Available online
Description
This programme uses models and animations to describe the construction and properties of graphs representing functions from R to R2, and R2 to R.
This programme uses models and animations to describe the construction and properties of graphs representing functions from R to R2, and R2 to R.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 09 |
| First transmission date: | 17-04-1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:12 |
| + Show more... | |
| Producer: | Jack Koumi |
| Contributor: | Peter Strain-Clark |
| Publisher: | BBC Open University |
| Keyword(s): | Animations; Codomain; Domain; Models |
| Footage description: | Peter Strain talks about functions which map joints from the plane R to the real line R. Jsing animations he describes how a function from R to R would be constructed, using height as the codomain and position on the plane as the domain. The resulting graph is a surface. He describes this surface in more detail by looking at sections of the curve. This first surface was obtained as x, y, mapped to the real number x + y. He then looks at other surfaces. When the function [x, y) maps to x² + y². This generates a continuous sloping plane. When the function (x, y) maps to xy. This generates a saddle shaped surface. Peter Strain now discusses functions from R to R², mapping points of R to positions on the plane. Using animation he shows how such a function would be constructed, with the domain along the horizontal axis and the codomain, which is a plane, perpendicular to it. Using models in the studio and animations he shows that these functions specify a position on a plane, which joined continuously give a curve in space. He then examines a straight curve in space in more detail to demonstrate the co-ordinate functions of f. He does this by viewing the model or the function from different angles. Viewing the model straight on gives the set of image positions in the codomain R². This is the path of the function. He now looks at another model representing R to R², starting from the path which is a circle he works back to show a spiral the co-ordinate functions of which are represented by the cosine and sine functions. Peter Strain now discusses functions which map R² to R². He describes the mapping of points as an affine transformation, which can be expressed in matrix notation. By using a series of dilations and rotations he obtains a good representation of the function. He then looks at a function R² to R² which maps horizontal and vertical lines to parabolas. Finally, he reviews the findings of the programme and gives a brief introduction to the idea of continuity. |
| Master spool number: | HOU3236 |
| Production number: | FOUM011J |
| Videofinder number: | 4096 |
| Available to public: | no |
