### 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 $.

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Answered: Find o for feDz e Taulor series centred…

Math

Advanced Math

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

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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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

### 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 $.

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