MS-A0201 Diff. int. 2 (TFM)

Luento 4: Ketjusäännöt ja lineaarinen approksimointi

Harri Hakula  /Kevät 2021

Ketjusääntö

z=f[u[x,y],v[y,t],t]

TreeForm[z]
D[z,t]

Ketjusääntö

z=f[u[x[t],y[t]],v[y[t],t],t]

TreeForm[z]
D[z,t]

Lineaarinen approksimointi

f[x_,y_]:=Sqrt[2x^2+Exp[2y]]
Show[
Plot3D[f[x,y],{x,-3,3},{y,-3,3},PlotStyle->Opacity[.2]],
Graphics3D[{PointSize[0.02],Point[{2,0,f[2,0]}]}],Graphics3D[{Red,PointSize[0.01],Point[{2.2,-0.2,f[2,-.2]}]}],ImageSize->Full
]

f[x_,y_]:=Sqrt[2x^2+Exp[2y]];
dx=D[f[x,y],x];
dy=D[f[x,y],y];
D1f[x_,y_]:=Evaluate[dx]
D2f[x_,y_]:=Evaluate[dy]

a=2;
b=0;Show[Plot3D[f[x,y],{x,1,3},{y,-1,1},PlotStyle->Opacity[.1]],
ContourPlot3D[z==f[2,0]+D1f[a,b](x-a)+D2f[a,b](y-b),{x,-5,5},{y,-5,5},{z,0,5},ContourStyle->Opacity[.3]],
Graphics3D[{PointSize[0.02],Point[{2,0,f[2,0]}]}],Graphics3D[{Red,PointSize[0.01],Point[{2.2,-0.2,f[2.2,-.2]}]}],ImageSize->Full]

a=2;
b=0;Show[Plot3D[f[x,y],{x,1.8,2.2},{y,-.4,0},PlotStyle->Opacity[.1]],
ContourPlot3D[z==f[2,0]+D1f[a,b](x-a)+D2f[a,b](y-b),{x,-5,5},{y,-5,5},{z,0,5},ContourStyle->Opacity[.3]],
Graphics3D[{PointSize[0.02],Point[{2,0,f[2,0]}]}],Graphics3D[{Red,PointSize[0.01],Point[{2.2,-0.2,f[2.2,-.2]}]}],ImageSize->Full]