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Determine the Taylor series for f(x) = e¹/² centered at c = 4 by first finding the general expression for f(n)(4).

Determine the Taylor series for f(x) = e¹/² centered at c = 4 by first finding the general expression for f(n)(4).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Determining the Taylor Series for \( f(x) = e^{x/2} \) Centered at \( c = 4 \)** To determine the Taylor series for \( f(x) = e^{x/2} \) centered at \( c = 4 \), we first need to find the general expression for \( f^{(n)}(4) \). ### Step-by-step Process: 1. **Compute the Derivatives**: We begin by computing the first few derivatives of \( f(x) \) to establish a pattern. 2. **Evaluate at \( c = 4 \)**: After finding the general expression for the \( n \)-th derivative, evaluate it at \( x = 4 \). 3. **Formulate the Taylor Series**: Use the general form of the Taylor series expansion centered at \( c = 4 \): \[ T(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(4)}{n!}(x - 4)^n \] Here, \( f^{(n)}(4) \) represents the \( n \)-th derivative of \( f(x) \) evaluated at \( x = 4 \). By following these steps, you can derive the Taylor series for \( f(x) = e^{x/2} \) centered at \( x = 4 \). This approach provides a systematic way to find the Taylor series for a function centered at any given point, especially useful in applications for approximating functions and analyzing them near specific values.
Transcribed Image Text:**Determining the Taylor Series for \( f(x) = e^{x/2} \) Centered at \( c = 4 \)** To determine the Taylor series for \( f(x) = e^{x/2} \) centered at \( c = 4 \), we first need to find the general expression for \( f^{(n)}(4) \). ### Step-by-step Process: 1. **Compute the Derivatives**: We begin by computing the first few derivatives of \( f(x) \) to establish a pattern. 2. **Evaluate at \( c = 4 \)**: After finding the general expression for the \( n \)-th derivative, evaluate it at \( x = 4 \). 3. **Formulate the Taylor Series**: Use the general form of the Taylor series expansion centered at \( c = 4 \): \[ T(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(4)}{n!}(x - 4)^n \] Here, \( f^{(n)}(4) \) represents the \( n \)-th derivative of \( f(x) \) evaluated at \( x = 4 \). By following these steps, you can derive the Taylor series for \( f(x) = e^{x/2} \) centered at \( x = 4 \). This approach provides a systematic way to find the Taylor series for a function centered at any given point, especially useful in applications for approximating functions and analyzing them near specific values.
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