Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 55
Capitolo 4, Problema 55

Sketch the graphs of the rational functions in Exercises 53–60.
y = (x2 + 1) / x                              

Guida verificata passo dopo passo
1
Identify the rational function: \( y = \frac{x^2 + 1}{x} \). This function is a rational function because it is the ratio of two polynomials.
Determine the domain of the function. The denominator \( x \) cannot be zero, so the domain is all real numbers except \( x = 0 \).
Find the vertical asymptote by setting the denominator equal to zero. Since \( x = 0 \) makes the denominator zero, there is a vertical asymptote at \( x = 0 \).
Simplify the function if possible. Here, \( y = \frac{x^2 + 1}{x} = x + \frac{1}{x} \). This helps in understanding the behavior of the function as \( x \to \infty \) and \( x \to -\infty \).
Analyze the behavior of the function as \( x \to \infty \) and \( x \to -\infty \). As \( x \to \infty \), \( y \approx x \), and as \( x \to -\infty \), \( y \approx x \). This suggests that the graph will resemble the line \( y = x \) for large values of \( |x| \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
10m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Rational Functions

A rational function is a ratio of two polynomials, where the numerator and denominator are polynomial expressions. Understanding the behavior of rational functions involves analyzing their asymptotes, intercepts, and overall shape. In this case, the function y = (x^2 + 1) / x is a rational function with a polynomial numerator and a linear denominator.
Video consigliato:
6:04
Intro to Rational Functions

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero, causing the function to approach infinity. For y = (x^2 + 1) / x, the vertical asymptote is at x = 0, since the denominator becomes zero at this point. This asymptote divides the graph into separate regions, influencing the function's behavior near x = 0.
Video consigliato:
Percorso guidato
3:40
Introduction to Cotangent Graph Example 1

End Behavior

End behavior describes how a function behaves as x approaches positive or negative infinity. For rational functions, this is often determined by the degrees of the numerator and denominator. In y = (x^2 + 1) / x, as x approaches infinity, the function behaves like y = x, since the highest degree term in the numerator dominates, indicating a linear growth.
Video consigliato:
03:07
Cases Where Limits Do Not Exist
Pratica correlata
Domanda del libro di testo

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

56. y = x² + 2/x

201
views
Domanda del libro di testo

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

60. y = 5 / (x⁴ + 5)

111
views
Domanda del libro di testo

In Exercises 49–52, graph each function. Then use the function’s first derivative to explain what you see.

y = 𝓍²/³ + (𝓍―1)²/³ 

207
views
Domanda del libro di testo

Sketch the graphs of the rational functions in Exercises 53–60.

y= (x + 1) / (x - 3)

228
views
Domanda del libro di testo

Applications


Classical accounts tell us that a 170-oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme’s speed exceeded 7.5 knots (sea or nautical miles per hour).

168
views
Domanda del libro di testo

The sum of two nonnegative numbers is 36. Find the numbers if

a. the difference of their square roots is to be as large as possible.

b. the sum of their square roots is to be as large as possible.

171
views