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Week 1, Short Day: Domains and Tangent Planes

Today we continue with the study of functions of two variables with their level curves, partial derivatives, gradients , directional derivatives, etc. From differentiable functions of one variable we know that the approximating first-degree polynomial is the tangent to the function in the chosen development point. Now we generalize this concept to two variables which will produce tangent planes. The chain rule along curves, that e.g. is used in the deduction of the formula for the directional derivative, is also new.

Today’s key Concepts

The approximating first-degree polynomial. Tangent plane. Domain. Set terminology: closed set, bounded set, unbounded set, closure, boundary, interior, exterior. The chain rule along curves.

Preparation and Syllabus

We will today continue with subjects from eNote 19 Functions of Two Variables and will also cover eNote 20 Gradients and Tangent Planes.

Maple Ressources

  • We will revisit the mtaylor command, this time with one more variable, to produce first-order approx. polynomials that now are equivalent to tangent planes.

Today’s two Maple demos are 21_Domain and 22_TangentPlane.

Activity Program

  • 13:00 – 14.00: $\,$ Lecture (aud. 42, b. 303A) (link to streaming)
  • 14.00 – 16.00: $\,$ Group exercises in the study areas (b. 302, bottom floor)
  • 16.00 – 17.00: $\,$ Weekly test

Group Exercises

  1. Level Curves
  2. Approximation of First Degree
  3. Fields in the (x,y)-Plane
  4. An Elevation Function
  5. Summary Exercise

Tip: If you want a printable version of the exercises without hints or answers, then go to the exercises page and click your browser’s print button.

Weekly Test

For all Weekly Tests, the following applies:

  • The test is an on-location test, meaning it can only be accessed in the study area.
  • No electronic aids are allowed (except for your own notes on e.g. a tablet).
  • The test can be accessed in the Möbius quiz system via a link on DTU Learn in the module for 01006 (in the top menu click “Möbius”).
  • Your solutions to the test questions must be typed into Möbius without in-between calculations or steps. The result is automatically evaluated by Möbius.
  • To ensure a smooth experience use the Firefox or Chrome browser, and disable any add-blocker.
  • Use a DTU network.
  • You may discuss the test questions with fellow students in your study group, but you have your own version of the test with scrambled numbers that you yourself must solve and enter into Möbius.
  • During the final hour on Fridays you have one attempt. Passing this attempt will grant you 1 bonus point. From Friday at 18:00 until Wednesday at 18:00 the test is reopened for repeated attempts. Passing during this phase will grant you ½ bonus point.

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