Question

Let \(w=f(x, y, z)=2 x+3 y+4 z\) which is defined for all \((x, y, z)\) in \(\mathbb{R}^{3} .\) Suppose we are interested in the partial derivative \(w_{x}\) on a subset of \(\mathbb{R}^{3}\), such as the plane \(P\) given by \(z=4 x-2 y .\) The point to be made is that the result is not unique unless we specify which variables are considered independent. a. We could proceed as follows. On the plane \(P\), consider \(x\) and \(y\) as the independent variables, which means \(z\) depends on \(x\) and \(y,\) so we write \(w=f(x, y, z(x, y)) .\) Differentiate with respect to \(x,\) holding \(y\) fixed, to show that \(\left(\frac{\partial w}{\partial x}\right)_{y}=18,\) where the subscript \(y\) indicates that \(y\) is held fixed. b. Alternatively, on the plane \(P\), we could consider \(x\) and \(z\) as the independent variables, which means \(y\) depends on \(x\) and \(z,\) so we write \(w=f(x, y(x, z), z)\) and differentiate with respect to \(x,\) holding \(z\) fixed. Show that \(\left(\frac{\partial w}{\partial x}\right)_{z}=8,\) where the subscript \(z\) indicates that \(z\) is held fixed. c. Make a sketch of the plane \(z=4 x-2 y\) and interpret the results of parts (a) and (b) geometrically. d. Repeat the arguments of parts (a) and (b) to find \(\left(\frac{\partial w}{\partial y}\right)_{x}\) \(\left(\frac{\partial w}{\partial y}\right)_{z},\left(\frac{\partial w}{\partial z}\right)_{x},\) and \(\left(\frac{\partial w}{\partial z}\right)_{y}\).

Short answer

a. \((\frac{\partial w}{\partial x})_{y}\) b. \((\frac{\partial w}{\partial x})_{z}\) c. \((\frac{\partial w}{\partial y})_{x}\) d. \((\frac{\partial w}{\partial y})_{z}\) e. \((\frac{\partial w}{\partial z})_{x}\) f. \((\frac{\partial w}{\partial z})_{y}\) Answer: a. 18 b. 8 c. 3 d. 3 e. 4 f. 4

Step-by-step solution

Rewrite the function in terms of the independent variables

As we will consider \(x\) and \(y\) as the independent variables, we need to write \(z\) in terms of \(x\) and \(y\), according to the plane \(P\). So we have \(z = 4x - 2y\). Now, rewrite the function \(w\), replacing \(z\) with \(4x - 2y\). \(w = 2x + 3y + 4(4x - 2y)\)

Differentiate with respect to \(x\) holding \(y\) fixed

Now we want to find the partial derivative with respect to \(x\). Since we are holding \(y\) fixed, we will treat it as a constant. \(\frac{\partial w}{\partial x} = \frac{\partial}{\partial x}(2x + 3y + 4(4x - 2y)) = 2 + 16 = 18\) Thus, \((\frac{\partial w}{\partial x})_{y} = 18\) b. Find \(\frac{\partial w}{\partial x}\) while holding \(z\) fixed

Rewrite the function in terms of the independent variables

Now we will consider \(x\) and \(z\) as the independent variables. So, we need to write \(y\) in terms of \(x\) and \(z\). From the plane \(P\), we can rewrite the equation as \(y = 2x - \frac{1}{2}z\). Now, replace \(y\) in the function \(w\). \(w = 2x + 3(2x - \frac{1}{2}z) + 4z\)

Differentiate with respect to \(x\) holding \(z\) fixed

Now find the partial derivative with respect to \(x\), while holding \(z\) fixed: \(\frac{\partial w}{\partial x} = \frac{\partial}{\partial x}(2x + 3(2x - \frac{1}{2}z) + 4z) = 2 + 6 = 8\) Thus, \((\frac{\partial w}{\partial x})_{z} = 8\) c. Sketch the plane and interpret the results geometrically d. Find other partial derivatives following the arguments of parts (a) and (b) Following the arguments of parts (a) and (b), we can find the other partial derivatives: \(\left(\frac{\partial w}{\partial y}\right)_{x}\) = 3 \(\left(\frac{\partial w}{\partial y}\right)_{z}\) = 3 \(\left(\frac{\partial w}{\partial z}\right)_{x}\) = 4 \(\left(\frac{\partial w}{\partial z}\right)_{y}\) = 4

Key concepts

Multivariable Calculus

In multivariable calculus, functions of more than one variable are examined. Unlike single-variable calculus where we deal with functions that have one input (like f(x)), multivariable functions have several inputs. An example is the function w = f(x, y, z), which involves three variables: x, y, and z.

The study of these functions extends the concepts of limits, continuity, derivatives, and integrals from single variable to multiple variables. Multivariable calculus is crucial in fields such as physics, engineering, economics, and anything involving optimization or where variables interact with each other.

An important application in multivariable calculus is to find the rate at which one variable affects another. To this end, we introduce partial derivatives, which are derivatives taken with respect to one variable at a time, holding the others constant. The exercise provided is a basic example of utilizing partial derivatives to understand how changes in variables affect a multivariable function.

Partial Differentiation

Partial differentiation is the process by which we find the derivative of a multivariable function considering one variable at a time. More intuitively, it can be thought of as examining how a function changes as we change just one of the variables, while keeping the others fixed.

Let's take our function w = f(x, y, z) from the exercise. When we want to find the rate at which w changes with respect to x while keeping y and z constant, we seek the partial derivative of w with respect to x, denoted as \(\frac{\partial w}{\partial x}\). The steps in the solution show this process concretely by first substituting z from the equation of plane P and then differentiating with respect to x, treating y as a constant.

It's important for students to grasp that the choice of which variables are held constant can change the outcome. The exercise illustrates this by showing different results when different variables are considered as the independent ones. This is akin to changing your viewpoint in a three-dimensional space to see how the landscape's appearance shifts.

Geometric Interpretation of Partial Derivatives

The geometric interpretation of partial derivatives can provide a visual understanding of how a function behaves as one variable changes while the others are held constant. Consider the plane z = 4x - 2y as in our exercise. The partial derivative of w with respect to x, \(\frac{\partial w}{\partial x}\), can be visualized as the slope of the function's graph in the x-direction at a fixed y.

When we calculated \(\left(\frac{\partial w}{\partial x}\right)_{y} = 18\), it tells us that for a small change in x while y is held constant, w increases at a rate of 18 units. Similarly, \(\left(\frac{\partial w}{\partial x}\right)_{z} = 8\) indicates a smaller slope when z is held constant, showing a less steep increase.

In essence, partial derivatives allow us to slice the graph of a multivariable function in planes parallel to the coordinate axes and then find the gradient of the resulting curve. This provides invaluable insights in optimizing processes and understanding multidimensional relationships. When instructing students, emphasizing visual tools such as graphs or contour maps can greatly aid in their comprehension of these abstract concepts.