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?
a. x² + √x
20
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 7.8.10.a
10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)
Guida verificata passo dopo passo1
Recall the definition of Big-O notation: A function \(f(x)\) is \(O(g(x))\) as \(x \to \infty\) if there exist positive constants \(C\) and \(x_0\) such that for all \(x > x_0\), \(|f(x)| \leq C |g(x)|\).
Identify the functions in the problem: \(f(x) = \frac{1}{x+3}\) and \(g(x) = \frac{1}{x}\).
Compare the behavior of \(f(x)\) and \(g(x)\) as \(x \to \infty\). Notice that \(\frac{1}{x+3}\) behaves very similarly to \(\frac{1}{x}\) because adding 3 becomes insignificant for very large \(x\).
To verify the Big-O relationship, consider the ratio \(\frac{|f(x)|}{|g(x)|} = \frac{\frac{1}{x+3}}{\frac{1}{x}} = \frac{x}{x+3}\). As \(x \to \infty\), this ratio approaches 1, which is a finite constant.
Since the ratio approaches a finite constant, there exist constants \(C\) and \(x_0\) such that \(|f(x)| \leq C |g(x)|\) for all \(x > x_0\), confirming that \(\frac{1}{x+3} = O\left(\frac{1}{x}\right)\) is true.
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:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Big O Notation
Big O notation describes the upper bound of a function's growth rate, focusing on its behavior as the input approaches infinity. It provides a way to classify functions according to their dominant terms, ignoring constant factors and lower-order terms.
Video consigliato:
Percorso guidato
04:22
Sigma Notation
Asymptotic Behavior of Functions
Asymptotic behavior studies how functions behave as the input variable grows very large. Understanding limits and dominant terms helps compare functions like 1/(x+3) and 1/x as x approaches infinity.
Video consigliato:
Percorso guidato
5:50Asymptotes of Hyperbolas
Equivalence of Functions in Big O Terms
Two functions f(x) and g(x) are related by f(x) = O(g(x)) if there exist constants C and x0 such that |f(x)| ≤ C|g(x)| for all x > x0. This means f(x) grows no faster than a constant multiple of g(x) for large x.
Video consigliato:
7:30Logarithms Introduction
Pratica correlata
Domanda del libro di testo
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
15. y = sin⁻¹√(1-u²), 0<u<1
22
views
Domanda del libro di testo
Find the volumes of the solids in Exercises 135 and 136.
135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis are
a. circles whose diameters stretch from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).
32
views
Domanda del libro di testo
23. What roles do the functions e^x and ln(x) play in growth comparisons?
20
views
Domanda del libro di testo
154. The linearization of log₃x
a. Find the linearization of
f(x) = log₃xatx = 3.
Then round its coefficients to two decimal places.
32
views
Domanda del libro di testo
In Exercises 41–44:
a. Find f⁻¹(x).
44. f(x) = 2x², x ≥ 0, a = 5
20
views