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Week 4, Long Day: Taylor’s Formulas

In highschool you have - maybe without realising it - worked with the approximating polynomial of the first degree. Its graph is a straight line which is the tangent to the function it is approximating at the so-called expansion point. Today we will introduce approximating polynomials of higher degrees. Our goal is to approximate difficult differentiable functions (or differentiable functions whose full extent is unknown) as polynomials, called approximating polynomials or Taylor polynomials, which often are much easier to work with. The higher the degree of the polynomial, the better the approximation usually is.

Introducing Maple

For this topic we need to work with graphical representations and plotting as well as longer and sometimes repetitive calculations, so today we will introduce the mathematical software Maple , an advanced mathematics and plotting tool that will be our companion throughout the rest of the course.

To save time on the day, please download and install Maple on your computer beforehand from DTU’s portal with licenced software. Today’s exercises will include guides that will walk you through an initial set up and the basic use of the software.

Today’s Key Concepts

Expansion/development point. Taylor’s formula and the remainder function. Approximating polynomials of degree $n$ and their graphs. Evaluation of remainder functions and error estimation. Taylor’s limit formula.

Preparation and Syllabus

Today’s material is based on eNote 4 Taylor’s approximation formulas.

Maple Ressources

  • taylor computes Taylor’s limit formula
  • mtaylor computes approximating polynomials
  • limit computes a limit value

If you are new to Maple, the following two Maple demos will help you to the Maple methods that are useful for the topics we have covered this far in the course: MapleDemo ComplexNumbers and MapleDemo Taylor.

Activity Program

  • 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: Your teachers are present in the study areas
  • 15.30 – 16.00: The teachers will answer conceptual questions to the homework

Exercises

  1. Approximating Polynomials. By Hand
  2. Introduction to Maple
  3. Illustration of Taylor Approximation
  4. Investigation of Approximating Polynomials
  5. Estimation of Errors by Approximation
  6. Introduction to Complex Numbers with Maple
  7. Approximations of Complex Functions of a Real Variable
  8. Taylor’s Limit Formula. By Hand

Homework Set 1

You must on Sunday Sep. 25, latest 23:55, upload your HW1 solution as a single .pdf file to your group module at DTU Learn under “Content” $\rightarrow$ “Assignment” $\rightarrow$ “Homework 1”. You may ask conceptual questions about the homework to your teacher from 15:30-16:00 today, but note that the teachers will not answer to details about methods and results.

Theme Exercise 1

On Friday Sep. 23 you will participate in Theme Exercise 1 (Theme1) of Advanced Engineering Mathematics 1.

The theme exercise will become available by the end of the Long Day (at about 17:00 today) and we advice you to start as soon as possible with a group of 4-6 that you may form yourselves. Read the description and rules of theme exercises on the agenda of Short day.