***************2-D  STUFF **************

The parametric equations are input here, with the limits on the parameter "t":

\(Clear[starttime, endtime, x, y, t]; 
x[t_]\  = \ Cos[t]; 

    y[t_]\  = \ Sin[t]; 
starttime\  = \ 0; 
endtime\  = \ 2*Pi; \)

The next cell will generate the graphics; directions for animating these are below.

\(\(
Clear[big, nplots]; 

    big = 1.1*
        Max[
		
          Table[N[Abs[x[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}], 
		
          Table[N[Abs[y[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}]
	]; 
nplots\  = \ 10; 

    Do[Show[
ParametricPlot[{x[t], y[t]}, {t, starttime, 
            starttime + \((i/nplots)\)*\((endtime - starttime)\)}, 
          AspectRatio -> 1, PlotRange -> {{\(-big\), big}, {\(-big\), big}}, 
          PlotStyle -> {RGBColor[1, 0, 0], Thickness[0.01]}, 
          DisplayFunction -> Identity], 
        Graphics[

          			{PointSize[0.04], 
            Point[{x[starttime + \((i/nplots)\)*\((endtime - starttime)\)], 

                						\ \ \ \ \ \ \ \ \ \ \ 
                y[starttime + \((i/nplots)\)*\((endtime - starttime)\)]}]
              \ \ \ }], DisplayFunction -> $DisplayFunction], 

      	{i, 1, nplots}]\)\)

To animate these graphics, close the cells by double-clicking on the middle cell bracket (only one graph will be showing).  While these cells are selected, go to the "Cell" menu above and choose "Animate Selected Graphics".

An extra copy of these cells is contained below, so you can edit them without fear of losing the originals.

*******************************************

The parametric equations are input here, with the limits on the parameter "t":

\(Clear[starttime, endtime, x, y, t]; 
x[t_]\  = \ Cos[2*t]; 

    y[t_]\  = \ 5*Sin[2*t]; 
starttime\  = \ 0; 
endtime\  = \ 2*Pi; 


    Clear[big, nplots]; 

    big = 1.1*
        Max[
		
          Table[N[Abs[x[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}], 
		
          Table[N[Abs[y[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}]
	]; 
nplots\  = \ 10; 

    Do[Show[
ParametricPlot[{x[t], y[t]}, {t, starttime, 
            starttime + \((i/nplots)\)*\((endtime - starttime)\)}, 
          AspectRatio -> 1, PlotRange -> {{\(-big\), big}, {\(-big\), big}}, 
          PlotStyle -> {RGBColor[1, 0, 0], Thickness[0.01]}, 
          DisplayFunction -> Identity], 
        Graphics[

          			{PointSize[0.04], 
            Point[{x[starttime + \((i/nplots)\)*\((endtime - starttime)\)], 

                						\ \ \ \ \ \ \ \ \ \ \ 
                y[starttime + \((i/nplots)\)*\((endtime - starttime)\)]}]
              \ \ \ }], DisplayFunction -> $DisplayFunction], 

      	{i, 1, nplots}]\)

*******************************************

The parametric equations are input here, with the limits on the parameter "t":

\(\(
Clear[starttime, endtime, x, y, t, a, b, c]; 

a = 7.0; 

    b = 6.0; 
c = 1.0; 

    x[t_]\  = \ \((a - b)\)\ Cos[t] + c\ Cos[\((1 - a\/b)\)\ t]; 

    y[t_]\  = \ \((a - b)\)\ Sin[t] + c\ Sin[\((1 - a\/b)\)\ t]; 

    starttime\  = \ 0; 
endtime\  = \ 12*Pi; 

Clear[big, nplots]; 

    big = 1.1*
        Max[
		
          Table[N[Abs[x[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}], 
		
          Table[N[Abs[y[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}]
	]; 
nplots\  = \ 10; 

    Do[Show[
ParametricPlot[{x[t], y[t]}, {t, starttime, 
            starttime + \((i/nplots)\)*\((endtime - starttime)\)}, 
          AspectRatio -> 1, PlotRange -> {{\(-big\), big}, {\(-big\), big}}, 
          PlotStyle -> {RGBColor[1, 0, 0], Thickness[0.01]}, 
          DisplayFunction -> Identity], 
        Graphics[

          			{PointSize[0.04], 
            Point[{x[starttime + \((i/nplots)\)*\((endtime - starttime)\)], 

                						\ \ \ \ \ \ \ \ \ \ \ 
                y[starttime + \((i/nplots)\)*\((endtime - starttime)\)]}]
              \ \ \ }], DisplayFunction -> $DisplayFunction], 

      	{i, 1, nplots}]\)\)

***************3-D STUFF **************

\(Clear[starttime, endtime, x, y, z, t]; 
x[t_]\  = \ 2.0 + 3*t; 

    y[t_]\  = \ 5.0*t; 
z[t_]\  = \ 7; 
starttime\  = \ \(-3\); 

    endtime\  = \ 5; 

Clear[big, nplots]; 

    big = 1.1*
        Max[
		
          Table[N[Abs[x[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}], 
		
          Table[N[Abs[y[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}], 
		
          Table[N[Abs[z[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}]
	]; 
nplots\  = \ 10; 

    Do[Show[
ParametricPlot3D[{x[t], y[t], z[t], 
            Hue[t/\((\((i/nplots)\)*\((endtime - starttime)\))\)]}, {t, 
            starttime, starttime + \((i/nplots)\)*\((endtime - starttime)\)}, 
          AspectRatio -> 1, 
          PlotRange -> {{\(-big\), big}, {\(-big\), big}, {\(-big\), big}}, 
          DisplayFunction -> Identity, ViewPoint -> {2, \(-1\), 1}], 
        Graphics3D[

          			{PointSize[0.04], 
            Point[{x[starttime + \((i/nplots)\)*\((endtime - starttime)\)], 

                						\ \ \ \ \ \ \ \ 
                y[starttime + \((i/nplots)\)*\((endtime - starttime)\)], 

                					\ \ \ \ \ \ \ \ 
                z[starttime + \((i/nplots)\)*\((endtime - starttime)\)]}]
              \ \ \ }], DisplayFunction -> $DisplayFunction], 

      	{i, 1, nplots}]\)

\(Clear[starttime, endtime, x, y, z, t]; 
x[t_]\  = \ Cos[t]; 

    y[t_]\  = \ Sin[t]; 
z[t_]\  = \ t/10; 
starttime\  = \ \(-2\)*Pi; 

    endtime\  = \ 2*Pi; 

Clear[big, nplots]; 

    big = 1.1*
        Max[
		
          Table[N[Abs[x[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}], 
		
          Table[N[Abs[y[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}], 
		
          Table[N[Abs[z[starttime + i/100*\((endtime - starttime)\)]]], {i, 
              0, 100}]
	]; 
nplots\  = \ 10; 

    Do[Show[
ParametricPlot3D[{x[t], y[t], z[t], 
            Hue[t/\((\((i/nplots)\)*\((endtime - starttime)\))\)]}, {t, 
            starttime, starttime + \((i/nplots)\)*\((endtime - starttime)\)}, 
          AspectRatio -> 1, 
          PlotRange -> {{\(-big\), big}, {\(-big\), big}, {\(-big\), big}}, 
          DisplayFunction -> Identity, ViewPoint -> {2, \(-1\), 1}], 
        Graphics3D[

          			{PointSize[0.04], 
            Point[{x[starttime + \((i/nplots)\)*\((endtime - starttime)\)], 

                						\ \ \ \ \ \ \ \ 
                y[starttime + \((i/nplots)\)*\((endtime - starttime)\)], 

                					\ \ \ \ \ \ \ \ 
                z[starttime + \((i/nplots)\)*\((endtime - starttime)\)]}]
              \ \ \ }], DisplayFunction -> $DisplayFunction], 

      	{i, 1, nplots}]\)

**************Other  STUFF *************

\(Clear[r, v, speed, a, t, ω]; 

    r[t_]\  = \ {Cos[ω*t], Sin[ω*t]}; 


    v[t_]\  = \ D[r[t], t]; 
speed[t_]\  = \ Simplify[Sqrt[v[t] . v[t]]]; 

    a[t_]\  = \ D[r[t], {t, 2}]; 

    Print["r", " = ", \ MatrixForm[r[t]]]; 

    Print["v", " = ", \ MatrixForm[v[t]]]; 

    Print["speed", " = ", \ speed[t]]; 

    Print["a", " = ", \ MatrixForm[a[t]]]; \)

\(Clear[r, v, speed, a, t, ω]; 

    r[t_]\  = \ {Cos[ω*t], Sin[ω*t], t/10}; 


    v[t_]\  = \ D[r[t], t]; 
speed[t_]\  = \ Simplify[Sqrt[v[t] . v[t]]]; 

    a[t_]\  = \ D[r[t], {t, 2}]; 

    Print["r", " = ", \ MatrixForm[r[t]]]; 

    Print["v", " = ", \ MatrixForm[v[t]]]; 

    Print["speed", " = ", \ speed[t]]; 

    Print["a", " = ", \ MatrixForm[a[t]]]; \)

\(\(
Clear[r, v, speed, a, t, ω]; 

    r[t_]\  = \ {Cos[ω*t], Sin[ω*t]}; 


    v[t_]\  = \ D[r[t], t]; 
speed[t_]\  = \ Simplify[Sqrt[v[t] . v[t]]]; 

    a[t_]\  = \ D[r[t], {t, 2}]; 

    Print["r", " = ", \ MatrixForm[r[t]]]; 

    Print["v", " = ", \ MatrixForm[v[t]]]; 

    Print["speed", " = ", \ speed[t]]; 

    Print["a", " = ", \ MatrixForm[a[t]]]; 
ω\  = \ 2; 

    ParametricPlot[r[t], {t, 0, 2}, AspectRatio -> Automatic]; \)\)