Math 1C with Jeff Anderson
Part 1: Multidimensional Differential Calculus
Lesson 12: Tangent Planes
©Jeffrey A. Anderson

Key Terms List: Briggs Section 12.7 p. 928
□  Definition: Equation of tangent plane for F(x,y,z) = 0 (p. 930)
□  Equation of tangent plane for z = f(x,y) (p. 931)
□  Definition: Linear Approximation of f(x,y) at (a,b) (p. 931)
□  Definition: The Differential dz (p. 933)

Visualizing Tangent Planes Using Mathematica

Example 12.7.2 p. 931

Let’s consider the function

f(x,y) = 2(x^2)+(y^2)

We begin this example by visualizing the surface given by f(x,y).

f[x_,y_]=32 -3x^2-4 y^2;

Let’s take a look at this function around the rectangular domain

D = {(x,y) : -2 ≤ x ≤ 2, -2 ≤ y ≤ 2 }

We will not define our PlotRange specifically and let Mathematica automatically determine the best viewing range:

Plot3D[f[x,y],{x,-3,3},{y,-3,3}]

Let’s control the PlotRange using the appropriate option:

Plot3D[
	f[x,y],
	{x,-4,4},
	{y,-4,4},
	PlotRange->{-5, 32},
	BoxRatios->{1, 1, 1}
]

Now, let’s take the partial derivatives of f with respect to x and y:

fx[x_,y_]=D[f[x,y],x];
fy[x_,y_]=D[f[x,y],y];

We could have equivalently defined the partial deriviatves using the limit definition:

Manipulate[
	x0=p[[1]];
	y0=p[[2]];
	Show[
	{
		Plot3D[
		{
		f[x,y],
		f[x0,y0]+fx[x0,y0] (x-x0)+fy[x0,y0] (y-y0)
		},
		{x,-3,3},
		{y,-3,3},
		BoxRatios->{1,1,1},
		PlotRange->{{-4,4},{-4,4},{0,40}},
		PlotStyle->{Directive[Opacity[0.6]],
					Directive[Orange,Opacity[0.6]]},
		ViewPoint->{2.5,-2,1},
		ClippingStyle->None],
		Graphics3D[
		{
		PointSize[Large],
		Point[{p[[1]],p[[2]],f[p[[1]],p[[2]]]}]
		}
		]
	}
	],
{{p,{2,2}},{-2,-2},{2,2}}
]