Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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

Briggs 3rd Edition

Ch. 11 - Power Series

Problema 11.1.69b

Capitolo 11, Problema 11.1.69b

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.

b. Estimate f(0.2) and give a bound on the error in the approximation.

f(x) = sin x ≈ x

Guida verificata passo dopo passo

Identify the function and the approximation given: the function is \(f(x) = \sin x\), and the approximation near zero is \(\sin x \approx x\).

To estimate \(f(0.2)\) using the approximation, substitute \(x = 0.2\) into the approximation: \(f(0.2) \approx 0.2\).

To find a bound on the error, recall the Taylor remainder theorem. The error in approximating \(\sin x\) by \(x\) near zero is given by the next term in the Taylor series expansion, which involves \(\frac{\cos c}{2} x^2\) for some \(c\) between 0 and \(x\).

Since \(|\cos c| \leq 1\) for all real \(c\), the maximum error bound is \(\left| R_2 \right| \leq \frac{|x|^3}{6}\), because the next term in the Taylor series for \(\sin x\) after \(x\) is \(-\frac{x^3}{3!}\).

Calculate the error bound by substituting \(x = 0.2\) into \(\frac{|x|^3}{6}\) to get the maximum possible error in the approximation.

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.

Small Angle Approximation

The small angle approximation states that for values of x near zero, sin(x) can be approximated by x. This simplifies calculations by replacing the sine function with a linear expression, which is accurate for small angles measured in radians.

Video consigliato:

Percorso guidato

04:57

Determining Error and Relative Error

Taylor Series Expansion

The Taylor series expresses a function as an infinite sum of terms calculated from its derivatives at a single point. For sin(x) near zero, the series starts as x - x³/6 + ..., and truncating after the first term gives the small angle approximation.

Video consigliato:

08:42

Taylor Series

Error Bound in Approximations

An error bound estimates the maximum difference between the true function value and its approximation. For Taylor approximations, the Lagrange remainder formula provides a way to calculate this bound, ensuring the approximation's accuracy is quantifiable.

Video consigliato:

Percorso guidato

04:57

Determining Error and Relative Error

Pratica correlata

Domanda del libro di testo

Taylor series

b. Write the power series using summation notation.

f(x) = 1/x, a = 1

118

views

Domanda del libro di testo

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.

b. Estimate f(0.2) and give a bound on the error in the approximation.

f(x) = ln (1 + x) ≈ x − x²/2

views

Domanda del libro di testo

Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.

b. The expected number of rounds (possessions by either team) required for the overtime to end is (1/6) ∑ₖ₌₁∞ k(5/6)ᵏ⁻¹. Evaluate this series.

views

Domanda del libro di testo

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = e²ˣ, a = 0

views

Domanda del libro di testo

Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.

b. Integrate the series to find a Taylor series for Si.

120

views