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Week 2, Long Day: Taylor’s Formulas

Today we will have a brief review of Taylor’s formulas for functions of one variable, and then we continue with functions of more variables. First we will make some approximations with approximating polynomials, and then we will investigate second-degree polynomials with multiple variables and consider how they can be reduced by the use of a coordinates coordinate change. For this we will revisit some of the theory about diagonalization of symmetric matrices. We will also briefly touch upon the problem of finding extrema, a topic that will be fully covers at next week’s Long Day.

Today’s Key Concepts

Taylor’s limit formula. Approximating second-degree polynomials. Approximation. Second-degree polynomial with multiple variables, and how they can be reduced. Quadratic form.

Preparation and Syllabus

Today’s text is eNote 21 Taylor’s Limit Formula for Functions of Two Variables, in particular Sections 21.1 and 21.2. Also we will revisit eNote 15 Symmetric Matrices and cover the remaining Sections 15.8 and 15.9.

Maple Ressources

We will again make good us of mtaylor for multi-variable functions. Today’s Mape demo is 23_TaylorTwoVariables.

Activity Program

  • 10.00 – 12.00: $\,$ Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.00 – 17.00: $\,$ Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: $\,$ Your teachers are present

Group Exercises:

  1. Recap: Functions of One Variable
  2. Taylor’s Formulas and Approximation. By hand
  3. Application of an Approximating Polynomial
  4. A Proper Local Maximum
  5. Diagonalization and Reduction of a Quadratic Form
  6. Teaser Exercise