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 4.8.17

Capitolo 4, Problema 4.8.17

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.

f(x) = tan x - 2x; x₀ = 1.2

Guida verificata passo dopo passo

Step 1: Understand Newton's Method. Newton's method is an iterative process to approximate the roots of a real-valued function. The formula for the next approximation is given by:

x_n + 1 = x_n - \frac{f (x_n)}{f' (x_n)}

Step 2: Calculate the derivative of the function. Given

f (x) = \tan (x) - 2 x

, the derivative is

f' (x) = \frac{1}{\cos^2 (x)} - 2

Step 3: Set up the iterative process. Start with the initial approximation

x_0 = 1.2

. Calculate

f (x_0)

and

f' (x_0)

Step 4: Compute the next approximation using the formula:

x_1 = x_0 - \frac{f (x_0)}{f' (x_0)}

. Repeat this process to find

x_2

x_3

, etc., until two successive approximations agree to five decimal places.

Step 5: Create a table to organize your results. For each iteration, record the values of

n

x_n

f (x_n)

f' (x_n)

, and the next approximation

x_n + 1

. Continue until the stopping criterion is met.

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 find approximate roots of a real-valued function. It starts with an initial guess and refines it using the formula x₁ = x₀ - f(x₀)/f'(x₀), where f' is the derivative of f. This process continues until the difference between successive approximations is sufficiently small, indicating convergence to a root.

Video consigliato:

Percorso guidato

4:26

Evaluating Composed Functions

Convergence Criteria

In numerical methods, convergence criteria determine when to stop the iterative process. For Newton's Method, this often involves checking if the absolute difference between two successive approximations is less than a specified tolerance level, such as 0.00001 for five decimal places. This ensures that the approximations are accurate enough for practical purposes.

Video consigliato:

Percorso guidato

05:44

Derivatives

Function and Derivative Evaluation

To apply Newton's Method, it is essential to evaluate both the function f(x) and its derivative f'(x) at each iteration. For the given function f(x) = tan x - 2x, the derivative f'(x) = sec² x - 2 must be computed. Accurate evaluation of these functions is crucial for obtaining correct approximations of the root.

Video consigliato:

Percorso guidato

4:26

Evaluating Composed Functions

Pratica correlata

Domanda del libro di testo

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

1/203

316

views

Domanda del libro di testo

Functions from derivatives Use the derivative f' to determine the x-coordinates of the local maxima and minima of f, and the intervals on which f is increasing or decreasing. Sketch a possible graph of f (f is not unique).

f'(x) = 10 sin 2x on [-2π, 2π]

274

views

Domanda del libro di testo

Trajectory high point A stone is launched vertically upward from a cliff 192 ft above the ground at a speed of 64 ft/s. Its height above the ground t seconds after the launch is given by s = -16t² + 64t + 192, for 0 ≤ t ≤ 6. When does the stone reach its maximum height?

258

views

Domanda del libro di testo

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = -12x⁵ + 75x⁴ - 80x³

193

views

Domanda del libro di testo

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

∫ (3x + 1) (4 - x) dx

views

Domanda del libro di testo

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

lim_x→ 0 (eˣ - 1) / (x² + 3x)

272

views