Section 14.1: Functions of Several Variables

Definition

A function f of two variables is a rule that assigns to each ordered pair of real numbers \((x, y)\) in a set \(D\) a unique real number denoted by \(f(x, y)\). The set \(D\) is the domain of \(f\) and its range is the set of values that \(f\) takes on, that is, \(f(x, y) \mid (x, y) \in D\).

Example

• Volume of a cylinder \(V(r,h) = \pi r^2 h\)
• Area of rectangle \(A(l,b) = lb\)

Definition

The domain of a function f is the collection of the points on which this function can be evaluated.

Example Question

Find and sketch the domain of \[f(x,y) = \frac{\sqrt{x+y+1}}{x-1}\].

Example Question

Find and sketch the domain of \(f(x, y) = x \ln(y^2 - x)\).

Example Question

Find and sketch the domain of \(g(x, y) = \sqrt{ 9 - x^2 - y^2}\).

Definition If \(f\) is a function of two variables with domain \(D\), then the graph of \(f\) is the set of all points \((x, y, z)\) in \(\mathbb{R}^3\) such that \(z = f(x, y)\) and \((x, y) \in D\).

Example

Sketch the graph of the function \(f(x, y) = 6 - 3x - 2y\).

Example

Sketch the graph of the function \(f(x, y) = \sqrt{9 - x^2 - y^2}\).

Exercise

Plot graphs of these functions on Geobegra or any plotting software you like.

• \(f(x, y) = (x^2 + 3y^2) e^{-x^2 - y^2}\)
• \(f(x, y) = \sin x + \sin y\)
• \(f(x, y) = \frac{\sin x \sin y}{xy}\)

Definition
The level curves of a function \(f\) of two variables are the curves with equations \(f(x, y) = k\), where \(k\) is a constant (in the range of \(f\)).

Example

Sketch the level curves of the function \(f(x, y) = 6 - 3x - 2y\) for the values \(k = -6, 0, 6, 12\).

Example Level curves

Sketch the level curves for the function \[ g(x,y) = \sqrt{ 9 - x^2 - y^2} \qquad \text{for} \qquad k = 0, 1, 2, 3. \]

Example Level Curves

Sketch some level curves for the function \[ h(x,y) = 4x^2 + y^2 + 1 \]

In Example 2 we considered the function \(W = f(T, v)\), where \(W\) is the wind-chill index, \(T\) is the actual temperature, and \(v\) is the wind speed. A numerical representation is given in Table 1 on page 889.
(a) What is the value of \(f(-15, 40)\)? What is its meaning?
(b) Describe in words the meaning of the question “For what value of \(v\) is \(f(-20, v) = -30\)?” Then answer the question.
(c) Describe in words the meaning of the question “For what value of \(T\) is \(f(T, 20) = -49\)?” Then answer the question.
(d) What is the meaning of the function \(W = f(-5, v)\)? Describe the behavior of this function.
(e) What is the meaning of the function \(W = f(T, 50)\)? Describe the behavior of this function.

The temperature-humidity index \(I\) (or humidex, for short) is the perceived air temperature when the actual temperature is \(T\) and the relative humidity is \(h\), so we can write \(I = f(T, h)\). The following table of values of \(I\) is an excerpt from a table compiled by the National Oceanic & Atmospheric Administration.

1. What is the value of \(f(95, 70)\)? What is its meaning?
   
2. For what value of \(h\) is \(f(90, h) = 100\)?
   
3. For what value of \(T\) is \(f(T, 50) = 88\)?
   
4. What are the meanings of the functions \(I = f(80, h)\) and \(I = f(100, h)\)? Compare the behavior of these two functions of \(h\).

A manufacturer has modeled its yearly production function \(P\) (the monetary value of its entire production in millions of dollars) as a Cobb-Douglas function
\[P(L, K) = 1.47L^{0.65}K^{0.35}\]
where \(L\) is the number of labor hours (in thousands) and \(K\) is the invested capital (in millions of dollars). Find \(P(120, 20)\) and interpret it.

Verify for the Cobb-Douglas production function
\[P(L, K) = 1.01L^{0.75}K^{0.25}\]
discussed in Example 3 that the production will be doubled if both the amount of labor and the amount of capital are doubled. Determine whether this is also true for the general production function
\[P(L, K) = bL^rK^{1 - r}\]

A model for the surface area of a human body is given by the function
\[S = f(w, h) = 0.1091w^{0.425}h^{0.725}\]
where \(w\) is the weight (in pounds), \(h\) is the height (in inches), and \(S\) is measured in square feet.
(a) Find \(f(160, 70)\) and interpret it.
(b) What is your own surface area?

The wind-chill index \(W\) discussed in Example 2 has been modeled by the following function:
\[W(T, v) = 13.12 + 0.6215T - 11.37v^{0.16} + 0.3965Tv^{0.16}\]
Check to see how closely this model agrees with the values in Table 1 for a few values of \(T\) and \(v\).

The wave heights \(h\) in the open sea depend on the speed \(v\) of the wind and the length of time \(t\) that the wind has been blowing at that speed. Values of the function \(h = f(v, t)\) are recorded in feet in Table 4.
(a) What is the value of \(f(40, 15)\)? What is its meaning?
(b) What is the meaning of the function \(h = f(30, t)\)? Describe the behavior of this function.
(c) What is the meaning of the function \(h = f(v, 30)\)? Describe the behavior of this function.

Let \(g(x, y) = \cos(x + 2y)\).
(a) Evaluate \(g(2, -1)\).
(b) Find the domain of \(g\).
(c) Find the range of \(g\).

Let \(F(x, y) = 1 + \sqrt{4 - y^2}\).
(a) Evaluate \(F(3, 1)\).
(b) Find and sketch the domain of \(F\).
(c) Find the range of \(F\).

Let \(f(x, y, z) = \sqrt{x} + \sqrt{y} + \sqrt{z} + \ln(4 - x^2 - y^2 - z^2)\).
(a) Evaluate \(f(1, 1, 1)\).
(b) Find and describe the domain of \(f\).

Let \(g(x, y, z) = x^3y^2\sqrt{10 - x - y - z}\).
(a) Evaluate \(g(1, 2, 3)\).
(b) Find and describe the domain of \(g\).

Find and sketch the domain of the function.
\(f(x, y) = \sqrt{x - 2} + \sqrt{y - 1}\)

Find and sketch the domain of the function.
\(f(x, y) = \sqrt[4]{x - 3y}\)

Find and sketch the domain of the function.
\(f(x, y) = \ln(9 - x^2 - 9y^2)\)

Find and sketch the domain of the function.
\(f(x, y) = \frac{\sqrt{x^2 + y^2 - 4}}{1 - x^2 - y^2}\)

Find and sketch the domain of the function.
\(g(x, y) = \frac{x - y}{x + y}\)

Find and sketch the domain of the function.
\(g(x, y) = \frac{\ln(2 - x)}{1 - x^2 - y^2}\)

Find and sketch the domain of the function.
\(f(x, y) = \frac{\sqrt{y - x^2}}{1 - x^2}\)

Find and sketch the domain of the function.
\(f(x, y) = \sin^{-1}(x + y)\)

Find and sketch the domain of the function.
\(f(x, y, z) = \sqrt{4 - x^2} + \sqrt{9 - y^2} + \sqrt{1 - z^2}\)

Find and sketch the domain of the function.
\(f(x, y, z) = \ln(16 - 4x^2 - 4y^2 - z^2)\)

Match the function with its graph (labeled I–VI). Give reasons for your choices.
(a) \(f(x, y) = \frac{1}{1 + x^2 + y^2}\)
(b) \(f(x, y) = \frac{1}{1 + x^2y^2}\)
(c) \(f(x, y) = \ln(x^2 + y^2)\)
(d) \(f(x, y) = \cos\sqrt{x^2 + y^2}\)
(e) \(f(x, y) = |xy|\)
(f) \(f(x, y) = \cos(xy)\)

Draw a contour map of the function showing several level curves.
\(f(x, y) = x^2 - y^2\)

Draw a contour map of the function showing several level curves.
\(f(x, y) = xy\)

Draw a contour map of the function showing several level curves.
\(f(x, y) = \sqrt{x} + y\)

Draw a contour map of the function showing several level curves.
\(f(x, y) = \ln(x^2 + 4y^2)\)

Draw a contour map of the function showing several level curves.
\(f(x, y) = ye^x\)

Draw a contour map of the function showing several level curves.
\(f(x, y) = y - \arctan{x}\)

Draw a contour map of the function showing several level curves.
\(f(x, y) = \sqrt[3]{x^2 + y^2}\)

Draw a contour map of the function showing several level curves.
\(f(x, y) = \frac{y}{x^2 + y^2}\)

Describe the level surfaces of the function.
\(f(x, y, z) = x + 3y + 5z\)

Describe the level surfaces of the function.
\(f(x, y, z) = x^2 + 3y^2 + 5z^2\)

Describe the level surfaces of the function.
\(f(x, y, z) = y^2 + z^2\)

Describe the level surfaces of the function.
\(f(x, y, z) = x^2 - y^2 - z^2\)