Sampling Code

(*Interactive code below for sampling*)
Manipulate[
 points = Table[{x, Sin[x]}, {x, 0, 20 Pi, sample}];
 Show[
  Plot[Sin[x],
   {x, 0, 20 Pi},
   Epilog -> {Red, PointSize -> 0.02, Point[points]},
   PlotStyle -> Gray
   ],
  ListLinePlot[points, PlotStyle -> Red]
  ], {sample, 0.1, 2 Pi}
 ]

(*Code for the row of sampling error examples*)
points1 = Table[{x, Sin[x]}, {x, 0, 20 Pi, 0.25}];
points2 = Table[{x, Sin[x]}, {x, 0, 20 Pi, 2.95}];
points3 = Table[{x, Sin[x]}, {x, 0, 20 Pi, 2 Pi}];
GraphicsRow[{
  Show[Plot[Sin[x], {x, 0, 20 Pi},
    Epilog -> {Red, PointSize -> 0.02, Point[points1]},
    PlotStyle -> Gray],
   ListLinePlot[points1, PlotStyle -> Red]
   ],
  Show[Plot[Sin[x], {x, 0, 20 Pi},
    Epilog -> {Red, PointSize -> 0.02, Point[points2]},
    PlotStyle -> Gray],
   ListLinePlot[points2, PlotStyle -> Red]
   ],
  Show[Plot[Sin[x], {x, 0, 20 Pi},
    Epilog -> {Red, PointSize -> 0.02, Point[points3]},
    PlotStyle -> Gray],
   ListLinePlot[points3, PlotStyle -> Red]
   ]
  }, ImageSize -> 1000]