Guide to Tangent Planes and Linear Approximation: Topics of Partial Derivatives

Overview

Up to this point, we have really highlighted the nuts and bolts of working with partial derivatives. Firstly, it began by elaborating the roles of equations comprised of multiple variables. We extrapolate on those insights in this article. Secondly, we took a brief detour to examine the nuances of limits and continuity in these types of functions. Finally, in our most recent article for this series, we actually presented the methodology for taking partial derivatives and its consequences. Here, we discuss one of the primary applications of partial derivatives: for tangent planes and linear approximations. Tangent planes and linear approximation applies widely throughout multivariate calculus, and beyond, in a variety of manners. So, let us begin our investigation.

Conceptualizing Tangent Planes

Calculus up until this point has often dealt with either looking at behavior of a function as it approaches a point on a line (derivatives) or computing area. We have been working presently with functions of two variables, which, if you recall, constitute a surface. Our elaborations of partial derivatives demonstrated we can use these tools to define a line through a point on this surface as well as compute the tangent of this point relative to the surface. Interestingly, at close approximations, this point of intersection appears as a plane. Let us proceed with this line of thought.

So, suppose that our multi-variable function z=f(x,y) defines a surface which we call ‘S’. Taking the partial derivative of our function ‘f’ with respect to either variables will yield two lines that follow the curve through this point. Then computing the tangent of both these curves yield two tangent lines that intersect this point. These two tangent lines, together, constitute the tangent plane to ‘s’ via the point intersected by the tangent lines. It would appear like this:

Now, if any other line on the surface intersects the point P, then the tangent of this line will exist on the tangent plane.

Computing the Tangent Plane

In many cases, the tangent plane is not immediately known, but must be found through careful computation. There is a rather convenient way of computing the tangent plane for most of these surfaces. Consider our scenario wherein we focus in on a point P(x0,y0,z0) on our surface ‘S’. Let us first focus on defining a function for a plane passing through this point ‘P’:

A(x-x_0)+B(y-y_0)+C(z-z_0)=0

This is a rather standard way to model planes in three dimensional space, but we can slightly remodel it to procure:

z-z_0=a(x-x_0)+b(y-y_0)

This equation above represents the equation for the tangent plane that intersects our point ‘P’. Now, let us attempt to put this in terms of our function comprised of multiple variables. If we have our original function ‘f’, then we can compute the tangent plane at the point P(x0,y0,z0) may be computed as:

z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)

Thus, the tangent plane for the surface through point ‘P’ may be defined by the partial derivatives of our multi-variable function.

Example of Computing the Tangent Plane

We now seek to apply this technique to identify the tangent plane to a surface through a particular point. Consider the following function:

z=2x^2+y^2

We will now attempt to identify the tangent plane to this surface through the point (1,1,3). Let us first begin by converting the function to the standard form we seek to operate in:

f(x,y)=2x^2+y^2

Our first task at hand is to compute the partial derivatives of our multivariable function ‘f’. Let us begin identifying the partial derivative with respect to ‘x’.

f_x(x,y)=4x\newline f_x(1,1)=4(1)\newline f_x(1,1)=4

So the partial derivative of ‘f’ with respect to ‘x’ returns a value of 4. Let us now attempt to compute the partial derivative of the function with respect to ‘y’.

f_y(x,y)=2y \newline f_y(1,1)=2(1) \newline f_y(1,1)=2

From computing this, we find that the partial derivative of ‘f’ with respect to ‘y’ returns a value of 2. Now that the partial derivatives, we can substitute these values into our function defining the tangent plane. The steps are demonstrated below:

z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)\newline z-3=(4)(x-1)+(2)(y-1) \newline z=4x+2y-3

This function succinctly demonstrates the tangent plane to our multi-variable function. The relationship between these components is demonstrated below:

Conceptualization of Linear Approximation

Previously, the tangent plane was computed by taking the partial derivative of a multivariable function through a point on the surface it defines. We can take this function for the tangent plane and use it to create a new, multi-variable function. It would appear as follows:

g(x,y)=4x+2y-3

This linear function ‘g’ is actually a decent estimation of the function f(x,y) when the point (x,y) is near (1,1). We actually call this a linearization of the function ‘f’ near the point (1,1). For this reason, we call the following function a linear approximation:

f(x,y)\approx 4x+2y-3

Methodology of Linearization

Our previous proofs have revealed that the tangent plane through a point (a,b,f(a,b) may be defined as:

z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)

Furthermore, we defined the linear function as:

g(x)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)

From this, we say that the linearization of the function ‘f’ at a point is a decent approximation of the line such that we can develop a linear approximation of the form:

f(x,y)\approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)

Incrementing

In early calculus, one of the most fundamental teachings was the notion that the change in the value of ‘x’ (Δx), ultimately procures a change in the value of y. The change in the value of y (Δy), can be computed by the function:

Δy=f(a+Δx)-f(a)

However, from our understanding of the derivative, we might also say that:

Δy=f'(a)Δx

For multi-variable equations, we know that z=f(x,y). In this case, a change in the x-dimension is quantified as Δx and a change in the y-dimension is modeled as Δy. If this is true, then we are also able to compute the increment of ‘z’ in a similar fashion such that:

Δz=f(a+Δx,b+Δy)-f(a,b)

Using this, we might make a formal definition of the increment that exists in the z-dimension. If we suppose that z=f(x,y), then we know that the function ‘f’ may be differentiated when we can confidently state that:

Δz=f(a+Δx,b+Δy)-f(a,b)+\epsilon_1Δx+ \epsilon_2Δy

The Take Away for Tangent Planes and Linear Approximation

The present article took great lengths towards demonstrating the use of partial derivatives to yield the tangent plane. We might take a moment to recall that the tangent plane is the plane that exists tangent to our surface defined by z=f(x,y) through a point ‘P’ on the surface. By using partial derivatives, the tangent plane can be readily computed. Furthermore, we learned how the tangent plane can be used as a candid linear approximation of the multivariable function at a particular point. For more resources on tangent planes and linear approximation, check out this article.

Hopefully this guide has been a useful resource. If you seek to continue on with this series documenting parameters of partial derivatives, check out the others below.

Modeling Functions of Multiple Variables

Limits and Continuity of Higher Variable Functions

A Guide to Executing Partial Derivatives