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Week 3, Short Day: Differentiability

In highschool you have probably seen the derivative introduced as the limit of a difference quotient:

$$f'(x_0)=\frac{f(x)-f(x_0)}{x-x_0}\,\,\,\mathrm{for}\,\,\,x\rightarrow x_0\,.$$

Today we will as an alternative approach introduce differentiability using epsilon functions, which will be needed later on for generalizing differentiability to functions of multiple variables. As a prelude we will differentiate complex functions of a single real variable. By doing this we become able to construct the derivative of an important case of the complex exponential function. Behind it all lie the usual computational rules for the derivative. It is crucial that you master these, and that you understand how differentiation in general is performed!

Today’s Key Concepts

Epsilon functions. Differentiation and rules of differentiation for real functions. Inverse functions and their derivatives. $\tan(x)$ and $\arctan(x)$. Complex functions of a single real variable and their derivatives. Differentiation of the complex exponential function.

Preparation and Syllabus

Today we finalise eNote 1 Complex Numbers by working with section 1.10. In addition, read eNote 3 Elementary Functions, sections 3.2 - 3.4 about differentiability of real functions.

Activity Program

  • 13.00 – 14.00: $\,$No live lecture today (lecturer is absent). See the pre-recorded lecture on this link, or show up as usual in Auditorium 42 at 13:00 where the recording will be shown on the screen.
  • 14.00 – 16.00: $\,$Group exercises in the study areas (b. 302, bottom floor)
  • 16.00 – 17.00: $\,$Weekly Test.

Group Exercises

  1. Today’s Wetware Exercise
  2. The Derivative
  3. Derivatives of Complex Functions
  4. To Differentiate an Inverse Function
  5. Tangent and Arctan
  6. Derivatives Directly from the Definition
  7. Quadratic Equations with Real Coefficients
  8. Quadratic Equations with Complex Coefficients
  9. Epsilon Functions (Advanced)

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 the Möbius quiz system via a link on DTU Learn in the module for 01006 (in the top menu click “Möbius”).
  • The TA will provide a code for Möbius test access.
  • You must be in full-screen mode so the test fills the whole screen.
  • 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.