Tangent Planes To Surfaces

Tangent Plane to f(x,y) = Log(2x+y) at (-1,3,0):

Define Function f(x,y) :

f[x_,y_]= Log[2*x+y];

Define Tangent Plane p(x,y) :

p[x_,y_]:=2*x+y-1;

Plot Surface z = f(x,y) near point of tangency (-1,3,0):

SurfPlot = Plot3D[f[x,y],{x,-2,0},{y,2,4},PlotStyle->Blue]

Plot Plane z = p(x,y) near same point.

TanPlot = Plot3D[p[x,y],{x,-2,0},{y,2,4},PlotStyle->Yellow]

Show Plots On Same Graph:

Show[TanPlot,SurfPlot,PlotRange->{-3,4},AxesLabel->{"x","y","z"}, PlotLabel->"Tangent Plane to the Surface z = Ln[2x+y] at (-1,3,0)"]

Tangent Plane to z = f(x,y) = (x^2)+(y^2) near (1,2,5) :

Define Function f(x,y) :

f[x_,y_]= x^2+y^2;

Define Tangent Plane p(x,y) :

p[x_,y_]:=2*x+4y-5;

Plot Surface z = f(x,y) :

SurfPlot = Plot3D[f[x,y],{x,-3,3},{y,-3,3},PlotStyle->Yellow]

Plot Plane z = p(x,y) near point of tangency (1,2,5) :

TanPlot = Plot3D[p[x,y],{x,0,2},{y,1,3},PlotStyle->Red]

Show Plots On Same Graph:

Show[TanPlot,SurfPlot,PlotRange->{0,9},AxesLabel->{"x","y","z"}, PlotLabel->"Tangent Plane to the Surface z = x^2 + 2y^2 at (1,2,5)"]