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## Multivariable calculus

### Course: Multivariable calculus>Unit 3

Lesson 1: Tangent planes and local linearization

# Tangent planes

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.

## 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

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.
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?
A plane passing through (2, 2, 2)
A plane passing through (2, 2, 2)
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
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.

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:

## 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.
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

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

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

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

## 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