t = Pi;

    
Plot[Sin[x] Cos[t], {x, 0, Pi}, PlotRange -> {-1.25, 1.25}, 
        PlotStyle -> {Thickness[0.015], RGBColor[0, 0, 1]}, 
        AspectRatio ->   .3, ImageSize -> 400, Axes -> False, 
        Frame -> True];

Clear[t]; x0 = 2; ff[x_, t_] := Sin[x] Cos[t]; 
    ffx[x_, t_] := Cos[x] Cos[t];
Table[
Plot[{ff[x, t]}, {x, 0, Pi}, 
          PlotRange -> {-1.25, 1.25}, 
          PlotStyle -> {{Thickness[0.015], RGBColor[0, 0, 1]}}, 
          AspectRatio ->   .3, ImageSize -> 400, 
          Axes -> False, Frame -> True, 
          Epilog -> {{PointSize[0.02], Red, 
                Point[{x0,  Sin[x0] Cos[t]}]}}, 
          PlotLabel -> N[t]], {t, 0, 2 Pi, (Pi/100)}];

Clear[t]; x0 = 2; ff[x_, t_] := Sin[x] Cos[t]; 
    ffx[x_, t_] := Cos[x] Cos[t];
Table[
Plot[{ff[x, t], 
            ffx[x0, t] (x - x0) + ff[x0, t]}, {x, 0, Pi}, 
          PlotRange -> {-1.25, 1.25}, 
          PlotStyle -> {{Thickness[0.015], 
                RGBColor[0, 0, 1]}, {Thickness[0.007], RGBColor[1, 0, 0]}}, 
          AspectRatio ->   .3, ImageSize -> 400, 
          Axes -> False, Frame -> True, 
          Epilog -> {{PointSize[0.02], Red, 
                Point[{x0,  Sin[x0] Cos[t]}]}}, 
          PlotLabel -> N[t]], {t, 0, 2 Pi, (Pi/100)}];

Plot3D[Sin[x] Cos[t], {x, 0, Pi}, {t, 0, 2 Pi}, 
        PlotRange -> {-1.1, 1.2}, ImageSize -> 400, 
        Axes -> True, 
        Ticks -> {Range[-2 Pi, 2 Pi, (Pi/2)], 
            Range[-2 Pi, 2 Pi, (Pi/2)], Automatic}];

ContourPlot[Sin[x] Cos[t], {x, 0, Pi}, {t, 0, 2 Pi}, 
        PlotRange -> {-1.1, 1.2}, ImageSize -> 300, 
        Axes -> True, Contours -> 25, 
        PlotPoints -> {100, 200}, 
        Ticks -> {Range[-2 Pi, 2 Pi, (Pi/2)], 
            Range[-2 Pi, 2 Pi, (Pi/2)], Automatic}];

Simplify[D[x Exp[(Sqrt[x y])], x]]

Fs[x_, t_] := Sin[x] Cos[t]

Fs[1, 1]

Clear[t, x, Fs];
Fs[x_, t_] := Sin[x] Cos[t]; t0 = 1; 
    x0 = 1;
bb[t_] = Pi Sin[t];
Plot[Fs[x, t0], {x, 0, Pi}, PlotRange -> {-1.25, 1.25}, 
        PlotStyle -> {Thickness[0.015], RGBColor[0, 0, 1]}, 
        Epilog -> {{PointSize[0.02], Red, 
              Point[{bb[t0], Fs[bb[t0], t0]}]}}, AspectRatio ->   .3, 
        ImageSize -> 400, Axes -> False, 
        Frame -> True];

Clear[t];
Clear[t, x, Fs];
Fs[x_, t_] := Sin[x] Cos[t]; Clear[t0];
bb[t_] = Pi Sin[t/2]^2;

    
Table[
Plot[Fs[x, t0], {x, 0, Pi}, 
            PlotRange -> {-1.25, 1.25}, 
            PlotStyle -> {Thickness[0.015], RGBColor[0, 0, 1]}, 
            Epilog -> {{PointSize[0.02], Red, 
                  Point[{bb[t0], Fs[bb[t0], t0]}]}}, 
            AspectRatio ->   .3, ImageSize -> 400, 
            Axes -> False, Frame -> True];, {t0, 0, 2 Pi, 
          (Pi/100)}];

Clear[t];
Clear[t, x, Fs];
Fs[x_, t_] := Sin[x] Cos[t]; Clear[t0];
bb[t_] = 0.3 + 2.7 Sin[t/2]^2;

    
Table[
Plot[Fs[x, t0], {x, 0, Pi}, 
            PlotRange -> {-1.25, 1.25}, 
            PlotStyle -> {Thickness[0.015], RGBColor[0, 0, 1]}, 
            Epilog -> {{PointSize[0.02], Red, 
                  Point[{bb[t0], Fs[bb[t0], t0]}]}}, 
            AspectRatio ->   .3, ImageSize -> 400, 
            Axes -> False, Frame -> True];, {t0, 0, 2 Pi, 
          (Pi/100)}];

Pictures for the website

Clear[t, st, t0, x0];
Clear[t, x, Fs, tbl];
Fs[x_, t_] := Sin[x] Cos[t]; 
    Clear[t0, x0];

\(StringSc[x0_, 
        t0_] := \ \ 
Module[{st, tbl, ar, df}, \ 
        st = Pi\/64; \ th1 = 0.005; ps1 =  .02; ar =  .3; is = 400; \ 
        df = \ $DisplayFunction (*\ $DisplayFunction\ ; \ 
          Identity\ \ *) ; 
tbl = 
          Range[Max[0, t0 - 10\ st], t0, 
            st]; 

If[
          t0 == 0, 
Plot[
            Evaluate[\(Fs[x, #] &\) /@ tbl], {x, 0, Pi}, 
            PlotRange -> {\(-1.15\), 1.15}, \ PlotPoints -> 300, 
            PlotStyle -> Evaluate[{Thickness[th1], GrayLevel[0]}], 
            Epilog -> {{PointSize[ps1], GrayLevel[0], 
                  Point[{x0, Fs[x0, 0]}]}}, AspectRatio -> \ ar, 
            ImageSize -> is, Axes -> False, Frame -> True, 
            FrameTicks -> {Range[0, Pi, Pi\/4], 
                Range[\(-1\), 1], {}, {}}, 
            PlotRegion -> {{\(-0.05\), 1}, {0,  .98}}, 
            DisplayFunction -> df], 


          Plot[Evaluate[\(Fs[x, #] &\) /@ tbl], {x, 0, Pi}, 
            PlotRange -> {\(-1.15\), 1.15}, \ PlotPoints -> 300, 
            PlotStyle -> 
              Evaluate[\({Thickness[th1], 
                      GrayLevel[\((\(t0 - #\)\/\(t0 - tbl[[1]]\))\)\^\(1/2\)]} &\) /@ tbl], 
            Epilog -> {\({PointSize[ps1], 
                      GrayLevel[\((\(t0 - #\)\/\(t0 - tbl[[1]]\))\)\^\(1/2\)], 
                      Point[{x0, Fs[x0, #]}]} &\) /@ tbl}, 
            AspectRatio -> \ ar, ImageSize -> is, 
            Axes -> False, Frame -> True, 
            FrameTicks -> {Append[
                  Range[0, Pi, Pi\/4], {x0, \*""(Subscript[x, 0])""}], 
                Range[\(-1\), 1], {}, {}}, 
            PlotRegion -> {{\(-0.05\), 1}, {0,  .98}}, 
            DisplayFunction -> df, 
            PlotLabel -> \ 
              TableForm[{"t", "=", \*""(Subscript[t, 0])"" (*\ 
                    NumberForm[N[t0 - 2  Pi], {4, 3}, 
                      NumberPadding -> {"", "0"}]*) }, 
                TableDirections -> Row, 
                TableSpacing ->  .1]\ , 
TextStyle -> {FontSize -> 8}
]]
]; 
    StringSc[2, 83  Pi/64]\)

Clear[StringPic1]; 
    StringPic1 = StringSc[1, 45  (Pi/64) + 2 Pi]

StringMovie = 
        Table[StringSc[1, t], {t, 2 Pi, 4  Pi, (Pi/64)}];

SetDirectory["C:\\Dropbox\\Work\\COURSES\\224"]

SetDirectory["C:\\Dropbox\\Work\\myweb\\Courses\\224_201210"]

Export["StringPic1.eps", StringPic1, "EPS"]

Export["M2n1.gif", M2n1, "GIF", 
        ConversionOptions -> {"ImageSize" -> 
              400}];

Export["StringMovie.gif", StringMovie, "GIF", 
      ConversionOptions -> {"Loop" ->  
            True, "ImageSize" -> 
            1200, "AnimationDisplayTime" -> 0.275}]

StringEP1 = StringSc[2, 83  Pi/64]

StringEP2 = StringSc[1, 22  Pi/64]

Export["StringEP1.eps", StringEP1, "EPS"]; 
    Export["StringEP2.eps", StringEP2, "EPS"]