3 kinds of tangent lines/planes

A confusing point is there's -3- ways- to think about the tangent
line.

These correspond to the 3 ways to think about a line or hyperplane:
- $y$ is a function of $x$
  $y=2x+1$
- $x,y$ are both parametrized by $t$
  $x(t)=t-1, y(t)=2t-1$
- it satisfies an equation
  $2x-y = -1$

Geometrically, the first is the worst,
but it's the most familiar.

These --really-- should be distinguished.

- $\bR \to \bR$ ($x \to y$)
  linear approximation
  
  Here $dy = f'(x_0) dx$,
  aka $y - y_0 = f'(x_0) (x-x_0)$
  aka $y = f(x_0) + f'(x_0) (x-x_0)$
  You can also do this for functions with several inputs

- $\bR \to \bR^2$ ($t \to x,y$)
  (or $\bR^2 \to \bR^3$)
  parametric
  $\gamma(t_0 + \Delta t) \approx \gamma(t_0) + \dot \gamma(t_0)\Delta t$

- several inputs, one output;
  implicit
  perpendicular to gradient


Here's how it should be taught:
[clarify this]
The motto is:
** the derivative is the change in output for change in input **
It should NOT be thought of as the slope of the tangent line until
  -secondarily-!!!!!!!
It's -easier- to teach the derivative if you start with parametric
paths, namely the circle!!

- derivatives should -first- be taught as velocities for paths in the
  plane -- this is far more visually understandable
  
  This gives a -parametric- description of the tangent line.

- second, they should be taught as linear approximations
  of a function $\bR \to \bR$,
  and you should say that this is just the -coordinates-
  of a 2-dimensional thing.

  This gives -first- the POV of ``change in output in terms of change
  in input'',
  and then gives a $y-y0=m(x-x0)$ description of the tangent line.

- third, they should be taught as linear approximations of functions
  with several inputs;
  also, if perp to gradient, then don't increase

- thus, fourth, the tangent plane to an -implicit- set can be given
  -implicitly-

The motto:
- given a function, you get a functional description of tangent plane
- given a parametric, you get a parametric description
- given an implicit, you get an implicit description

...and these are all (obviously) related:
- functional is also just parametric, with $x(t)=t$
- implicit is as ``when functional doesn't change''
