/*! 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 35 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 35

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{5-3 x}{x-2}\)

Short Answer

Expert verified
The function \(y=\frac{5-3 x}{x-2}\) has an x-intercept at \(x=\frac{5}{3}\), and a y-intercept at x = 0. It has a vertical asymptote at x = 2 and a horizontal asymptote at y = -3, with no relative extrema or inflection points. The domain is \(x 鈭� R, x 鈮� 2\).

Step by step solution

01

Find the intercepts

The x-intercept is found by setting y = 0 and solving for x. Setting \(\frac{5-3x}{x-2}=0\) gives x = \(\frac{5}{3}\). The y-intercept is found by setting x = 0, but in this case there is a vertical asymptote at x = 0, so the function does not intercept the y-axis.
02

Determine the extrema

The extrema of the function are found by taking the derivative of the function and setting it equal to zero. The derivative of \(y=\frac{5-3 x}{x-2}\) is \(-\frac{1}{(x-2)^2}\). Setting this equal to 0 does not yield any real values, so there are no relative extrema.
03

Find the points of inflection

The function's second derivative is \(+\frac{2}{(x-2)^3}\) where it does not equal 2 (which is a point of discontinuity). But setting this equal to zero also does not yield any real values, implying that there are no points of inflection.
04

Find the asymptotes

A vertical asymptote exists where the function is undefined. As \((x-2) 鈮� 0\), x = 2 is a vertical asymptote. A horizontal asymptote is found by taking the limit of the function as x approaches positive and negative infinity. Thus, y = -3 is a horizontal asymptote.
05

Define the domain of the function

The domain of a function is the set of all potential x values. The function is undefined at x = 2, therefore the domain of the function is \(x 鈭� R, x 鈮� 2\).

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!

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

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{3}+3 x^{2}+3 x+2\)

The side of a square is measured to be 12 inches, with a possible error of \(\frac{1}{64}\) inch. Use differentials to approximate the possible error and the relative error in computing the area of the square.

The cost and revenue functions for a product are \(C=25.5 x+1000\) and \(R=75.5 x\) (a) Find the average profit function \(\bar{P}=\frac{R-C}{x}\). (b) Find the average profits when \(x\) is 100,500 , and 1000 . (c) What is the limit of the average profit function as \(x\) approaches infinity? Explain your reasoning.

The cost \(C\) (in dollars) of producing \(x\) units of a product is \(C=1.35 x+4570\) (a) Find the average cost function \(\bar{C}\). (b) Find \(\bar{C}\) when \(x=100\) and when \(x=1000\). (c) What is the limit of \(\bar{C}\) as \(x\) approaches infinity?

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-x^{3}+x-2\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.