with(plots): with(VectorCalculus): c1:=plot([cos(t),sin(t),t=0..2*Pi],x=-2..2,y=-2..2, color=black): c2:=plot([8*cos(t^2),8*sin(t^2),t=0..sqrt(2*Pi)], x=-10..10,y=-10..10,color=black): tan1:=arrow(<1,0>,<0,1>,length=1, shape=arrow,width=[7,relative], head_width=.4, color=blue): tan2:=arrow(<0,1>,<-1,0>,length=1, shape=arrow, head_width=.4, color=blue): tan3:=arrow(<-1,0>,<0,-1>,length=1, shape=arrow, head_width=.4, color=blue): tan4:=arrow(<0,-1>,<1,0>,length=1, shape=arrow, head_width=.4, color=blue): tan81:=arrow(<8,0>,<0,1>,length=1, shape=arrow, head_width=.4, color=red): tan82:=arrow(<8*cos(Pi/16),8*sin(Pi/16)>,<-8*sin(Pi/16),8*cos(Pi/16)>,length=1, shape=arrow, head_width=.4, color=red): tan83:=arrow(<8*cos(Pi/8),8*sin(Pi/8)>,<-8*sin(Pi/8),8*cos(Pi/8)>,length=1, shape=arrow, head_width=.4, color=red): tan84:=arrow(<8*cos(3*Pi/16),8*sin(3*Pi/16)>,<-8*sin(3*Pi/16),8*cos(3*Pi/16)>,length=1, shape=arrow, head_width=.4, color=red): Consider two circles: the unit circle and a circle of radius 8. Consider how the tangent vectors for each change over a fixed length of each curve. On each, we look at a length of curve 2\317\200 units long -- that's the whole unit circle, but only 1/8 of the larger circle -- and at tangent vectors placed every \317\200/2 units. display(c1,c2,tan1, tan2, tan3, tan4, tan81,tan82,tan83,tan84);

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Differentiating T(t): T:=t-> <2*t,3,3*t^2>/sqrt(4*t^2+9+9*t^4); diff(T(t),t); subs(t=1,%);