Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 2 - Limits

Problema 2.47

Capitolo 2, Problema 2.47

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?

f(x)=(2x−3)^2/3

Guida verificata passo dopo passo

The function $ f(x) = (2x - 3)^{2/3} $ is a composition of a power function and a linear function. The power function $ g(x) = x^{2/3} $ is continuous for all $ x \geq 0 $.

The inner function $ 2x - 3 $ is a linear function, which is continuous everywhere. Therefore, the domain of $ 2x - 3 $ is all real numbers.

Since $ g(x) = x^{2/3} $ is continuous for $ x \geq 0 $, the composition $ f(x) = (2x - 3)^{2/3} $ is continuous where $ 2x - 3 \geq 0 $.

Solve $ 2x - 3 \geq 0 $ to find the interval of continuity. This gives $ x \geq \frac{3}{2} $.

Since $ f(x) $ is defined for $ x \geq \frac{3}{2} $, check continuity from the right at $ x = \frac{3}{2} $. The function is continuous from the right at this endpoint.>

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.

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. Understanding continuity is essential for determining where a function behaves predictably without breaks or jumps.

Video consigliato:

05:34

Intro to Continuity

Endpoints of Intervals

Endpoints of intervals are the boundary points that define the start and end of an interval on the real number line. When analyzing continuity, it is crucial to evaluate the behavior of the function at these endpoints, as a function may be continuous from the left or right at these points, indicating one-sided limits exist and match the function's value.

Video consigliato:

03:34

Finding Global Extrema (Extreme Value Theorem) Example 4

Piecewise Functions and Roots

The function f(x) = (2x - 3)^(2/3) is a piecewise function that involves roots and powers. Understanding how to handle fractional exponents is important, as they can introduce points of discontinuity. Specifically, the behavior of the function near the root (where the expression inside the parentheses equals zero) must be analyzed to determine continuity and the nature of the function at that point.

Video consigliato:

Percorso guidato

05:36

Piecewise Functions

Pratica correlata

Domanda del libro di testo

Determine the following limits.

lim x→1 x^3 − 7x^2 + 12x / 4 − x

379

views

Domanda del libro di testo

Determine the following limits.

lim x→∞ sin x / e^x

470

views

Domanda del libro di testo

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2−25 / x−5; a=5

290

views

Domanda del libro di testo

Evaluate each limit.

$\lim_{x \to 3} \sqrt{x^{2} + 7}$