Local Linearity and Tangent Planes

Section 6.6 Local Linearity and Tangent Planes

Objectives

Understand how to geometrically identify when surfaces are locally linear
Identify and compute the necessary information to give the tangent plane to a surface.
Use information about the change on surfaces to compute the total differential
Use the total differential to compute approximations to nonlinear functions

Subsection 6.6.1 Local Linearity

In the lesson on Local Linearity, you saw examples of where zooming into an surface produced a plot that was linear as well as examples where the surface did not become linear.

Activity 6.6.1. Making Locally Linear Plots.

Use a computer program to graph 1) \(f(x,y)=xy^2+1\) and 2) its tangent plane at the point \((2,1,f(2,1))\) in each one of the following rectangles. Make sure to show how you found the equation of the tangent plane. In each case, print the resulting graph (which should have at least the surface, the tangent plane, and the point labeled.)

(a)

\(2 \leq x \leq 3\) and \(1 \leq y \leq 3\)

(b)

\(2 \leq x \leq 2.1\) and \(1 \leq y \leq 1.2\)

(c)

\(2 \leq x \leq 2.01\) and \(1 \leq y \leq 1.02\)

(d)

Write a few sentences about what you observe in terms of the relationships between the graphs for the previous three parts.

Subsection 6.6.2 Tangent Planes

In the lesson, we primarily examined tangent planes for multivariable functions of the form \(z=f(x,y)\text{.}\) We learned that the tangent plane for these functions at a point \((a,b)\) is given by

\begin{equation*} z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b) (y-b) \text{.} \end{equation*}

Activity 6.6.2.

(a)

Use the notion of vertical change on a plane to find the equation of the plane represented by the following figure:

(b)

Use the notion of vertical change on a plane to find the equation of the plane represented in the following table:

Table 6.6.1.

	y=0	y=3	y=6
x=2	18	20	22
x=4	15	17	19
x=6	12	14	16

(c)

The following is a graph of the tangent plane to the graph of \(z=f(x,y)\) at the point \((2,7,8)\text{.}\) Find \(f_x (2,7)\text{,}\) \(f_y (2,7)\text{,}\) and then given the equation of the tangent plane.

(d)

Find the equation of the tangent plane to the graph of \(f(x,y)=\frac{1}{x+2y}\) at the point \((2,1,f(2,1))\text{.}\)

Activity 6.6.3.

We would like to extend the tangent plane idea to implicit surfaces. That is, surfaces that can be represented by an equation of the form

\begin{equation*} f(x,y,z)=0\text{.} \end{equation*}

Consider the surface given by the equation

\begin{equation*} x^2 - y^2 + z^2-zx - 2 = 0\text{.} \end{equation*}

Our goal is to find the tangent plane at the point \((-2,2,1)\text{.}\)

(a)

Find the gradient for the function defining the implicit surface.

Hint.

Consider equation as a function of three variables given by \(f(x,y,z) = x^2-y^2+z^2-zx-2\) and find

\begin{equation*} \nabla f = \langle f_x, f_y, f_z \rangle\text{.} \end{equation*}

(b)

Find the value of the gradient at the given point \((-2,2,1)\) and add the gradient to the graph below.

(c)

What do you notice about the gradient and the its relationship to the surface? Why is this the case?

Hint.

Recall the geometric properties of the gradient, in particular, those that apply to contour plots. Now consider the equation

\begin{equation*} x^2 - y^2 + z^2-zx - 2 = 0 \end{equation*}

as a level curve of the function \(f\text{.}\)

(d)

Use the gradient and the point to find the equation of the tangent plane.

Subsection 6.6.3 Total Differential

Activity 6.6.4.

Let \(f(x,y)=x^3y\text{.}\)

(a)

Find the equation of the tangent plane to the graph of \(f\) at the point \((1,2)\) and express it in the form \(z-f(a,b)=f_x(a,b) (x-a)+f_y(a,b) (y-b)\text{.}\)

(b)

Use your equation from the previous part to find the differential of \(f\) at the point \((1,2)\text{.}\)

(c)

Evaluate the differential found in the previous part for \(dx=0.2\) and \(dy=-0.3\text{.}\) Also evaluate the differential for \(dx=3\) and \(dy=4\text{.}\)

Activity 6.6.5.

The following is a table of the tangent plane to the graph of \(z=f(x,y)\) at the point \((1,2,5)\text{.}\)

Table 6.6.2.

	y=1	y=2	y=3
x=0	4	7	10
x=1	2	5	8
x=2	0	3	6

(a)

Find the equation of the tangent plane to the graph of \(f\) at the point \((1,2)\) and express it in the form \(z-f(a,b)=f_x(a,b) (x-a)+f_y(a,b) (y-b)\text{.}\)

(b)

Use ihe information from the previous part to find the differential of \(f\) at the point \((1,2)\text{.}\)

(c)

Evaluate the differential found in the previous part for \(dx=0.2\) and \(dy=-0.3\text{.}\) Also evaluate the differential for \(dx=-5\) and \(dy=1\text{.}\)

Activity 6.6.6.

The following is a table of the tangent plane to the graph of \(z=f(x,y)\) at the point \((3,0,11)\text{.}\)

(a)

Find the equation of the tangent plane to the graph of \(f\) at the point \((3,0)\) and express it in the form \(z-f(a,b)=f_x(a,b) (x-a)+f_y(a,b) (y-b)\text{.}\)

(b)

Use your equation from the previous part to find the differential of \(f\) at the point \((3,0)\text{.}\)

(c)

Evaluate the differential found in the previous part for \(dx=0.2\) and \(dy=-0.3\text{.}\) Also evaluate the differential for \(dx=-4\) and \(dy=1\text{.}\)

Activity 6.6.7.

Let \(f(x,y)=xy^2+1\text{.}\)

(a)

Find \(df(x,y)\text{.}\)

(b)

Find \(df(2,1)\text{.}\)

(c)

Fill in the following table.

Table 6.6.3.

Horizontal Change	Use the differential to find the vertical change on the tangent plane to the graph of f at the point	Vertical change on the graph of the function	% error (from actual output of \(f\))
dx=1, dy=2
dx=0.1, dy=0.2
dx=0.01, dy=0.02

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