Arguably, the three central concepts in the theory of differential calculus are that of a function, that of a tangent and that of a limit. Here we introduce functions and tangents.

## Functions

When looking at differential calculus, two good ways to think about functions are via algebraic geometry and interdependent variables. Neither give the proper, abstract, definition of a function, but both give a nice way of thinking about them.

### Algebraic Geometry Approach

Let us set up the plane, . We choose a distinguished point called the origin and a distinguished direction which we call ‘positive ‘. Draw a line through the origin in the direction of positive . This is the -axis. Choose a unit distance for the -direction.

Now, perpendicular to the -axis, draw a line through the origin. This is the -axis. By convention positive is anti-clockwise of positive . Choose a unit distance for the -direction.

This is the plane, :

Now points on the plane can be associated with a pair of numbers . For example, the point a distance one along the positive and five along the negative can be denoted by the coordinates (1,-5):

Similarly, I can take a pair of numbers, say (-1,3), and this corresponds to a point on the plane.

This gives a duality:

points on the plane $lates \Leftrightarrow$ pairs of numbers

Now consider the completely algebraic objects

.

## Recent Comments