Domanda del libro di testo
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)
23
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 7.8.4g
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
Guida verificata passo dopo passo1
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.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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.
Video consigliato:
Percorso guidato
04:16
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.
Video consigliato:
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.
Video consigliato:
Percorso guidato
5:50Asymptotes of Hyperbolas
Pratica correlata
Domanda del libro di testo
9. True, or false? As x→∞,
g. ln(x) = o(ln(2x))
32
views
Domanda del libro di testo
82. Use the definitions of the hyperbolic functions to find each of the following limits.
h. lim(x→0^-) coth x
33
views
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
27
views
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)
21
views
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))
19
views