Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

2,819,387 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This section introduces functions of several variables and presents multiple ways to represent them—verbally, numerically, algebraically, and visually. Students learn how to determine the domain and range, sketch graphs and level curves, and interpret contour maps. Real-world examples such as temperature distribution, wind-chill index, and economic production functions illustrate the relevance of these concepts. A clear understanding of these topics is crucial for visualizing complex surfaces and for applications in science, engineering, and economics.

Learning Objectives

1

Describe functions of two or more variables from verbal, numerical, algebraic, and visual perspectives.

2

Determine and analyze the domain and range of functions of several variables.

3

Graph functions using techniques such as arrow diagrams, surface plots, and contour maps (level curves).

4

Interpret real-world applications like temperature distributions, wind-chill index, and the Cobb–Douglas production function.

Key Concepts

CONCEPT

DEFINITION

Function of Two Variables

A rule that assigns each ordered pair (x, y) in a domain D ⊆ ℝ² a unique real number, often expressed as z = f(x, y).

Domain

The set of all input pairs (x, y) for which the function f(x, y) is well-defined.

Range

The set of all output values that the function f(x, y) can take, i.e., { f(x, y) | (x, y) ∈ D }.

Graph of a Function

A visualization of f(x, y) as a surface in ℝ³, where each point (x, y, z) satisfies z = f(x, y).

Level Curves (Contour Lines)

Curves obtained by setting f(x, y) = k for various constants k, representing all points in the domain where the function attains the same value.

Cobb–Douglas Production Function

An economic model typically expressed as P = bL^αK^β, representing total production P as a function of labor L and capital K.

Contour Map

A map that uses level curves to indicate regions of constant function value; commonly used in topographic maps to show elevation or in meteorology for isothermals and isobars.

Example Problems

Example 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.

Example 2

The $ \textit{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. (a) What is the value of $ f(95, 70) $? What is its meaning? (b) For what value of $ h $ is $ f(90, h) = 100 $? (c) For what value of $ T $ is $ f(T, 50) = 88 $? (d) 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 $.

Example 3

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.

Example 4

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 $$ P(L, K) = bL^{\alpha}K^{1 - \alpha} $$

Example 5

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?

Step-by-Step Explanations

QUESTION

Find the domain and range of the function t(x, y) = √(9 − x² − y²).

STEP-BY-STEP ANSWER:

Step 1: Recognize that the square root is defined when its argument is nonnegative; therefore, set 9 − x² − y² ≥ 0.
Step 2: Rearrange the inequality to x² + y² ≤ 9. This describes a disk in the xy-plane with radius 3 centered at the origin.
Step 3: The domain of t is all points (x, y) such that x² + y² ≤ 9.
Step 4: Since square roots yield nonnegative outputs, the range is 0 ≤ t(x, y) ≤ 3, with the maximum value 3 occurring at (0,0).
Final Answer: Domain = {(x, y) ∈ ℝ² | x² + y² ≤ 9} and Range = [0, 3].

Determining Domain and Range for t(x, y) = √(9 − x² − y²)

QUESTION

Describe how the level curves of the function f(x, y) = √(9 − x² − y²) can be used to visualize its graph.

STEP-BY-STEP ANSWER:

Step 1: Set f(x, y) = k, where 0 ≤ k ≤ 3, which gives √(9 - x² - y²) = k.
Step 2: Square both sides to obtain 9 - x² - y² = k², and then rearrange to get x² + y² = 9 - k².
Step 3: Notice that for each constant k, the level curve is a circle centered at the origin with radius √(9 - k²).
Step 4: As k increases from 0 to 3, the circles shrink from a maximum radius of 3 down to a point (when k = 3).
Final Answer: Level curves are concentric circles whose radii decrease as the function value increases, helping to visualize the upper hemisphere of a sphere.

Sketching Level Curves for f(x, y) = √(9 − x² − y²)

Common Mistakes

Failing to consider all restrictions on the domain, such as excluding values that make a denominator zero or make a radicand negative.
Confusing the independent variables (inputs) with the dependent variable (output) when analyzing graphs and level curves.
Misinterpreting level curves as complete representations of a surface rather than as slices at constant heights.
Overlooking that not every function of two variables can be easily written in an explicit formula; some are given only by data tables or verbal descriptions.