Textbook Question
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
g. x^3 e^(-x)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.8.4g
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
g. (1.1)^x
Verified step by step guidance1
Recall that to compare the growth rates of functions as \(x \to \infty\), we analyze their behavior by considering limits or known growth hierarchies: polynomial, exponential, logarithmic, etc.
The function given is \(g(x) = (1.1)^x\), which is an exponential function with base greater than 1.
Since \(x^2\) is a polynomial function and \((1.1)^x\) is an exponential function, exponential functions grow faster than any polynomial function as \(x \to \infty\).
Therefore, \(g(x) = (1.1)^x\) grows faster than \(x^2\) as \(x \to \infty\).
To summarize: \((1.1)^x\) grows faster than \(x^2\), so it does not grow at the same rate or slower.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Growth Rates of Functions
Growth rates describe how functions behave as the input becomes very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace. For example, polynomial functions like x² grow slower than exponential functions like (1.1)^x as x approaches infinity.
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Intro To Related Rates
Polynomial vs. Exponential Functions
Polynomial functions are expressions involving powers of x, such as x², and grow at a rate proportional to a power of x. Exponential functions, like (1.1)^x, grow by repeatedly multiplying by a constant base, leading to much faster growth than any polynomial as x becomes large.
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6:13Exponential Functions
Limits and Asymptotic Behavior
Limits describe the behavior of functions as the input approaches infinity. Analyzing limits helps compare growth by examining the ratio of two functions as x→∞. If the limit of their ratio is zero, one grows slower; if infinite, it grows faster; if finite and nonzero, they grow at the same rate.
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5:50Asymptotes of Hyperbolas
Related Practice
Textbook Question
9. True, or false? As x→∞,
g. ln(x) = o(ln(2x))
1
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Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
h. lim(x→0^-) coth x
1
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Textbook Question
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
g. ln(ln x)
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Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ sin(2x) cos(3x) dx
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
i. lim(x→-∞) csch x