>
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?