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 11.3.33a

Capitolo 11, Problema 11.3.33a

Taylor series

a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x) = 2ˣ, a = 1

Guida verificata passo dopo passo

Recall the definition of the Taylor series of a function \(f(x)\) centered at \(a\): \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\] where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).

Identify the function and the center: here, \(f(x) = 2^x\) and \(a = 1\). We will need to find the derivatives of \(f(x)\) and evaluate them at \(x=1\).

Find the first four derivatives of \(f(x)\): - \(f(x) = 2^x\) - \(f'(x) = 2^x \ln(2)\) - \(f''(x) = 2^x (\ln(2))^2\) - \(f^{(3)}(x) = 2^x (\ln(2))^3\) - \(f^{(4)}(x) = 2^x (\ln(2))^4\)

Evaluate each derivative at \(x = 1\): - \(f(1) = 2^1\) - \(f'(1) = 2^1 \ln(2)\) - \(f''(1) = 2^1 (\ln(2))^2\) - \(f^{(3)}(1) = 2^1 (\ln(2))^3\) - \(f^{(4)}(1) = 2^1 (\ln(2))^4\)

Write the first four nonzero terms of the Taylor series using the formula: \[f(x) \approx \sum_{n=0}^3 \frac{f^{(n)}(1)}{n!} (x - 1)^n = \sum_{n=0}^3 \frac{2 \cdot (\ln(2))^n}{n!} (x - 1)^n,\] which explicitly is: \[2 + 2 \ln(2)(x-1) + \frac{2 (\ln(2))^2}{2!} (x-1)^2 + \frac{2 (\ln(2))^3}{3!} (x-1)^3.\]

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.

Taylor Series Definition

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. Each term involves the nth derivative evaluated at the center point, multiplied by (x - a)^n and divided by n!. This series approximates the function near the center a.

Video consigliato:

08:42

Taylor Series

Derivatives of Exponential Functions

For the function f(x) = 2^x, derivatives involve the natural logarithm of the base. Specifically, the nth derivative of 2^x is (ln 2)^n times 2^x. Understanding this pattern is essential to compute the terms of the Taylor series accurately.

Video consigliato:

04:50

Derivatives of General Exponential Functions

Evaluating the Series at the Center Point

To find the Taylor series terms centered at a = 1, each derivative must be evaluated at x = 1. These values are then used in the formula for each term, ensuring the series accurately approximates the function near x = 1.

Video consigliato:

Percorso guidato

06:00

Geometric Series

Pratica correlata

Domanda del libro di testo

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.

∞

Σ (x - 1)ᵏ/(k5ᵏ)

k = 1

views

Domanda del libro di testo

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.

a. √(1 + 2x)

A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²