Visualizing Mathematics with GeoGebraLaajuus (3 ECTS)

Objective

Knowledge and understanding:
The students will experience the impact of visualization on understanding key concepts in algebra and calculus such as functions, equations, differentiation, Taylor series, and Fourier series through the activities using mathematical software GeoGebra. They will also recognize these mathematics concepts usually come from simple ideas and can be connected to each other into a structured knowledge system. The bird-eyed view of the mathematics concepts gained in this course will help the students to start learning mathematics rigorously or to construct mathematics models in their own specialties.

Skills:
The students will master the skills of first-order approximation, and how to use GeoGebra for visualizing more mathematical concepts.

Content

- Introduction to GeoGebra, a mathematics software for visualizing mathematics functions.
- Transforming graphs of linear and quadratic functions by changing value of the coefficients.
- Analyzing symbolic formulas and connecting them with graphic features of the graphs.
- Visualizing the derivative of a target function as a step-graph using line-graph approximation of the target function; also inducing the derivative of power functions.
- Taylor series approximation of exponential function and trigonometric function.
- Visualizing a trigonometric function as a helix in 3D space; Relating combining and differentiating trigonometric functions to a phase shift or rotation of the helix.
- Visualizing Euler’s formula as rotation of the helix.
- Visualizing average value of the function, which is the multiplication of two trigonometric functions.
- Fourier series approximation of triangular and square waves; the idea of spectrum analysis.
- Relating a plane in 3D space with its vector equations: the one with two parameters, the other with the unit normal vector.

Qualifications

The students are expected to have basic knowledge of algebra, for example, linear function, quadratic function, and how to draw the graphs of these functions on a paper with a pencil.