Domanda del libro di testo
1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
f. (e^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 7.8.3g
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)
Guida verificata passo dopo passo1
Identify the function given: \(g(x) = x^{3} e^{-x}\).
Recall that as \(x \to \infty\), the exponential term \(e^{-x}\) approaches 0 very rapidly, while the polynomial term \(x^{3}\) grows without bound but at a much slower rate compared to exponentials.
Analyze the behavior of \(g(x)\) by considering the dominant terms: since \(e^{-x}\) decays faster than any polynomial grows, the product \(x^{3} e^{-x}\) tends to 0 as \(x \to \infty\).
Compare this behavior to \(x^{2}\): since \(g(x)\) tends to 0 and \(x^{2}\) tends to infinity, \(g(x)\) grows slower than \(x^{2}\) as \(x \to \infty\).
Conclude that \(g(x)\) 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 dominant terms and limits at infinity.
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04:16
Intro To Related Rates
Exponential Decay vs Polynomial Growth
Exponential decay functions like e^(-x) decrease rapidly to zero as x approaches infinity, often overpowering polynomial growth terms. When combined, the exponential decay can cause the entire function to approach zero, affecting the overall growth rate compared to pure polynomials.
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09:29Exponential Growth & Decay
Limit Comparison for Growth Rates
To compare growth rates, we use limits of the ratio of two functions as x approaches infinity. If the limit is zero, the numerator grows slower; if infinite, it grows faster; if finite and nonzero, they grow at the same rate. This method rigorously classifies growth behavior.
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Percorso guidato
04:16Intro To Related Rates
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Domanda del libro di testo
2. Express the following logarithms in terms of ln 5 and ln 7.
f. (ln35 + ln(1/7))/(ln25)
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Domanda del libro di testo
5. 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. 1/x
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Domanda del libro di testo
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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Domanda del libro di testo
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
g. e^(cos(x))
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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?
g. (1.1)^x
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