Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 4.2.22

Capitolo 4, Problema 4.2.22

Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

g(t) = 1/(1 − t) + √(1 + t) − 3.1, (−1, 1)

Guida verificata passo dopo passo

First, understand that finding the zero of a function means finding the value of t for which g(t) = 0. We need to show that g(t) has exactly one zero in the interval (-1, 1).

Consider the function g(t) = \(\frac{1}{1 - t}\) + \(\sqrt{1 + t}\) - 3.1. We need to analyze the behavior of this function within the interval (-1, 1).

Check the continuity of g(t) in the interval (-1, 1). The function is continuous in this interval because both \(\frac{1}{1 - t}\) and \(\sqrt{1 + t}\) are continuous for t in (-1, 1).

Use the Intermediate Value Theorem, which states that if a function is continuous on a closed interval [a, b] and takes different signs at the endpoints, then it has at least one zero in the interval (a, b). Evaluate g(t) at the endpoints of the interval: g(-1) and g(1).

Calculate the derivative g'(t) to determine the behavior of g(t) within the interval. If g'(t) does not change sign, it indicates that g(t) is either strictly increasing or strictly decreasing, which implies exactly one zero in the interval (-1, 1).

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.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function, f, takes on values of opposite signs at two points, a and b, then it must have at least one root in the interval (a, b). This theorem is crucial for proving the existence of a zero in a given interval, as it ensures that the function crosses the x-axis.

Video consigliato:

Percorso guidato

06:11

Fundamental Theorem of Calculus Part 1

Continuity of Functions

A function is continuous on an interval if there are no breaks, jumps, or holes in its graph within that interval. For the function g(t) = 1/(1 − t) + √(1 + t) − 3.1, it is important to verify its continuity on the interval (−1, 1) to apply the Intermediate Value Theorem effectively. Discontinuities would invalidate the theorem's application.

Video consigliato:

05:34

Intro to Continuity

Behavior of Rational and Radical Functions

Understanding the behavior of rational and radical functions is essential for analyzing g(t). The function 1/(1 − t) is undefined at t = 1, and √(1 + t) is defined for t ≥ -1. Analyzing these components helps determine the function's behavior and continuity over the interval (−1, 1), ensuring it meets the conditions for the Intermediate Value Theorem.

Video consigliato:

06:13

Limits of Rational Functions with Radicals

Pratica correlata

Domanda del libro di testo

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

g(x) = {−x, 0 ≤ x < 1

x − 1, 1 ≤ x ≤ 2

245

views

Domanda del libro di testo

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(4secx tanx − 2 sec²x)dx

views

Domanda del libro di testo

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

5. y=x+sin(2x), -2π/3≤x≤2π/3

317

views

Domanda del libro di testo

101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.

306

views

Domanda del libro di testo

Sketch the graphs of the rational functions in Exercises 53–60.

𝓍⁴ ― 1

y = ------------------

𝓍²

196

views

Domanda del libro di testo

Initial Value Problems

Solve the initial value problems in Exercises 71–90.