> 
Curve Examples with Maple

> with(plots);
> plot( [cos(2*t),sin(2*t), t=0..Pi],-3..3,-3..3);
That shows the obvious thing:  a circle traversed once.
Now let's see a tangent line at some point, say  t=0.   At t=0, the curve
is at the point  (1,0), as this little computation shows:

> [cos(0),sin(0)];
The velocity vector at any time  t  is given by the derivatives.

> vel := [diff(cos(t),t), diff(sin(t),t) ];
Thus   the velocity vector  at time  t=0   is given by substituting  t=0  into
this expression:
 
> subs(t=0,vel);
Let's get a numerical value:

> evalf( % );

Now we can find the tangent line, since we have a point on it and a vector
giving direction.       The tangent line   has vector equation    

             X = (1,0) + t(0,1)

In scalar form, the line has equations   x=1,  y=t.   Now let's plot
both the tangent line and the curve together.


> plot( { [cos(t),sin(t),t=0..2*Pi], [1,t,t=0..1]} );
 


Let's do something similar with a space curve.     First, here's the curve
itself:


> spacecurve( [cos(t),sin(t),t, t=0..10] , color=black,axes=framed );

Does the curve look reasonable?    If so, let's think about drawing a tangent line, 
say at the point t=Pi/2.     

> subs( t=Pi/2, [cos(t),sin(t),t] );
> evalf(%);
> vel := diff( [cos(t),sin(t),t], t );


Note that diff  worked as you'd think --- it differentiated each of the parts
separately.


> subs( t=Pi/2, vel);
> evalf(%);


Now we have the necessary information to write down the tangent line.   It has
the form    X = (0,1,Pi/2) + t (-1,0,1).    In scalar form, that's

     x=-t, y=1, z=Pi/2+t

Let's try it out graphically:


> spacecurve({[cos(t),sin(t),t,t=0..2*Pi],[-t,1,Pi/2+t,t=-1..1]}, axes = framed,color=black );
> 


Looks plausible.       

Do these:

1.   (2-d problem)  Let the curve be the unit circle, as  above.   Use
      Maple to draw both the curve and the tangent line at the ``north pole.''

 
2.   (2-d problem)  Let the curve be the unit circle, as above.   Use
      Maple to draw both the curve and the tangent line at  t=1.   What's 
     the speed at   t=1?


3.  ( 2-d problem)  Use Maple to draw both the curve and the tangent line  at t=0   for
      the curve   

             X(t) = ( cos(t), sin(2*t) )  ,  t = 0 .. 2*Pi .
                              
      What's the speed at t=0?

4.  ( 3-d problem)  Use Maple to draw both the curve and the tangent line  at t=0   for
      the curve   

             X(t) = ( cos(t), sin(2*t) , t)  ,  t = 0 .. 2*Pi .

      (This curve is ALMOST, but not quite, like the one illustrated above.)
                         
      What's the speed at t=0?

5.  ( 3-d problem)  Use Maple to draw both the curve and the tangent line  at t=1   for
      the curve   

             X(t) = ( cos(t), sin(2*t) , t)  ,  t = 0 .. 2*Pi .

      (This curve is ALMOST, but not quite, like the one illustrated above.)
                         
      What's the speed at t=0?