Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section expands the ideas of single-variable calculus to functions of several variables. Key topics include the definition of functions in higher dimensions, determining the domain, and understanding regions via interior and boundary points. It also introduces graphical techniques such as level curves and surfaces, with real-world applications demonstrating the importance of multivariable analysis.

Learning Objectives

1

Explain the concept of functions of several variables and their domains and ranges.

2

Describe how to determine interior, boundary, open, closed, bounded, and unbounded regions in the plane and in space.

3

Analyze and sketch level curves and level surfaces for given functions.

4

Apply multivariable function concepts to real-world problems such as temperature variation and mortgage calculations.

Key Concepts

CONCEPT

DEFINITION

Function of Several Variables

A rule that assigns a unique real number to each ordered n-tuple (x1, x2, ..., xn) of real numbers.

Domain

The set of all input values for which the function is defined, typically including restrictions to avoid complex numbers or division by zero.

Range

The set of all possible output values obtained by applying the function to its domain.

Interior Point

A point in a region R that is the center of a disk (in the plane) or solid ball (in space) lying entirely within R.

Boundary Point

A point where every neighborhood around it contains both points that are inside and outside the region R.

Open Region

A region that consists entirely of interior points.

Closed Region

A region that contains all its boundary points.

Bounded Region

A region that fits inside a disk (or ball in space) of finite radius.

Level Curve

The set of points (x, y) for which a function f(x, y) equals a constant c, typically producing curves in the plane.

Level Surface

The set of points (x, y, z) in space where a function of three variables equals a constant c.

Example Problems

Example 1

In Exercises $1-4,$ find the specific function values. $$ \begin{array}{ll}{f(x, y)=x^{2}+x y^{3}} \\ {\text { a. } f(0,0)} & {\text { b. } f(-1,1)} \\ {\text { c. } f(2,3)} & {\text { d. } f(-3,-2)}\end{array} $$

Example 2

In Exercises $1-4,$ find the specific function values. $$ f(x, y)=\sin (x y) $$ $$ \begin{array}{ll}{\text { a. } f\left(2, \frac{\pi}{6}\right)} & {\text { b. } f\left(-3, \frac{\pi}{12}\right)} \\ {\text { c. } f\left(\pi, \frac{1}{4}\right)} & {\text { d. } f\left(-\frac{\pi}{2},-7\right)}\end{array} $$

Example 3

In Exercises $1-4,$ find the specific function values. $$ f(x, y, z)=\frac{x-y}{v^{2}+z^{2}} $$ $$ \begin{array}{ll}{\text { a. } f(3,-1,2)} & {\text { b. } f\left(1, \frac{1}{2},-\frac{1}{4}\right)} \\ {\text { c. } f\left(0,-\frac{1}{3}, 0\right)} & {\text { d. } f(2,2,100)}\end{array} $$

Example 4

In Exercises $1-4,$ find the specific function values. $$ f(x, y, z)=\sqrt{49-x^{2}-y^{2}-z^{2}} $$ $$ \begin{array}{ll}{\text { a. } f(0,0,0)} & {\text { b. } f(2,-3,6)} \\ {\text { c. } f(-1,2,3)} & {\text { d. } f\left(\frac{4}{\sqrt{2}}, \frac{5}{\sqrt{2}}, \frac{6}{\sqrt{2}}\right)}\end{array} $$

Example 5

In Exercises $5-12,$ find and sketch the domain for each function. $$ f(x, y)=\sqrt{y-x-2} $$
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Step-by-Step Explanations

QUESTION

Describe the domain of the function f(x, y) = 2y - x².

STEP-BY-STEP ANSWER:

Step 1: Recognize that f(x, y) is defined where the expression 2y - x² gives a real number.
Step 2: Identify any restrictions. Here, to ensure meaningful outputs especially when the function represents a real-world quantity, one might require that y - x²/ ? is non-negative, if interpreting it as a physical measure. In this exercise, the problem statement indicates y must be at least x² (i.e., y ≥ x²) for a real outcome.
Step 3: Conclude that the domain is the set of all points (x, y) such that y ≥ x².
Final Answer: The domain of f(x, y) = 2y - x² is { (x, y) ∈ ℝ² | y ≥ x² }.

Determining the Domain of a Function

QUESTION

Find and describe the level curves for f(x, y) = 100 - x² - y².

STEP-BY-STEP ANSWER:

Step 1: Set the function equal to a constant c: 100 - x² - y² = c.
Step 2: Rearrange the equation to obtain x² + y² = 100 - c.
Step 3: Recognize that for each constant c with c ≤ 100, the equation x² + y² = constant represents a circle centered at the origin with radius √(100 - c).
Step 4: Note that different values of c yield different circles, and if c = 100 then the level curve reduces to the single point (0,0).
Final Answer: The level curves of f(x, y) are circles with radius √(100 - c), provided that 100 - c is non-negative.

Finding Level Curves of a Function

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Common Mistakes

  • Confusing the domain restrictions with the function's overall range.
  • Overlooking the effect of the direction of approach when evaluating limits in multiple dimensions.
  • Miscalculating the boundaries by not considering that boundary points may not belong to the region.
  • Assuming that conditions for open and closed sets in single-variable functions directly apply without modifications in higher dimensions.