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Week 3, Long Day: Extremum Investigations

Today we will find minimum og maximum of functions of multiple variables, both local and global, so highest and lowest function values. In short we will optimize! There is plenty to do and much of what we have learned this far in the course will now come in use, not least Taylor’s formulas, Hessian matrices, eigenvalues, orthogonal matrices, quadratic surfaces and more.

Today’s Key Concepts

Stationary point. Local and global extremum. Proper extremum. Properties of continuous function on closed, bounded and connected set. Boundary investigation. Elliptic and hyperbolic paraboloids.

Preparation and Syllabus

Today’s topics will return to eNote 21 Taylor’s Limit Formula for Functions of Two Variables, in particular Definition 21.9 and Section 21.3 about extremum investigations.

Maple Ressources

Today’s Maple demo is 25_Extrema.

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. Teasing Exercise
  2. Application of Hessian Matrix. By Hand
  3. Local Extrema for a Function of Two Variables
  4. Global Maximum and Global Minimum
  5. Boundary Investigation where the Boundary is Parametrized
  6. Global Extrema for a Function of Three Variables
  7. Supplementary Exercise

Note about Theme Exercise 5

On Friday Theme Exercise 5 about functions of multiple variables will take place, the first Theme exercise of the Spring smester. The Theme exercise will be uploaded today, Wednesday, at 17:00 at DTU Learn, and you are adviced to form groups and work on the exercise before showing up for the Short Day on Friday where the Theme test will take place.