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

Guida verificata passo dopo passo

Identify the dominant term in the function as \( x \to \infty \). For the function \( f(x) = x^{2} + \sqrt{x} \), the terms are \( x^{2} \) and \( \sqrt{x} = x^{1/2} \).

Compare the growth rates of each term to \( x^{2} \). Since \( x^{2} \) grows faster than \( x^{1/2} \), the \( x^{2} \) term dominates the behavior of the function for large \( x \).

Determine if the function grows faster, slower, or at the same rate as \( x^{2} \) by considering the limit \( \lim_{x \to \infty} \frac{f(x)}{x^{2}} = \lim_{x \to \infty} \frac{x^{2} + \sqrt{x}}{x^{2}} \).

Simplify the limit expression to \( \lim_{x \to \infty} \left(1 + \frac{\sqrt{x}}{x^{2}}\right) = \lim_{x \to \infty} \left(1 + x^{-3/2}\right) \).

Evaluate the limit: since \( x^{-3/2} \to 0 \) as \( x \to \infty \), the limit equals 1, indicating that \( f(x) \) grows at the same rate as \( x^{2} \).

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:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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 pace as a reference function, such as x², by focusing on dominant terms and ignoring lower-order terms.

Video consigliato:

Percorso guidato

04:16

Intro To Related Rates

Dominant Term in Polynomial Functions

In polynomials, the term with the highest exponent dominates the function's behavior for large x. For example, in x² + √x, the x² term grows faster than √x as x→∞, so the overall growth rate is determined by x², making it grow at the same rate as x².

Video consigliato:

07:00

Taylor Polynomials

Big-O and Big-Theta Notation

Big-O and Big-Theta notations formalize growth comparisons by classifying functions based on upper bounds (Big-O) or tight bounds (Big-Theta). Saying a function grows at the same rate as x² means it is Θ(x²), while growing faster or slower corresponds to different asymptotic classes.

Video consigliato: