Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problema 4.8.5

Capitolo 4, Problema 4.8.5

Let ƒ(x) = 2x³ - 6x² + 4x. Use Newton’s method to find x₁ given that x₀ = 1.4. Use the graph of f (see figure) and an appropriate tangent line to illustrate how x₁ is obtained from x₀ . <IMAGE>

Guida verificata passo dopo passo

Newton's method is an iterative process used to approximate the roots of a real-valued function. The formula for Newton's method is: x_{n+1} = x_n - \(\frac{f(x_n)}{f'(x_n)}\).

First, we need to find the derivative of the function f(x) = 2x^3 - 6x^2 + 4x. The derivative, f'(x), is calculated as follows: f'(x) = \(\frac{d}{dx}\)(2x^3 - 6x^2 + 4x) = 6x^2 - 12x + 4.

Next, substitute x_0 = 1.4 into the function f(x) and its derivative f'(x) to find f(1.4) and f'(1.4).

Calculate f(1.4) = 2(1.4)^3 - 6(1.4)^2 + 4(1.4) and f'(1.4) = 6(1.4)^2 - 12(1.4) + 4.

Finally, use the Newton's method formula to find x_1: x_1 = 1.4 - \(\frac{f(1.4)}{f'(1.4)}\). This will give you the next approximation of the root, x_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.

Newton's Method

Newton's Method is an iterative numerical technique used to approximate the roots of a real-valued function. It starts with an initial guess and uses the function's derivative to find successively better approximations. The formula for updating the guess is x₁ = x₀ - f(x₀)/f'(x₀), where f' is the derivative of f. This method is particularly effective when the initial guess is close to the actual root.

Video consigliato:

Percorso guidato

4:26

Evaluating Composed Functions

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In the context of Newton's Method, the derivative is crucial for determining the slope of the tangent line at the current guess, which guides the next approximation towards the root.

Video consigliato:

Percorso guidato

05:44

Derivatives

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. In Newton's Method, the tangent line at the point (x₀, f(x₀)) is used to find the next approximation x₁. The intersection of this tangent line with the x-axis provides a new estimate for the root, illustrating how the method converges to the actual solution.

Video consigliato:

Percorso guidato

05:13

Slopes of Tangent Lines

Pratica correlata

Domanda del libro di testo

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→π/2⁻ (π/2 - x) sec x

180

views

Domanda del libro di testo

{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.

f(x) = e⁻ˣ - x; x₀ = ln 2

210

views

Domanda del libro di testo

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (4√x - (4 /√x)) dx

105

views

rank

Domanda del libro di testo

{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.

y = 1/x and y = 4 - x²

239

views

Domanda del libro di testo

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

g(s) = 1 / (s² + 1)

views

Domanda del libro di testo

Maximum-area rectangles Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.)

262

views