• Using (x,y) we can specify points, in 2D.

• Can specify a line using
  

  y = mx+c,

  where m is the slope and c is y(0).
• Given an equation like this we can draw it to see the line that it determines.
• Given a line we can determine the equation which describes it.
• The slope is how steeply the line is rising. For every step in the x-direction how many steps in the y-direction must we go up:
  

  m = Dy Dx .

For example if the line is flat, the slope is 0 since Dy = 0. Note that to compute the slope we need two points (x₁,y₁) and (x₂,y₂). Once we pick two points on the line we can get the slope:

m = y2-y1 x2-x1 .

It doesn't matter which two points on the line we take, which makes sense since otherwise it would not be well defined.

• Here the slope is not well defined. It depends on which two points we take. This should be the case since the slope of the curve is always changing!
• We will now tackle this issue, on how to make sense of this problem. In short:
  • The idea of limits needs to be introduced, then we can take the limit as one point goes to the other to effectively have the slope at a point!

• Finally, we are interested in the slope because it lets us determine the max/min of functions and lets us deal with the rate of change of things as well the things themselves.