Textbook Question
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.
392
views
Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 2, Problem 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.
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
5:53Graph 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.
Recommended video:
04:57Determining Error and Relative Error
Related Practice
Textbook Question
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
Textbook Question
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
Textbook Question
The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>
lim x→−2 h(x)
208
views
Textbook Question
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.
234
views
Textbook Question
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)?
361
views