MANE 3351
Lecture 6, September 8
Classroom Management
Agenda
- Attendance
- Raspberry Pi VM update
- No lab today
Resources
Handouts
- Lecture 6 Slides
- Lecture 6 Marked Slides
Calendar
| Lecture | Date | Content |
|---|---|---|
| 1 | August 23 | Welcome to Class, Syllabus |
| 2 | August 25 | Error Analysis |
| 3 | August 30 | Introduction to Jupyter Notebook |
| 4 | September 1 | Markdown |
| 5 | September 6 | Labor Day Holiday |
| 6 | September 8 | Taylor Polynomials, Homework 1 |
Assignments
Taylor Series
Introduction to Taylor Series
Cheney and Kincaid [^1] provide some commonly used Taylor series.
Example
To find \(e^8\), recall \(e^x=\sum_{k=0}^\infty\frac{x^k}{k!}\)
- \(e^8=1\)
- \(e^8=1+x=1+8\)
- \(e^8=1+8+\frac{x^2}{2!}=1+8+\frac{64}{2}\)
- \(e^8=1+8+\frac{64}{2}+\frac{x^3}{3!}=1+8+\frac{64}{2}+\frac{512}{6}\)
Jupyter Notebook Demonstration
Taylor Series Expansion about a Point
Source: textbook, page 14
Error Analysis of Taylor Series
- Note that \(f(x)=T_n(x)+R_n(x)\)
- Absolute error is \(|f(x)-T_n(x)|=|R_n(x)|\)
- Absolute error depends on three factors:
- \(|x-x_0|^{n+1}\)
- \(\frac{1}{(n+1)!}\)
- \(|f^{(n+1)}(\xi)|\)
- An error bound can be found by finding an upper bound on \(|f^{(n+1)}(\xi)|\).
Error Analysis using Jupyter Notebooks
- Examine Cells 3 and 4 in today's Jupyter Notebook Lecture materials
[1]: Cheney and Kincaid (2004), Numerical Mathematics and Computing, 5th edition.