Evaluate Exact error for Py (Y2) estimating f (Yz) where flx)= et, of degrée 4 atx=1) for fx)=ex Py(x) Tagler Polynomial

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Title: Understanding Exact Error in Taylor Polynomial Approximation**

**Objective:**
We aim to evaluate the exact error for the approximation of \( f(x) \) using the fourth-degree Taylor polynomial \( P_4(x) \) at \( x = \frac{1}{2} \). 

**Function:**
\[ f(x) = e^x \]

**Taylor Polynomial:**
\[ P_4(x) \]
*Degree: 4*
- The Taylor polynomial is centered at \( x = 1 \).

**Task:**
Evaluate the exact error for \( P_4\left(\frac{1}{2}\right) \).

**Explanation:**
To find this error, we use the formula for the remainder (error) in Taylor series approximation, which involves the (n+1)th derivative of the function and the term involving \( (x-c)^{n+1} \). For this example, the exact computation steps and integration considerations should be carefully addressed to understand the behavior and accuracy of the approximation at the given point.
Transcribed Image Text:**Title: Understanding Exact Error in Taylor Polynomial Approximation** **Objective:** We aim to evaluate the exact error for the approximation of \( f(x) \) using the fourth-degree Taylor polynomial \( P_4(x) \) at \( x = \frac{1}{2} \). **Function:** \[ f(x) = e^x \] **Taylor Polynomial:** \[ P_4(x) \] *Degree: 4* - The Taylor polynomial is centered at \( x = 1 \). **Task:** Evaluate the exact error for \( P_4\left(\frac{1}{2}\right) \). **Explanation:** To find this error, we use the formula for the remainder (error) in Taylor series approximation, which involves the (n+1)th derivative of the function and the term involving \( (x-c)^{n+1} \). For this example, the exact computation steps and integration considerations should be carefully addressed to understand the behavior and accuracy of the approximation at the given point.
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