8. For each function below, determine the number of terms needed to calculate the error on the Taylor series centered at c, to within E ≤ 0.0001 (10-4) of the value of the function at a. Calculate your estimate and compare to the true value. a. f(x) = e²x, c = 0, a = 1 b. f(x)=√√8 + x, c = 2, a = 2.5
8. For each function below, determine the number of terms needed to calculate the error on the Taylor series centered at c, to within E ≤ 0.0001 (10-4) of the value of the function at a. Calculate your estimate and compare to the true value. a. f(x) = e²x, c = 0, a = 1 b. f(x)=√√8 + x, c = 2, a = 2.5
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 8: Taylor Series and Error Estimation**
For each function below, determine the number of terms needed to calculate the error on the Taylor series centered at \( c \), to within \( E \leq 0.0001 \) \((10^{-4})\) of the value of the function at \( a \). Calculate your estimate and compare to the true value.
a. \( f(x) = e^{2x}, \, c = 0, \, a = 1 \)
b. \( f(x) = \sqrt[3]{8 + x}, \, c = 2, \, a = 2.5 \)
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This problem involves using the Taylor series to approximate functions and requires calculating how many terms are needed to ensure the approximation error is within a specified tolerance. For each function, you'll use the Taylor series expansion formula centered at a given point \( c \) and evaluate at \( a \). A comparison between the estimated value using the Taylor series and the true value should be performed to confirm the accuracy.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa098e07d-b15e-4284-9177-08ea985b9919%2Fcc56878b-332b-4205-a057-0bb78fef1436%2Ft6ojybn_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 8: Taylor Series and Error Estimation**
For each function below, determine the number of terms needed to calculate the error on the Taylor series centered at \( c \), to within \( E \leq 0.0001 \) \((10^{-4})\) of the value of the function at \( a \). Calculate your estimate and compare to the true value.
a. \( f(x) = e^{2x}, \, c = 0, \, a = 1 \)
b. \( f(x) = \sqrt[3]{8 + x}, \, c = 2, \, a = 2.5 \)
---
This problem involves using the Taylor series to approximate functions and requires calculating how many terms are needed to ensure the approximation error is within a specified tolerance. For each function, you'll use the Taylor series expansion formula centered at a given point \( c \) and evaluate at \( a \). A comparison between the estimated value using the Taylor series and the true value should be performed to confirm the accuracy.
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