### Problem 1:**Objective:**Find the Taylor series for $ f(x) = e^{5x} $ centered at $ c = -3 $.**Instruction:**Use a table to solve the problem.This problem involves finding the Taylor series expansion of the function $ f(x) = e^{5x} $ about the center $ x = -3 $. A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point.To solve this, you will need to construct a table that includes the necessary derivatives of $ f(x) $ evaluated at $ x = -3 $. Then, use the formula for the Taylor series to write the expansion.### Steps to Solve:1. **Calculate the derivatives:** - $ f(x) = e^{5x} $ - First derivative: $ f'(x) = 5e^{5x} $ - Second derivative: $ f''(x) = 25e^{5x} $ - Third derivative: $ f^{(3)}(x) = 125e^{5x} $ - Continue this pattern to find as many derivatives as needed. 2. **Evaluate at $ x = -3 $:** - $ f(-3) = e^{-15} $ - $ f'(-3) = 5e^{-15} $ - $ f''(-3) = 25e^{-15} $ - $ f^{(3)}(-3) = 125e^{-15} $ - And so on...3. **Construct the Taylor series:** The Taylor series for a function $ f(x) $ centered at $ c $ is given by: \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x - c)^n \] For this problem, $ c = -3 $.4. **Write the series:** Substitute the evaluated derivatives into the Taylor series formula to find the expansion.Using these steps and the constructed table, you can find the Taylor series for $ f(x) = e^{5x} $ centered at $ c = -3 $.

Answered: Image /qna-images/answer/b460deb9-bf55-43bc-b70c-91af7f3c670e.jpg

Answered: Find o for feDz e Taulor series centred…

Find o for feDz e Taulor series centred Jat c= -3. Use toble to solve the problem. a.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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