Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.8.2.c
Capitolo 7, Problema 7.8.2.c

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)

Guida verificata passo dopo passo
1
Recall that the function \(e^x\) grows exponentially as \(x \to \infty\), which means it increases faster than any polynomial or root function.
Analyze the given function \(\sqrt{1 + x^4}\). For large \(x\), the term \(x^4\) dominates inside the square root, so \(\sqrt{1 + x^4} \approx \sqrt{x^4} = x^2\).
Since \(x^2\) is a polynomial function, it grows slower than the exponential function \(e^x\) as \(x \to \infty\).
Therefore, \(\sqrt{1 + x^4}\) grows slower than \(e^x\) as \(x \to \infty\).
To summarize, \(\sqrt{1 + x^4}\) 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:
6m

Concetti 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 any polynomial function as x→∞.
Video consigliato:
Percorso guidato
04:16
Intro To Related Rates

Exponential vs. Polynomial Growth

Exponential functions (e.g., e^x) increase much faster than polynomial functions (e.g., x^n) as x becomes very large. Polynomials grow at a rate proportional to a power of x, while exponentials grow proportionally to a constant raised to the power of x, leading to faster growth.
Video consigliato:
09:29
Exponential Growth & Decay

Asymptotic Comparison Using Limits

To compare growth rates, we use limits such as lim(x→∞) f(x)/g(x). If the limit is zero, f grows slower than g; if infinite, f grows faster; if finite and nonzero, they grow at the same rate. This method helps classify functions like √(1+x^4) relative to e^x.
Video consigliato:
Percorso guidato
07:45
Limit Comparison Test
Pratica correlata
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?

c. √x

25
views
Domanda del libro di testo

10. True, or false? As x→∞,

c. 1/x - 1/x² = o(1/x)

25
views
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

4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

c. x²e^(-x)

21
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