Domanda del libro di testo
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
e. x^3 - x^2
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.8.3e
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
e. x ln(x)
Guida verificata passo dopo passo1
Recall that to compare growth rates of functions as \(x \to \infty\), we often use limits of their ratios. Specifically, for two functions \(f(x)\) and \(g(x)\), we consider \(\lim_{x \to \infty} \frac{f(x)}{g(x)}\).
Here, we want to compare \(f(x) = x \ln(x)\) with \(g(x) = x^{2}\). So, set up the limit: \(\lim_{x \to \infty} \frac{x \ln(x)}{x^{2}}\).
Simplify the expression inside the limit: \(\frac{x \ln(x)}{x^{2}} = \frac{\ln(x)}{x}\). Now, analyze the behavior of \(\frac{\ln(x)}{x}\) as \(x \to \infty\).
Since \(\ln(x)\) grows slower than any positive power of \(x\), the limit \(\lim_{x \to \infty} \frac{\ln(x)}{x} = 0\). This means \(x \ln(x)\) grows slower than \(x^{2}\) as \(x \to \infty\).
Therefore, \(x \ln(x)\) grows slower than \(x^{2}\) as \(x\) approaches infinity.
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Asymptotic Growth Rates
Asymptotic growth rates describe how functions behave as the input grows very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same rate as a reference function, such as x², by analyzing their dominant terms for large x.
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Intro To Related Rates
Big-O and Big-Theta Notation
Big-O notation classifies functions by their upper bound growth rate, while Big-Theta notation describes functions that grow at the same rate asymptotically. These notations help compare functions like x ln(x) and x² by formalizing their relative growth as x approaches infinity.
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Behavior of Logarithmic and Polynomial Functions
Logarithmic functions like ln(x) grow slower than any positive power of x, while polynomial functions grow at rates determined by their degree. Understanding that x ln(x) grows slower than x² because ln(x) grows slower than x is key to comparing their growth rates.
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5:26Graphs of Logarithmic Functions
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