Average Rate of Change

Code

graphics[t1_,t2_]:=(
v1=r[t1];
v2 = r[t2];
point1={Blue,PointSize[0.02],Point[v1]};
point2={Red,PointSize[0.02],Point[v2]};
rateVector=(v2-v1)/(t2-t1);
rateArrow={Blue,Thickness[0.005],Arrow[{v1,v1+rateVector}]};Show[Graphics3D[{point1,point2,rateArrow,Text[x,{9,0,0}],Text[y,{0,9,0}],Text[z,{0,0,13}]}],ParametricPlot3D[r[s],{s,0,4 Pi},PlotStyle->{Orange,Thickness[0.005]}],Axes->True,AxesOrigin->{0,0,0},PlotRange->{{-8,8},{-8,8},{0,12}},Boxed->False,ViewPoint->{20,4,8}]
)

Graphics

r[t_]:={8 Cos[t],8 Sin[t],t};
t1 = 1.95Pi;
t2 = 1.95Pi+0.5;

graphics[t1,t2]

Derivative at a Point as a Limit

Code

graphics[tValue_,h_]:=(v1=r[tValue];
v2 = r[tValue+h];
point1={Blue,PointSize[0.02],Point[v1]};
point2={Red,PointSize[0.02],Point[v2]};
rateVector=(v2-v1)/h;
rateArrow={Blue,Thickness[0.005],Arrow[{v1,v1+rateVector}]};Show[Graphics3D[{point1,point2,rateArrow,Text[x,{9,0,0}],Text[y,{0,9,0}],Text[z,{0,0,13}]}],ParametricPlot3D[r[s],{s,0,4 Pi},PlotStyle->{Orange,Thickness[0.005]}],Axes->True,AxesOrigin->{0,0,0},PlotRange->{{-8,8},{-8,8},{0,12}},Boxed->False,ViewPoint->{20,4,8}])

Graphics

r[t_]:={8 Cos[t],8 Sin[t],t};
t = 2Pi;

frames=Table[Image[graphics[t,h]],{h,1,0.01,-0.04}];
ListAnimate[frames]

The Derivative Function

Code

graphics[tValue_]:=(v1=r[tValue];
v2 = r'[tValue];
v3 = r''[tValue];
point={Blue,PointSize[0.02],Point[v1]};
originArrow=If[showDerivAtOrigin,{Blue,Thickness[0.005],Arrow[{{0,0,0},v2}]},{}];
graphArrow=If[showDerivOnGraph,{Blue,Thickness[0.005],Arrow[{v1,v1+v2}]},{}];
secondDerivArrow=If[showSecondDeriv,{Red,Thickness[0.005],Arrow[{v1,v1+v3}]},{}];
(*text = If[showSecondDeriv,{"The ",Style["first",Blue]," and ",Style["second",Red]," derivative at "},{"The ",Style["first derivative",Blue]," at "}];*)
Show[Graphics3D[{point,graphArrow,originArrow,secondDerivArrow,Text[x,{6,0,0}],Text[y,{0,6,0}],Text[z,{0,0,6}]}],ParametricPlot3D[r[s],{s,parameterLeft,parameterRight},PlotStyle->{Orange,Thickness[0.008]}],Axes->True,AxesOrigin->{0,0,0},Boxed->False,PlotRange->plotRange,ViewPoint->{20,4,8}])

Graphics

r[t_]:={t^2,3-t^2,t^3};
showDerivOnGraph=True;
showDerivAtOrigin=False;
showSecondDeriv = False;

(*Display Parameters*)
plotRange={{-5,5},{-5,5},{-5,5}};
parameterLeft=-2;
parameterRight=2;

Show[graphics[1],ParametricPlot3D[{1+2t,2-2t,1+3t},{t,-2,2},PlotStyle->Blue]]

Derivatives and Parameterizations

Code

graphics[tValue_]:=(
v1=r[tValue];
v2 = r'[tValue];
v3 = r''[tValue];
r2[t_]:=r[secondParameterizationSpeed*t];
u1 = r2[tValue];
u2 = r2'[tValue];
u3 = r2''[tValue];
point1={Blue,PointSize[0.02],Point[v1]};
derivArrow1=If[showFirstDeriv,{Blue,Thickness[0.005],Arrow[{v1,v1+v2}]},{}];
secondDerivArrow1=If[showSecondDeriv,{Red,Thickness[0.005],Arrow[{v1,v1+v3}]},{}];
point2={RGBColor[0,0,0.5],PointSize[0.02],Point[u1]};
derivArrow2=If[showFirstDeriv,{RGBColor[0,0,0.5],Thickness[0.005],Arrow[{u1,u1+u2}]},{}];
secondDerivArrow2=If[showSecondDeriv,{RGBColor[0.5,0,0],Thickness[0.005],Arrow[{u1,u1+u3}]},{}];
(*text = If[second,If[first,"The first and second derivatives","The second derivatives"],If[first,"The first derivatives",""]];*)
Show[Graphics3D[{point1,point2,derivArrow1,derivArrow2,secondDerivArrow1,secondDerivArrow2,Text[x,{3.2,0,0}],Text[y,{0,3.2,0}],Text[z,{0,0,3.2}]}],ParametricPlot3D[r[s],{s,0,4 Pi},PlotStyle->{Orange,Thickness[0.008]}],Axes->True,AxesOrigin->{0,0,0},PlotRange->{{-3,3},{-3,3},{-3,3}},Boxed->False,ViewPoint->{20,4,8}]
)

Graphics

secondParameterizationSpeed controls how fast the second parameterization is compared to r(t) itself.

r[t_]:={Cos[t/2],Sin[t/2],Sin[t]};
showFirstDeriv=True;
showSecondDeriv =False;
secondParameterizationSpeed=2;

frames=Table[Image[graphics[t]],{t,0,4 Pi,Pi/9}];
ListAnimate[frames]