I teach AP Calculus, both AB and BC, and as many of you (I think ???) I am trying to determine the most effective way of introducing Euler' Method and slopefields. I took Diff Eq in the 70's, but I don't recall a lot of attention given to slopefields ... or Euler's method for that matter.
I would appreciate any comments, corrections, suggestions, etc. that you may care to make.
It seems to me that a slopefield is a diagram of many local linear approximations of the solutions of a given DE through selected points representing associated boundary values. Euler's method seems to take one particular boundary value and construct an approximation to that particular solution ... which also uses local linear approximations to determine successive data points for the approximate solution. This method seems to improve its accuracy as the increment of evaluation decreases.