Textbook Question
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ ∞ (4x³ - 2x² + 6) / (πx³ + 4)
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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 4, Problem 29
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = 3x/(x² - 1)
Verified step by step guidance1
Identify the domain of the function f(x) = \( \frac{3x}{x^2 - 1} \). The denominator x² - 1 cannot be zero, so solve x² - 1 = 0 to find the values of x that are not in the domain.
Determine the vertical asymptotes by setting the denominator equal to zero and solving for x. This will give you the x-values where the function is undefined.
Find the horizontal asymptote by analyzing the behavior of f(x) as x approaches infinity or negative infinity. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
Calculate the first derivative f'(x) to find the critical points and determine intervals of increase and decrease. Use the quotient rule: \( f'(x) = \frac{(x^2 - 1)(3) - (3x)(2x)}{(x^2 - 1)^2} \). Simplify and solve f'(x) = 0 to find critical points.
Calculate the second derivative f''(x) to determine concavity and points of inflection. Use the quotient rule again on f'(x) and solve f''(x) = 0 to find potential inflection points. Analyze the sign of f''(x) to determine concavity intervals.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Behavior
Understanding the behavior of the function involves analyzing its domain, range, and asymptotic behavior. For the function f(x) = 3x/(x² - 1), we need to identify where the function is defined and where it may have vertical asymptotes, which occur where the denominator equals zero.
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Graphs of Exponential Functions
Finding Asymptotes
Asymptotes are lines that the graph approaches but never touches. For rational functions like f(x) = 3x/(x² - 1), vertical asymptotes can be found by setting the denominator to zero, while horizontal asymptotes can be determined by analyzing the degrees of the numerator and denominator as x approaches infinity.
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05:21Finding Limits by Direct Substitution
Critical Points and Intervals
Critical points are where the derivative of the function is zero or undefined, indicating potential local maxima, minima, or points of inflection. Analyzing these points, along with the intervals of increase and decrease, helps in sketching the overall shape of the graph and understanding its key features.
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04:50Critical Points
Related Practice
Textbook Question
Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = (x⁴/2) - 3x² + 4x + 1
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Textbook Question
Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 3x/(x² + 3)
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Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x²/(x - 2)
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Textbook Question
Demand and elasticity The economic advisor of a large tire store proposes the demand function D(p) = 1800/p-40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p.
c. Find the elasticity function on the domain of the demand function.
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Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = (x2 + 12)/(2x + 1)
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