> 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?