Skip to main content
Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 2.5.55c
Capitolo 2, Problema 2.5.55c

Complete the following steps for the given functions. 


c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

Guida verificata passo dopo passo
1
Identify the function: \( f(x) = \frac{4x^3 + 4x^2 + 7x + 4}{x^2 + 1} \). This is a rational function where the degree of the numerator is higher than the degree of the denominator.
Determine the vertical asymptotes by setting the denominator equal to zero: \( x^2 + 1 = 0 \). Since this equation has no real solutions, there are no vertical asymptotes.
Find the horizontal or oblique asymptote. Since the degree of the numerator (3) is greater than the degree of the denominator (2), there is no horizontal asymptote. Instead, perform polynomial long division to find the oblique asymptote.
Perform polynomial long division of \( 4x^3 + 4x^2 + 7x + 4 \) by \( x^2 + 1 \) to find the quotient, which represents the oblique asymptote.
Use a graphing utility to plot the function \( f(x) \) and the oblique asymptote. Then, sketch the graph by hand, ensuring to correct any discrepancies observed in the computer-generated graph.

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:
3m

Concetti chiave

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

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They can be vertical, horizontal, or oblique. Vertical asymptotes occur where the function is undefined, typically at values that make the denominator zero. Horizontal asymptotes indicate the behavior of the function as x approaches infinity, showing the end behavior of the graph.
Video consigliato:
Percorso guidato
5:37
Introduction to Cotangent Graph

Graphing Rational Functions

Graphing rational functions involves plotting the function defined as the ratio of two polynomials. Key steps include identifying intercepts, asymptotes, and the behavior of the function at critical points. Understanding the degree of the numerator and denominator helps predict the end behavior and the presence of horizontal asymptotes.
Video consigliato:
Percorso guidato
5:53
Graph of Sine and Cosine Function

Error Correction in Graphing

Error correction in graphing involves comparing a computer-generated graph with a hand-drawn sketch to identify discrepancies. This process requires understanding the function's characteristics, such as asymptotes and intercepts, to ensure accuracy. By analyzing the graph's behavior at critical points, one can refine the sketch to better represent the function.
Video consigliato:
Percorso guidato
04:57
Determining Error and Relative Error
Pratica correlata
Domanda del libro di testo

Complete the following steps for the given functions. 


c. Graph ff and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.


f(x)=x2−3x+6f\(\left\)(x\(\right\))=\(\frac{x^2-3}{x+6}\)

358
views
Domanda del libro di testo

Determine whether the following statements are true and give an explanation or counterexample.


c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.

393
views
Domanda del libro di testo

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→2^− f(x)

253
views
Domanda del libro di testo

Complete the following steps for the given functions. 


c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.


f(x)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

328
views
Domanda del libro di testo

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→^3− h(x)

273
views
Domanda del libro di testo

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


d. For what values of t on the interval [0, 9] is the instantaneous velocity positive (the projectile moves upward)?

362
views