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Use Manipulate to create an interactive plot that shows the parametrically-defined curve described by {x[t], y[t]} = {t Sin[t], t^2/15} for {t,0,10} along with a point of tangency, tangent vector, and tangent line to the curve for any choice of t between 0 and 10.

Clear[t, x, y, P, velvector, vel, scalefactor]; 
scalefactor = 0.5; 
P[t_] = {t Sin[t], t^2/15}; 
curveplot = ParametricPlot[P[t], {t,0,10}, PlotStyle -> Thickness[0.01], 
AxesLabel -> {"x", "y","z"}];
vel[t_] = D[P[t], t]; 
velvector[t_] := Vector[vel[t], Tail -> P[t], VectorColor -> Red, ScaleFactor 
-> scalefactor];
Manipulate[Show[curveplot, velvector[t],{t,0,10}, PlotRange -> All]]

Something not right here, it failed.

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    $\begingroup$ ...and do you have any code showing your attempt at solving this? $\endgroup$ – J. M. will be back soon Jun 28 '16 at 2:01
  • $\begingroup$ Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! $\endgroup$ – Michael E2 Jun 28 '16 at 2:17
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    $\begingroup$ Most users on this site will not be too interested in just doing a problem for someone, especially if it's homework (many of us are educators, so we don't look kindly on people asking us to do their homework). What exactly about this problem is giving you trouble? Is it how to use Manipulate (see the documentation)? Is it how to plot a point, line, and vector on top of a graph (construct the objects using math and use Graphics to make the pictures, then use Show: see the docs)? Is it how to make a parametric plot to begin with (use the docs)? Be specific, and show your work. $\endgroup$ – march Jun 28 '16 at 2:37
  • $\begingroup$ ​Clear[t, x, y, P, velvector, vel, scalefactor]; scalefactor = 0.5; P[t_] = {t Sin[t], t^2/15}; curveplot = ParametricPlot[P[t], {t,0,10}, PlotStyle -> Thickness[0.01], AxesLabel -> {"x", "y","z"}] vel[t_] = D[P[t], t]; velvector[t_] := Vector[vel[t], Tail -> P[t], VectorColor -> Red, ScaleFactor -> scalefactor]; From here I have my vector and my curve. How do i plot it in parametrization of using manipulate with the constraint of t? Manipulate[Show[curveplot, velvector[t],{t,0,3}, PlotRange -> All]] $\endgroup$ – user41325 Jun 28 '16 at 3:02
  • $\begingroup$ Please edit your question to include the code in your comment. $\endgroup$ – J. M. will be back soon Jun 28 '16 at 3:26
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I guess what you are looking for is something like this

scalefactor = 0.5;

P[t_] = {t Sin[t], t^2/15};

curveplot[t0_] := ParametricPlot[P[t], {t, 0, t0},
                  PlotStyle -> Thickness[0.01], AxesLabel -> {"x", "y"}]

vel[t_] = D[P[t], t];
velvector[t0_] := Graphics[{Red, Arrow[{P[t0], P[t0] + scalefactor vel[t0]}]}]

Manipulate[Show[curveplot[t], velvector[t], PlotRange -> {{0, 2}, {0, 1}},
           AspectRatio -> 1], {t, 1, 10}]

enter image description here

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