Is finding a tangent plan similar/related to the use of a Jacobian to approximate nonlinear functions?

The short answer is: yes. In our case, the Jacobian is the gradient vector.

How can you guarantee that the graph of a linear function passes through a particular point (x0,y0,z0) in space? One clean way to do this is to write our linear function as: g(x,y)=a(x−x0)+b(y−y0)+z0 ------------------------------------------- Can someone please explain this part to me? May be this obvious but I do not understand why it is always true. An illustrative example will be really helpful

That's because if you plug in x_0 and y_0 into the equation, the equation becomes: g(x,y)=a(x_0 - x_0)+b(y_0 − y_0)+z_0 =z_0 In other words, if you plug in x_o and y_0, the first two terms go to zero and so the whole thing evaluates to z_0, which is exactly just a way of saying that the function passes through x_0, y_0 and z_0. It's a bit like if you wanted to write a one-variable linear function that passes through, say, the point (1, 2). What you would do is you would use the point-slope form and write: y - 2 = m(x - 1) Just by looking at the equation, you know that this line would pass through (1, 2). But to make it look more like the two-variable case, you could write it as: y = m(x - 1) + 2 If x = 1, then the equation becomes y = 2, which is equivalent to saying that the line passes though the point (1, 2). Just like what I said earlier about the two-variable case.

someone please explain to me why the equation g(x,y)=ax+by+c satisfying the slope constant

Because the partial derivatives with respect to X and Y of this particular function are respectively a and b (two constant), therefore they are independent from the specific point in which they are calculated (i.e. they don’t depend on the two variable x and y). Given that the two partial derivatives express the slope of the function in the direction of x axis and y axis respectively, as long as they are constant in any point of the function so is the slope. Hope it helps.

Main content

Multivariable calculus

Course: Multivariable calculus > Unit 3

Lesson 1: Tangent planes and local linearization

Tangent planes

Google Classroom

Just as the single variable derivative can be used to find tangent lines to a curve, partial derivatives can be used to find the tangent plane to a surface.

Background

Partial derivatives

What we're building to

A tangent plane to a two-variable function f, left parenthesis, x, comma, y, right parenthesis is, well, a plane that's tangent to its graph.

The equation for the tangent plane of the graph of a two-variable function f, left parenthesis, x, comma, y, right parenthesis at a particular point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis looks like this:
T, left parenthesis, x, comma, y, right parenthesis, equals, f, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, plus, f, start subscript, start color #0c7f99, x, end color #0c7f99, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, left parenthesis, start color #0c7f99, x, end color #0c7f99, minus, x, start subscript, 0, end subscript, right parenthesis, plus, f, start subscript, start color #bc2612, y, end color #bc2612, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, left parenthesis, start color #bc2612, y, end color #bc2612, minus, y, start subscript, 0, end subscript, right parenthesis

The task at hand

Think of a scalar-valued function with a two-coordinate input, like this one:

f, left parenthesis, x, comma, y, right parenthesis, equals, minus, x, squared, minus, y, squared, plus, 3

Intuitively, it's common to visualize a function like this with its three-dimensional graph.

Remember, you can describe this graph more technically by describing it as a certain set of points in three-dimensional space. Specifically, it is all the points that look like this:

left parenthesis, x, comma, y, comma, f, left parenthesis, x, comma, y, right parenthesis, right parenthesis, equals, left parenthesis, x, comma, y, comma, minus, x, squared, minus, y, squared, plus, 3, right parenthesis

Here, x and y can range over all possible real numbers.

A tangent plane to this graph is a plane which is tangent to the graph. Hmmm, that's not a good definition. This is hard to describe with words, so I'll just show a video with various different tangent planes.

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Key question: How do you find an equation representing the plane tangent to the graph of the function at some specific point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, f, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, right parenthesis in three-dimensional space?

Representing planes as graphs

Well, first of all, which functions g, left parenthesis, x, comma, y, right parenthesis have graphs that look like planes?

The slope of a plane in any direction is constant over all input values, so both partial derivatives g, start subscript, x, end subscript and g, start subscript, y, end subscript would have to be constants. The functions with constant partial derivatives look like this:

g, left parenthesis, x, comma, y, right parenthesis, equals, a, x, plus, b, y, plus, c

Here, a, b, and c are each some constant. These are called linear functions. Well, technically speaking they are affine functions since linear functions must pass through the origin, but it's common to call them linear functions anyway.

Question: How can you guarantee that the graph of a linear function passes through a particular point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis in space?

One clean way to do this is to write our linear function as

g, left parenthesis, x, comma, y, right parenthesis, equals, start color #11accd, a, end color #11accd, left parenthesis, x, minus, x, start subscript, 0, end subscript, right parenthesis, plus, start color #e84d39, b, end color #e84d39, left parenthesis, y, minus, y, start subscript, 0, end subscript, right parenthesis, plus, z, start subscript, 0, end subscript

[Wait, this doesn't look like ax+by+c]

Concept check: With g defined this way, compute g, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis.

(Choice A)
left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis
(Choice B)
z, start subscript, 0, end subscript

[Explain]

Writing g, left parenthesis, x, comma, y, right parenthesis like this makes it clear that g, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, equals, z, start subscript, 0, end subscript. This guarantees that the graph of g must pass through left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis:

left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, g, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, right parenthesis, equals, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis

The other constants start color #11accd, a, end color #11accd and start color #e84d39, b, end color #e84d39 are free to be whatever we want. Different choices for start color #11accd, a, end color #11accd and start color #e84d39, b, end color #e84d39 result in different planes passing through the point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, z, start subscript, 0, end subscript, right parenthesis. The video below shows how those planes change as we tweak start color #11accd, a, end color #11accd and start color #e84d39, b, end color #e84d39:

Khan Academy video wrapper

See video transcript

Equation for a tangent plane

Back to the task at hand. We want a function T, left parenthesis, x, comma, y, right parenthesis that represents a plane tangent to the graph of some function f, left parenthesis, x, comma, y, right parenthesis at a point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, f, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, right parenthesis, so we substitute f, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis for z, start subscript, 0, end subscript in the general equation for a plane.

T, left parenthesis, x, comma, y, right parenthesis, equals, start color #0d923f, f, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0d923f, plus, start color #0c7f99, a, end color #0c7f99, left parenthesis, x, minus, x, start subscript, 0, end subscript, right parenthesis, plus, start color #bc2612, b, end color #bc2612, left parenthesis, y, minus, y, start subscript, 0, end subscript, right parenthesis

As you tweak the values of start color #0c7f99, a, end color #0c7f99 and start color #bc2612, b, end color #bc2612, this equation will give various planes passing through the graph of f at the desired point, but only one of them will be a tangent plane.

Of all the planes passing through left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, f, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, right parenthesis, the one tangent to the graph of f will have the same partial derivatives as f. Pleasingly, the partial derivatives of our linear function are given by the constants a and b.

Try it! Take the partial derivatives of the equation for T, left parenthesis, x, comma, y, right parenthesis above.
[Solution]

Therefore setting start color #0c7f99, a, equals, f, start subscript, x, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #0c7f99 and start color #bc2612, b, equals, f, start subscript, y, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, end color #bc2612 will guarantee that the partial derivatives of our linear function T match the partial derivatives of f. Well, at least they will match for the input left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, but that's the only point we care about. Putting this together, we get a usable formula for the tangent plane.

\begin{aligned} \quad \large T(x, y) = \greenE{f(x_0, y_0)} + \blueE{f_x(x_0, y_0)}(x-x_0) + \redE{f_y(x_0, y_0)}(y-y_0) \end{aligned}

Example: Finding a tangent plane

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See video transcript

Problem:

Given the function

f, left parenthesis, x, comma, y, right parenthesis, equals, sine, left parenthesis, x, right parenthesis, cosine, left parenthesis, y, right parenthesis,

find the equation for a plane tangent to the graph of f above the point left parenthesis, start fraction, pi, divided by, 6, end fraction, comma, start fraction, pi, divided by, 4, end fraction, right parenthesis.

The tangent plane will have the form

\begin{aligned} \quad T(x, y) = {f_x(x_0, y_0)}(x-x_0) + {f_y(x_0, y_0)}(y-y_0) + f(x_0, y_0) \end{aligned}

Step 1: Find both partial derivatives of f.

f, start subscript, x, end subscript, left parenthesis, x, comma, y, right parenthesis, equals

f, start subscript, y, end subscript, left parenthesis, x, comma, y, right parenthesis, equals

[Answer]

Step 2: Evaluate the function f as well as both these partial derivatives at the point left parenthesis, start fraction, pi, divided by, 6, end fraction, comma, start fraction, pi, divided by, 4, end fraction, right parenthesis:

f, left parenthesis, pi, slash, 6, comma, pi, slash, 4, right parenthesis, equals

f, start subscript, x, end subscript, left parenthesis, pi, slash, 6, comma, pi, slash, 4, right parenthesis, equals

f, start subscript, y, end subscript, left parenthesis, pi, slash, 6, comma, pi, slash, 4, right parenthesis, equals

[Answer]

Putting these three numbers into the general equation for a tangent plane, you can get the final answer

T, left parenthesis, x, comma, y, right parenthesis, equals

[Answer]

Summary

A tangent plane to a two-variable function f, left parenthesis, x, comma, y, right parenthesis is, well, a plane that's tangent to its graph.

The equation for the tangent plane of the graph of a two-variable function f, left parenthesis, x, comma, y, right parenthesis at a particular point left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis looks like this:

T, left parenthesis, x, comma, y, right parenthesis, equals, f, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, plus, f, start subscript, start color #0c7f99, x, end color #0c7f99, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, left parenthesis, start color #0c7f99, x, end color #0c7f99, minus, x, start subscript, 0, end subscript, right parenthesis, plus, f, start subscript, start color #bc2612, y, end color #bc2612, end subscript, left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, right parenthesis, left parenthesis, start color #bc2612, y, end color #bc2612, minus, y, start subscript, 0, end subscript, right parenthesis

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Aaron Hargrove
Posted 7 years ago. Direct link to Aaron Hargrove's post “Is finding a tangent plan...”
Is finding a tangent plan similar/related to the use of a Jacobian to approximate nonlinear functions?
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(8 votes)
Answer
- bmesszabo
  Posted 7 years ago. Direct link to bmesszabo's post “The short answer is: yes....”
  The short answer is: yes. In our case, the Jacobian is the gradient vector.
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  (8 votes)
abvn2
Posted 4 years ago. Direct link to abvn2's post “How can you guarantee tha...”
How can you guarantee that the graph of a linear function passes through a particular point (x0,y0,z0) in space?

One clean way to do this is to write our linear function as:
g(x,y)=a(x−x0)+b(y−y0)+z0
-------------------------------------------
Can someone please explain this part to me? May be this obvious but I do not understand why it is always true. An illustrative example will be really helpful
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Christopher
  Posted 3 years ago. Direct link to Christopher's post “That's because if you plu...”
  That's because if you plug in x_0 and y_0 into the equation, the equation becomes:
  
  g(x,y)=a(x_0 - x_0)+b(y_0 − y_0)+z_0
  =z_0
  
  In other words, if you plug in x_o and y_0, the first two terms go to zero and so the whole thing evaluates to z_0, which is exactly just a way of saying that the function passes through x_0, y_0 and z_0.
  
  It's a bit like if you wanted to write a one-variable linear function that passes through, say, the point (1, 2). What you would do is you would use the point-slope form and write:
  
  y - 2 = m(x - 1)
  
  Just by looking at the equation, you know that this line would pass through (1, 2). But to make it look more like the two-variable case, you could write it as:
  
  y = m(x - 1) + 2
  
  If x = 1, then the equation becomes y = 2, which is equivalent to saying that the line passes though the point (1, 2). Just like what I said earlier about the two-variable case.
  Button navigates to signup page
  (5 votes)
bababiba
Posted 7 years ago. Direct link to bababiba's post “someone please explain to...”
someone please explain to me why the equation g(x,y)=ax+by+c satisfying the slope constant
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Marco
  Posted 6 years ago. Direct link to Marco's post “Because the partial deriv...”
  Because the partial derivatives with respect to X and Y of this particular function are respectively a and b (two constant), therefore they are independent from the specific point in which they are calculated (i.e. they don’t depend on the two variable x and y). Given that the two partial derivatives express the slope of the function in the direction of x axis and y axis respectively, as long as they are constant in any point of the function so is the slope. Hope it helps.
  Button navigates to signup page
  (5 votes)
Gianlu
Posted 4 years ago. Direct link to Gianlu's post “hey guys.. i actually und...”
hey guys.. i actually understood how to get the tangent function.. at the end, how does should we interpret the T(x,y)?

should be the entire equation as it is, i mean what x and y need for, should make a new equation of value with x and y being the position of the tangent plane?
thanks gian :)
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(2 votes)
Answer
Nikhishah Ramesh
Posted 3 years ago. Direct link to Nikhishah Ramesh's post “Find an equation of the t...”
Find an equation of the tangent plane to the surface

z =
√
xy

at the point (1, 1, 1).
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(1 vote)
Answer
- 𝜏 Is Better Than 𝝅
  Posted 3 years ago. Direct link to 𝜏 Is Better Than 𝝅's post “z=(x+y)/2”
  z=(x+y)/2
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  (3 votes)
hayyafatima172
Posted 7 years ago. Direct link to hayyafatima172's post “here we are finding tange...”
here we are finding tangent planes but the equation used is of linearization function please answer me fastly
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(1 vote)
Answer
- Steven Messmer
  Posted 6 years ago. Direct link to Steven Messmer's post “The tangent plane is a li...”
  The tangent plane is a linearization of a three dimensional surface
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  (1 vote)
davidpp
Posted 7 years ago. Direct link to davidpp's post “how do i find a tangent p...”
how do i find a tangent plane for a function thats defined as follows: R^2 ---> R^2
more specifically:
(x,y)--> (sqrt(x^2+y^2 , arctan(y/x))

please help as fast as possible
thanks in advance
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(1 vote)
Answer
- Serkan
  Posted 7 years ago. Direct link to Serkan's post “Your question might be in...”
  Your question might be in a wrong page, an equation for f(x,y) and a specific coordinate are needed to calculate the tangent plane.
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  (1 vote)
Josh Parkinson
Posted 5 years ago. Direct link to Josh Parkinson's post “I have a surface equation...”
I have a surface equation of: z(x,y) = √(12(〖sec〗^2 x/y-1))+ln⁡(9/10 (x^2/π^2 +y^2/4)).
I need to find the tangent plane to the surface at the point P(π/3, 2).
I can get halfway through this problem to find z_0 = 2 but cannot find the constants f_x or f_y.
Any help would be greatly appreciated.
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(1 vote)
Answer
gurungdebendra82
Posted 7 years ago. Direct link to gurungdebendra82's post “so this the equation of ...”
so this the equation of the plane with two input varibles. kind of a headache . when already we define a plane to be represented as an input of three variables.... is there a convenient way to see it as of the form ax + by +cz= d or should we not attempt to do that.
Button navigates to signup pageComment on gurungdebendra82's post “so this the equation of ...”
(1 vote)
Answer
- Grant
  Posted 7 years ago. Direct link to Grant's post “It's perfectly fine to pu...”
  It's perfectly fine to put it in the form ax+by+cz=d.
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  (1 vote)