/*! 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 57 Find an equation of the slant as... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 57

Find an equation of the slant asymptote. Do not sketch the curve. \(y=\frac{x^{2}+1}{x+1}\)

Short Answer

Expert verified
The equation of the slant asymptote is \( y = x - 1 \).

Step by step solution

01

Set Up Long Division for Polynomials

To find the slant asymptote, you should perform polynomial long division on the function \( y = \frac{x^2 + 1}{x + 1} \). Start by setting up long division where \( x^2 + 1 \) is the dividend and \( x + 1 \) is the divisor.
02

Perform the Division

Divide \( x^2 \) by \( x \) to get \( x \). Multiply \( x \) by \( x + 1 \) to get \( x^2 + x \). Subtract \( x^2 + x \) from \( x^2 + 1 \) to get a remainder of \( 1 - x \).
03

Continue Division Process

Divide \( -x \) by \( x \) to get \( -1 \). Multiply \( -1 \) by \( x + 1 \) to get \( -x - 1 \). Subtract \( -x - 1 \) from \( 1 - x \) to get a remainder of \( 2 \).
04

Write the Division Result

The division yields \( x - 1 + \frac{2}{x + 1} \). The quotient \( x - 1 \) represents the equation of the slant asymptote, as the remainder becomes negligible for very large values of \( x \).
05

Conclusion

The slant asymptote is the linear part of the quotient from the division, which is \( y = x - 1 \).

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.

Polynomial Long Division
Polynomial long division is a method used to divide one polynomial by another. It is very similar to the long division algorithm used for numbers.
The whole purpose is to simplify expressions where a polynomial (the dividend) is divided by another polynomial (the divisor).
This method helps find both the quotient and the remainder of the division.
  • Start by dividing the first term of the dividend by the first term of the divisor.
  • Multiply the entire divisor by this newly found term, writing the result beneath the dividend.
  • Subtract this result from the dividend to find the new dividend.
  • Repeat these steps using the new dividend until no smaller degree terms can be divided.
Polynomial long division is particularly useful when dealing with rational functions, especially when determining asymptotic behavior like slant asymptotes.
Rational Functions
Rational functions are quotients of two polynomials, where the numerator and the denominator are both polynomials, and the denominator is non-zero.
They can be expressed in the form:\[y = \frac{P(x)}{Q(x)}\]where \(P(x)\) and \(Q(x)\) are polynomials.
Rational functions are key in many applications like finding asymptotes and analyzing graphs. Their graphs can exhibit complex behavior, including multiple horizontal, vertical, or slant asymptotes.
  • If the degree of the numerator is greater than that of the denominator, it might indicate the presence of a slant asymptote.
  • The simplification and division of these functions can reveal critical asymptotic characteristics.
Asymptotic Behavior
Asymptotic behavior describes how a function behaves as the input grows very large or very small. It's like predicting the path a function travels in extreme conditions.
When it comes to rational functions, you are often interested in one of the following:
  • Horizontal asymptotes occur when the degrees of the numerator and denominator are equal.
  • Vertical asymptotes occur when the denominator is zero, and the function can't be simplified.
  • Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator.
Slant asymptotes give us a linear path that the graph of the function approaches as the input becomes very large (both positively and negatively). In the given example, after using polynomial long division, the slant asymptote is described by the quotient.
Long Division of Polynomials
Long division of polynomials is vital for simplifying and understanding rational expressions. It can be used to divide complex expressions into simpler parts.
Imagine you have the polynomial:\[y = \frac{x^2 + 1}{x + 1}\]where you need to perform long division to find out how this fraction behaves.
  • It begins with aligning and preparing the polynomials similar to number long division.
  • Successive division reveals a more straightforward division of the function, yielding both a quotient (a polynomial) and a remainder.
  • The quotient here becomes very important; when the rational function's degree suggests a slant asymptote, the quotient supplies this linear asymptote.
In this exercise, division identifies the quotient \(x - 1\), pointing to the slant asymptote. The remainder becomes negligible as \(x\) becomes large.

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

A manufacturer has been selling 1000 flat-screen TVs a week at \(\$ 450\) each. A market survey indicates that for each \(\$ 10\) rebate offered to the buyer, the number of TVs sold will increase by 100 per week. (a) Find the demand function. (b) How large a rebate should the company offer the buyer in order to maximize its revenue? (c) If its weekly cost function is \(C ( x ) = 68,000 + 150 x\) , how should the manufacturer set the size of the rebate in order to maximize its profit?

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x)=x^{3}+c x\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f "\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x)=x^{6}-15 x^{5}+75 x^{4}-125 x^{3}-x\)

An oil refinery is located on the north bank of a straight river that is 2\(\mathrm { km }\) wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6\(\mathrm { km }\) east of the refinery. The cost of laying pipe is \(\$ 400,000 / \mathrm { km }\) over land to a point \(P\) on the north bank and \(\$ 800,000 / \mathrm { km }\) under the river to the tanks. To minimize the cost of the pipeline, where should \(P\) be located?

(a) Show that if the profit \(P ( x )\) is a maximum, then the mar- ginal revenue equals the marginal cost. (b) If \(C ( x ) = 16,000 + 500 x - 1.6 x ^ { 2 } + 0.004 x ^ { 3 }\) is the cost function and \(p ( x ) = 1700 - 7 x\) is the demand function, find the production level that will maximize profit.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied 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.