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Tangent Planes of Graphs

Recall from single-variable calculus that we can use a tangent line to approximate the behavior of a function near a point. Given a point on a curve , the tangent line at is given by the following:

We can extend this idea to multivariable functions. Instead of a tangent line, we use a tangent plane to approximate the behavior of a function near a point.

Before even attempting to find the tangent plane, let's explore 3D planes a bit more to build some intuition.

Table of Contents​

Understanding and Controlling 3D Planes in Space​

There are many ways to represent a plane in 3D space. We have previously explored planes in the context of linear algebra, where a plane is defined by a normal vector and a point, i.e. .

Let the plane be a function of and :

Consider a point in 3D space, . Then, consider a plane such that the plane passes through the point . There could be many such planes, so we will need to specify more information to uniquely determine the plane.

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The first property of this plane function, is that it must pass through the point . This can be expressed as:

Next, consider the slope of the plane. If we slice the plane along the -axis, we will see a line, and likewise for the -axis (check above for a visualization). This can then be expressed as:

Where is some arbitrary constant. We can use these equations to try to get the equation for the plane.

First, consider the first equation . This means that is equal to plus something that doesn't depend on :

Likewise, the second equation means that is equal to plus something that doesn't depend on :

Combining these two equations, we would get something like:

What should be? We know that , so we can substitute these values in:

This gives us a system of equations that we can solve to find the values of , , and . For example, given these following slopes:

Then, we can find the equation of the plane:

Given that , we can plug in these values to find :

Thus, the equation of the plane is:

Another way to think about this is through some function transformation. Going back to what we had:

Let's rewrite using the following:

Where is a new constant. This is equivalent to the previous equation (the partial derivatives are the same), but once we plug in , we get:

Meaning that . We can generalize this. Given a point , and the slopes and , the equation of the plane is:

This is essentially the 3D equivalent of the slope-intercept form of a line, .

Computing the Tangent Plane​

Now that we have a better understanding of 3D planes, let's compute the tangent plane of a function at a point .

The tangent plane can be defined using the slopes of the function at that point, and the point itself. Hence, the equation of the tangent plane is:

This is the 3D equivalent of the tangent line in single-variable calculus.

Notice that the first two terms resemble a dot product between the gradient of the function and the vector . If we set and , then the equation of the tangent plane can be written as:

Which is even closer to the slope-intercept form of a line:

Linearity of Tangent Planes​

A linear function, for this context, is a function that can be written in the form . In other words, the output of the function is a linear combination of the inputs.

Usually, the input of the function is collected into a vector like . The output is then written as a dot product of that vector with a coefficient vector:

In a tangent plane, however, we add a constant term to the linear function. Now the output is a linear combination of the inputs plus a constant:

This is no longer a linear function, but an affine function.

Local Linearization​

Recall, in single-variable calculus, that the tangent line is used to approximate the behavior of a function near a point.

In this case, we can do the same. Suppose we have a nonlinear function . We can approximate the function near a point using a tangent plane.

Recall from Equation that the equation of the tangent plane is:

And as such, this can be used to approximate the function near the point .

Example Problem: Tangent Plane of a Multivariable Function​

The function is given. Find the equation of the tangent plane at the point .

To find the equation of the tangent plane, we need to find the partial derivatives of at the point :

Plugging in the values , we get:

Thus, the equation of the tangent plane is as follows, using the point :

Example Problem: Approximating an Expression​

Solve for an approximate value for the following expression:

(Source)

To solve this problem, we need to find a function that approximates the given expression. Let's define the function as follows:

Recall that, to find the local linearization of a function, we need:

  1. The value of
  2. The partial derivatives of at

We can set , , and , a close approximation to the values in the expression. Then, we can find the value of the function at :

This itself serves as a close approximation to the original expression, but we can get closer! Next, we need to find the partial derivatives of at , the easiest of which is :

For the other partial derivatives, we need to use the chain rule to find the derivatives of the inner functions:

Finally, for , another application of the chain rule is needed:

We have three components in these partial derivatives. Evaluating them one by one at :

Hence the partial derivatives at are:

Recall the equation of the tangent plane for local linearization:

Plugging in the values, we get:

Finally, using the values , , and , we can find the approximate value of the expression:

Thus, the approximate value of the expression is .

The Actual Value

The actual value for the expression is . The approximation is very close (within ), showing the power of local linearization in approximating functions.

Summary​

We have explored the concept of tangent planes in 3D space, and how they can be used to approximate the behavior of a function near a point.

  • The equation of a plane in 3D space can be written as .

    • and are the slopes of the plane along the and axes respectively.
    • is a constant that determines the height of the plane.

  • The equation of the tangent plane of a function at a point is given by:

  • The equation of the tangent plane can also be written in vector form as:

  • The tangent plane is an affine function, not a linear function.

  • The tangent plane can be used to approximate the behavior of a function near a point, a concept known as local linearization.

Next, we will learn about quadratic approximations, another method to approximate functions near a point. Eventually, we will generalize even further to the Multivariable Taylor Series, which can approximate functions to any degree near a point.