Linearization: Tangent Planes and Differentials

This section relies on the ideas of planes from and first partial derivatives as measuring the tilt of a surface plot at a particular location from . Some instructors will only cover a subset of the linearization tools covered here (tangent plane, total differential, linearization for approximation, differentiability, locally linear) but these tools are introduced and defined in such a way that each conceptually flows from the idea of locally linear functions. To cover all the topics in this section will likely take a couple of class meetings.

We have moved examples of functions that are not differentiable to the later subsection, . This subsection does not have activities because we feel that the deeper ideas involved in showing a function is not differentiable are not usually suitable for students at this level. There is an extension of the example function to to show that a function that has partial derivatives existing is NOT the same as the function being differentiable. There are several exercises that work through a more complete definition of differentiability including .

Throughout single-variable calculus, the tangent line serves as a geometric and conceptual tool to understand the derivative at a point. While the value of a derivative at a point measures the instantaneous rate of change, we visualize the value of the derivative as the slope of the line tangent to the function's graph. This allowed us to see the derivative's value in terms of an important behavior of the function's graph.

In , we saw that a curve in space, given as the graph of a vector-valued function of one variable, is locally linear and approximated well by a tangent line provided that the function is differentiable. So the vector-valued function of one variable that describes a curve in space will be differentiable if the curve looks like a line on a small scale.

In this section, we will explore the various ways in which we can extend to functions of two variables the related ideas of differentiability and local linearity. Geometrically, this means we will use a tangent plane as a flat analog for surfaces of the form \(z=f(x,y)\).For graphs in the cartesian plane, there was only one type of flat graph: a line. When we generalized to higher dimensions, we saw how both lines and planes can be considered flat graphs in three dimensions. However, we noted that lines are 1-dimensional flat graphs because there is only direction to move along the graph. On the other hand, planes are 2-dimensional flat graphs. We will also introduce a couple of new tools to help us algebraically and numerically describe the change in the output of \(f(x,y)\).

To start our investigation, let's consider the graph of a nice two-variable function on a small scale. We will look at \[\begin{aligned}\end{aligned}\] whose graph is shown in Figure.

You should examine the behavior of \(f\) near the input \((x_0,y_0) = (1,1)\), which has output of \(f(1,1)=4.5\). You can change how closely Figure shows the graph of \(z=f(x,y)\) on smaller and smaller scales around the point \((1,1,4.5)\). With zoom set to 0, the plot clearly reveals that the surface is curved. However, once you increase the zoom level to 3 or higher, the surface looks more like a tilted plane than a curved surface.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

As the preview activity suggests, the graph of most functions we have encountered will be approximated well by a plane tangent to the graph at a point of interest. In , we will talk more about the technical conditions for the tangent plane to be a good approximation to the surface, but for now we will use the following general formula for the tangent plane from our final result of Preview Activity.

If \(f\) is a function of \(x\) and \(y\) for which both \(f_x\) and \(f_y\) exist and are continuous in an open disk containing the point \((x_0,y_0)\), then the equation of the plane tangent to the graph of \(f\) at the point \((x_0,y_0,f(x_0,y_0))\) is \[\begin{aligned}\end{aligned}\].

Having the equation for a tangent plane in the form given by \(z = f(x_0,y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0)\) allows us to quickly identify important information about the function \(f\) at the point \((x_0,y_0)\). For example, if a function \(f\) has a tangent plane given by \(z = 7 - 2(x-3) + 4(y+1)\), then we can immediately read the following information from the given form of the tangent plane equation:

• \(f(3,-1) = 7\),

• \(f_x(3,-1)=-2\) is the slope of the trace to both \(f\) and the tangent plane in the \(x\)-direction at \((3,-1)\), and

• \(f_y(3,-1) = 4\) is the slope of the trace of both \(f\) and the tangent plane in the \(y\)-direction at \((3,-1)\).

In single-variable calculus, an important use of the tangent line is to approximate a differentiable function. Near the point \(x_0\), the tangent line to the graph of \(f\) at \(x_0\) is close to the graph of \(f\) for input values close to \(x_0\), as shown in Figure. Adjust the Zoom slider to examine a small region around the highlighted point to see how the tangent line is a good approximation for a small region around that point. In fact, the smaller the scale around the point, the better the approximation is.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In single-variable calculus, we let \(L\) denote the function whose graph is the tangent line, and thus \[\begin{aligned}\end{aligned}\]. Furthermore, \(f(x) \approx L(x)\) near \(x_0\). We call \(L\) the linearization of \(f\).

Similarly, the tangent plane to the graph of \(f(x,y)\), a locally linear function of two variables, at a point \((x_0,y_0)\) provides a good approximation near \((x_0, y_0)\). We define the linearization, \(L\), to be the two-variable function whose graph is the tangent plane. Thus, \[\begin{aligned}\end{aligned}\] is the linearization of \(f\) at \((x_0,y_0)\). Note that \(f(x,y)\approx L(x,y)\) for points near \((x_0, y_0)\), as illustrated in Figure.

Using a function's linearization when you already have an algebraic expression for the function provides little added value. In many applications the function of interest does not have a known algebraic formula. For instance, if you were working for a mining company and were trying to map a pocket of some resource underground, it would be very expensive to drill a bunch of samples to make a chart with amounts of the resource at a comprehensive grid of locations. Instead, you could be more selective in obtaining data via other sampling techniques and then use linearization to make estimates for intermediate locations. Linearization is a valuable first step in estimating values between different input locations. In later math courses, you may encounter more sophisticated ways to estimate between data points and discuss the advantages and drawbacks of these ideas.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In our equation for a tangent plane, we gave the general formula for the equation of the tangent plane at a particular point on the graph of \(z=f(x,y)\), but we had a condition involving partial derivatives that we needed to satisfy in order for the tangent plane to make sense. We will briefly discuss some ways in which a function can fail to be locally linear; in other words, we will look at several examples of functions of two variables where the tangent plane is either not defined or is not a good approximation of the surface near a particular point.

In single-variable calculus, one of the first functions you studied that was continuous but had a point where the graph was not locally linear was the absolute value function \(f(x) = |x|\). When zooming in on the graph of \(f\) near the point \((0,0)\), the graph maintains a sharp corner, and thus \(f\) is not locally linear at the point on its graph with \(x=0\).

Remember that a two-variable function \(f\) is locally linear near \((x_0,y_0)\) provided that the graph of \(f\) looks like a plane (its tangent plane) when viewed on a small scale near \((x_0,y_0)\). Determining when a function of two variables is locally linear at a point involves more nuance than in the single-variable setting, as the next example illustrates.

If \(f\) is a function of \(x\) and \(y\) for which both \(f_x\) and \(f_y\) exist and are continuous in an open disk containing the point \((x_0,y_0)\), then \(f\) is continuously differentiable at \((x_0,y_0)\).

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.