Domanda del libro di testo
10. True, or false? As x→∞,
c. 1/x - 1/x² = o(1/x)
25
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 7.8.1.c
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?
c. √x
Guida verificata passo dopo passo1
Recall the growth rates of common functions as \(x \to \infty\). The exponential function \(e^x\) grows faster than any polynomial or root function.
Identify the function given: \(\sqrt{x}\), which is equivalent to \(x^{1/2}\), a root function and a type of power function.
Compare the growth of \(\sqrt{x}\) to \(e^x\). Since \(e^x\) grows exponentially and \(\sqrt{x}\) grows polynomially, \(e^x\) grows faster than \(\sqrt{x}\) as \(x \to \infty\).
Conclude that \(\sqrt{x}\) grows slower than \(e^x\) as \(x \to \infty\).
Summarize: \(\sqrt{x}\) does not grow faster than or at the same rate as \(e^x\); it grows 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 approaches infinity. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace. For example, exponential functions like e^x grow faster than polynomial or root functions as x→∞.
Video consigliato:
Percorso guidato
04:16
Intro To Related Rates
Exponential Functions
Exponential functions have the form a^x, where the variable is in the exponent. The function e^x is a fundamental exponential function that grows rapidly as x increases. Its growth rate surpasses that of any polynomial or root function for large x.
Video consigliato:
6:13Exponential Functions
Polynomial and Root Functions
Polynomial functions involve variables raised to constant powers, while root functions are fractional powers (e.g., √x = x^(1/2)). These functions grow slower than exponential functions like e^x as x→∞, meaning their values increase but at a much slower rate.
Video consigliato:
07:00Taylor Polynomials
Pratica correlata
Domanda del libro di testo
In Exercises 41–44:
c. Evaluate df/dx at x = a and df⁻¹/dx at x = f(a) to show that
(df⁻¹/dx)|ₓ₌f(a) = 1 / (df/dx)|ₓ₌a
44. f(x) = 2x², x ≥ 0, a = 5
22
views
Domanda del libro di testo
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
19
views
Domanda del libro di testo
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2
24
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?
c. √(1+x^4)
20
views
Domanda del libro di testo
88. Given that x>0, find the maximum value, if any, of
c. x^(1/x^n) (n a positive integer)
20
views