}{2} \), where \( k \) is any integer (e.g., \( k = 0, \pm1, \pm2, \ldots \)).
03

Determine the domain of \(\sec x\)

Since \( \sec x \) is the reciprocal of \( \cos x \), \( \sec x \) will be undefined wherever \( \cos x = 0 \). Therefore, \( \sec x \) is undefined at odd multiples of \( \frac{\pi}{2} \), and its domain is all real numbers except at these points.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Secant Function
The secant function, denoted as \( \sec x \), is a fundamental trigonometric function. It is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This relationship means that wherever the cosine function is defined, the secant function can also be defined, except where the cosine equals zero because division by zero is undefined.
For example, if \( \cos x = 0.5 \), then \( \sec x = \frac{1}{0.5} = 2 \). However, the secant function isn't always defined. It's specifically undefined when \( \cos x = 0 \). Understanding this relationship is crucial, as it forms the basis for determining the domain of the secant function, which we will explore further.
Cosine Function
The cosine function, \( \cos x \), is a periodic function with a cycle of \( 2\pi \), meaning it repeats its values every \(2\pi\) radians. It is defined for all real numbers, and its values range from -1 to 1. Importantly, the cosine function reaches zero at specific points. These critical points occur at odd multiples of \( \frac{\pi}{2} \), such as \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \) or generally described by the expression \( x = \frac{(2k+1)\pi}{2} \), where \( k \) is any integer.
The significance of these points is that wherever \( \cos x \) equals zero, the reciprocal or secant function is undefined. These zeros of the cosine function create the critical boundaries for the domain of the secant function.
Domain of a Function
The domain of a function refers to all possible input values (or \( x \)-values) for which the function is defined. In simpler terms, it represents all the values you can plug into a function without causing an undefined expression. For the secant function, \( y = \sec x \), the domain consists of all real numbers except where the cosine function is zero.
Since \( \sec x = \frac{1}{\cos x} \), if \( \cos x = 0 \), then \( \sec x \) is undefined, as division by zero is not allowed. Therefore, for the secant function, the domain includes all real numbers except at \( x = \frac{(2k+1)\pi}{2} \), where \( k \) is any integer.
  • The secant function has vertical asymptotes (lines where the function shoots up to infinity) at these excluded x-values.
  • Every excluded point corresponds to a zero of the cosine function.
Understanding the domain helps graph the function correctly and anticipate its behavior across the x-axis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship in a coordinate system whose axes are accordingly transformed so that a straight line results. $$ y=2^{-x} ; \text { base } 2 $$

Brown lemming densities in the tundra areas of North America and Eurasia show cyclic behavior: Every three to four years, lemming densities build up very rapidly, and they typically crash the next year. Sketch a graph that describes this situation.

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=4 \times 10^{5 x} $$

Continuation of Problem 86) Estimating \(v_{\max }\) and \(K_{m}\) from the Lineweaver-Burk graph as described in Problem 86 is not always satisfactory. A different transformation typically yields better estimates (Dowd and Riggs, 1965 ). Show that the Michaelis-Menten equation can be written as $$ \frac{v_{0}}{s_{0}}=\frac{v_{\max }}{K_{m}}-\frac{1}{K_{m}} v_{0} $$ and explain why this transformation results in a straight line when you graph \(v_{0}\) on the horizontal axis and \(\frac{v_{0}}{s_{0}}\) on the vertical axis. Explain how you can estimate \(v_{\max }\) and \(K_{m}\) from the graph.

use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $$ y=5 \times 2^{4 x} $$
See all solutions

Recommended explanations on Biology Textbooks

Biological Organisms

Read Explanation

Ecosystems

Read Explanation

Responding to Change

Read Explanation

Energy Transfers

Read Explanation

Substance Exchange

Read Explanation

Biology Experiments

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.