Modern Calculus and Analytic Geometry

By Richard A. Silverman

A self-contained text for an introductory course, this volume places strong emphasis on physical applications. Key elements of differential equations and linear algebra are introduced early and are consistently referenced, all theorems are proved using elementary methods, and numerous worked-out examples appear throughout. The highly readable text approaches calculus from the student's viewpoint and points out potential stumbling blocks before they develop. A collection of more than 1,600 problems ranges from exercise material to exploration of new points of theory — many of the answers are found at the end of the book; some of them worked out fully so that the entire process can be followed. This well-organized, unified text is copiously illustrated, amply cross-referenced, and fully indexed.

• Language: English
• Publisher: Dover Publications
• Release date: Apr 15, 2014
• ISBN: 9780486793986

Book preview

ERRATA

CHAPTER 1 SETS AND FUNCTIONS

We begin by acquiring some basic mathematical vocabulary indispensable to our later work. The topics considered in this chapter have much in common with what is known nowadays as the new math.

1. SETS

A collection of objects of any kind is called a set, and the objects themselves are called elements or members of the set. Sets are usually denoted by capital letters and their elements by small letters. If x is an element of a set A, we write x A(not to be confused with the Greek letter epsilon) is read is an element of. Other ways of reading x A are "x is a member of A x belongs to A or A contains x." By x A we mean that x is not an element of A. The words class and family are often used as synonyms for set.

One way of describing a set is to write its elements between curly brackets. Thus {1} is the set whose only element is the number 1, and {a, b} is the set consisting of the elements a and b (and no others). Similarly, {{1}, {a, b}} is the set whose only elements are the sets {1} and {a, b}, while {1, 2, . . .} is the set of all positive integers, with the dots indicating the infinitely many missing integers.

If every element of a set A is also an element of a set B, we write A ⊂ B and say that A is a subset of B or A is contained in B. Note that A ⊂ B means something quite different from A B. The fact that A ⊂ B can also be expressed by writing B ⊃ A, which is read as "B contains A." We say that two sets A and B are equal and write A = B if A and B have the same members. Clearly A = B if and only if A ⊂ B and B ⊂ A. Otherwise we write A ≠ B. If A ⊂ B but A ≠ B, we say that A is a proper subset of B. Thus the set of all equilateral triangles is a proper subset of the set of all regular polygons.

Example 1. Clearly

REMARK (Some elementary logic). Let P and Q be two statements. Then the three assertions

mean exactly the same thing, and so do the three assertions

We note in passing that another way of expressing (2) is to say that "Q is false if P is false" (why?). Combining (1) and (2), we find that the three assertions

"P implies Q and conversely,"

"P is a necessary and sufficient condition for Q,"

"Q is true if and only if P is true"

are equivalent, i.e., have the same meaning. You will often see the word iff in advanced mathematics books. It is pronounced if and only if.

WARNING. The following statement appears a few lines above:

It is understood that every such statement is a definition. In other words, not only do we say that A = B if A and B have the same members, but we do not say A = B unless A and B do in fact have the same members. Purists would prefer the phrase if and only if in (3) instead of the word if. This is rather pedantic, since (3) is clearly intended as a definition, as indicated by the phrase we say that. On the other hand, in the statement

"A = B if and only if A ⊂ B and B ⊂ A"

the full phrase if and only if cannot be replaced by either if or only if without changing the meaning, in fact without weakening the statement.

If a set has no members at all, it is said to be empty. For example, the set of even prime numbers greater than 2 is empty and so is the set of female American presidents. Every element x of an empty set belongs to any given set E in the trivial sense that there are no such x at all and hence no need to verify that x belongs to E.

A set is often specified by stating properties that uniquely determine its elements. Thus

{3, 4, . . .} = {all integers greater than 2}

= {all x such that x is an integer and 2x > 4}

= {x | x is a positive integer, x² > 4},

where in the last expression the vertical line stands for such that and the comma for and (we also omit the superfluous word all).

Example 2. Clearly

{x | x E) = E,

{x | x is a real number, x,

{x | x is an even prime number} = {2}.

Given two sets A and B, by the difference B − A we mean the set of elements which belong to B but not to A. More concisely,

In any given context, there is always an underlying universal set containing every set under discussion. Let U be this universal set. (For example, U is the plane in plane geometry and three-dimensional space in solid geometry.) Then by the complement of A, written Ac, we mean the set U − A. Thus if U is the rectangle shown in Figure 1.1 and A is the circular disk, Ac is the shaded area lying outside A. Here U contains A, as it must. More generally, suppose B does not contain A. Then B − A is the shaded area shown in Figure 1.2, where A and B themselves are disks. Diagrams of this sort are called Venn diagrams.

FIGURE 1.1

FIGURE 1.2

Example 3. If U .

Example 4. Taking the complement of any set A twice restores the original set, i.e., (Ac)c = A.

The set of all elements belonging to either of two given sets A and B (or to both) is called the union of A and B, written A ∪ B and read "A cup B." The set of all elements belonging to both A and B is called the intersection of A and B, written A ∩ B and read "A cap B." More concisely,

In terms of Venn diagrams, A ∪ B is the shaded area in Figure 1.3 and A ∩ B is the shaded area in Figure 1.4 (A and B themselves are again disks). Two sets A and B , are said to be disjoint. This is not to be confused with the concept of distinct sets, distinct being a synonym for unequal. Thus the set of odd numbers and the set of prime numbers are distinct but not disjoint.

FIGURE 1.3

FIGURE 1.4

Problem Set 1

1. Are disjoint sets necessarily distinct?

2. Write all elements of the set of all subsets of A = {a, b}.

3. Which of the following sets are empty:

;

b) Integers with odd squares;

c) Right triangles whose side lengths are integers;

d) Regular polygons containing a right angle;

e) Regular polygons containing an angle of 45°?

4. Is getting a grade of higher than 50 (out of 100) on the final examination a necessary condition for passing this course? Is it a sufficient condition?

5. Suppose P is a necessary and sufficient condition for Q. Is Q a necessary and sufficient condition for P?

6. Prove the following formulas:

a) A ∪ B = B ∪ A, A ∩ B = B ∩ A;

b) A ∪ A = A, A ∩ A = A;

;

d) (A ∪ B) ∪ C = A ∪ (B ∪ C), ∩ B) ∩ C = A ∩ (B ∩ C);

e) A ⊂ A ∪ B, A ∩ B ⊂ A;

f) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C);

g) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

7. Is the intersection of two nonparallel lines in three dimensions always nonempty?

8. Suppose that in a survey of 100 students, it is found that 48 take calculus, 30 take French, 21 take music, 3 take calculus and music but not French, 5 take calculus and French but not music, 8 take music and French but not calculus, and 2 take all three courses.

a) How many students take none of these courses?

b) How many take only calculus?

c) How many take calculus if and only if they take music?

Hint. Use a Venn diagram of the kind shown in Figure 1.5.

FIGURE 1.5

9. In the preceding problem, is it possible for 70 students to take calculus without changing the other data?

10. Prove that

a) (A ∪ B)c = Ac ∩ Bc;

b) (A ∩ B)c = Ac ∪ Bc;

c) B − A = B ∩ Ac.

Interpret these formulas with appropriate Venn diagrams.

11. By the symmetric difference A Δ B of two sets A and B is meant the set consisting of all elements in exactly one of the sets A and B. Write A Δ B in the {x | . . .} notation. Find A Δ A . Fill in the missing entries in the formulas

*12. Let A be the set of all sets which are members of themselves (e.g., the set of all sets), and let B be the set of all sets which are not members of themselves (e.g., the set of all cats). Show that the naive belief that every set belongs to either A or B leads to a contradiction.

Hint. Is B a member of itself?

Comment. This is Russell’s paradox, devised by the English logician and philosopher Bertrand Russell (1872–1970).

2. ORDERED n-TUPLES. CARTESIAN PRODUCTS

In a set of the form {a, b}, the order in which the elements a and b are written does not matter, i.e., {a, b} = {b, a}. We now consider sets consisting of two elements a and b (say) in which one of the elements (namely a) is designated as the first and the other (namely b) is designated as the second. Such a set is called an ordered pair and is written (a, b), with ordinary parentheses instead of curly brackets. Since the order in which the elements are written now matters, we have (a, b) ≠ (b, a) unless b = a. Note that (a, a) means an ordered pair whose first and second elements are the same, whereas {a, a} can be replaced by {a}. Continuing in this vein, we can define ordered triples (a, b, c), ordered quadruples (a, b, c, d), and more generally, ordered n-tuples (a1, a2, . . . , an).

DEFINITION 1.1. Two ordered n-tuples

(a1, a2, . . . , an)    and    (b1, b2, . . . , bn)

are equal if and only if

a1 = b1, a2 = b2, . . . , an = bn.

Example 1. Let n be a positive integer. Then by the number n!, read "n factorial," we mean the product

n! = n(n − 1) · · · 2 · 1.

Thus

1! = 1,   2! = 2 · 1 = 2,   3! = 3 · 2 · 1 = 6, . . . ,

and so on. From n distinct symbols a1, a2, . . . , an we can form precisely n! ordered n-tuples with distinct elements. In fact, the first element can be chosen in n distinct ways and after fixing the first element, the second element can be chosen in n − 1 distinct ways. But this means that the first two elements can be chosen in n(n − 1) distinct ways. Continuing this argument, we eventually find that the next to the last element can be chosen in just two ways, while the last element is uniquely determined by the choice of the other n − 1 elements. Thus there are n(n − 1) · · · 2 · 1 = n! n-tuples in all. If the n-tuples need not have distinct elements, i.e., if repetition of the elements is allowed, then we can form nn distinct n-tuples, since this time not only the first element but also the second element, the third element, and so on can be chosen in n distinct ways.

Given two sets A and B, the set of all ordered pairs (a, b) such that a belongs to A and b belongs to B is called the Cartesian product of A and B† written A × B and read "A cross B." In symbols,

The generalization of (1) to the case of n factors is the Cartesian product

In other words, A1 × A2 × · · · × An is the set of all ordered n-tuples (a1, a2, . . . , an) such that a1 belongs to A1, a2 belongs to A2, and so on.

Example 2. Given m distinct symbols a1, a2, . . . , am and n distinct symbols b1, b2, . . . , bn, let

Then A × B consists of the mn distinct ordered pairs

Similarly, A × A consists of m² distinct ordered pairs and B × B consists of n² distinct ordered pairs (list them).

Example 3 (A preview of n-space). Let R be the set of all real numbers (here we anticipate the description of R given in Chapter 2). Then R × R is the set of all ordered pairs (x, y) where x and y are real numbers. This set is called 2-dimensional space or simply 2-space and is usually denoted by R² (read "R two"), by analogy with ordinary multiplication (R² = R × R). An ordered pair (x, y) is called a point in 2-space for reasons that will be apparent later (see Sec. 14). Similarly, 3-dimensional space or simply 3-space is the set

of all ordered triples (x, y, z) of real numbers, and (x, y, z) is called a point in R³. More generally, n-dimensional space or simply n-space is the set

of all ordered n-tuples (x1, x2, . . ., xn) of real numbers, and (x1, x2,. . ., xn) is called a point in Rn.

At this stage, you should be tempted to ask What is 1-space? The only reasonable answer is R itself, with the elements of R called points (see Sec. 12).

REMARK. Thinking of (x1, x2, . . . , xn) as a point in n-space, we call the numbers x1, x2, . . . , xn the coordinates of (x1, x2, . . ., xn). In this language, Definition 1.1 can be paraphrased as follows: Two points in n-space are equal if and only if their corresponding coordinates are equal.

Problem Set 2

1. Let A = {1, 3} and B = {2, 4, 6}. Which of the ordered pairs

(1, 1), (3, 6), (6, 1), (4, 4), (1, 4), (1, 3)

belong to A × B? Which belong to A × A?

2. Let A = {1, 3, 5} and B = {2, 3, 5, 6}. Write all ordered pairs in the set

(A × B) ∩ (B × A).

3. Prove that the operation × figuring in the definition of a Cartesian product is noncommutative, i.e., that A × B is not necessarily equal to B × A. Under what circumstances does A × B equal B × A?

4. ?

*5. Define the ordered pair (a, b) entirely in terms of ordinary sets.

Hint. Consider the set {{a}, {a, b}}.

6. Let A be the set {0, 1}. How many points does the space

contain?

*7. Let I = {1, 2, . . .} be the set of all positive integers, and let An be the subset of I³ = I × I × I consisting of all ordered triples (x, y, zI³ which satisfy the equation

xn + yn = zn,

where n is itself a positive integer. Does A1 coincide with I³? Is A2 a proper subset of I³?

Comment. Fermat’s last theorem, which asserts that An is empty if n > 2, remained unproved for 350 years. It was finally proved by Andrew Wiles in 1994.

FIGURE 1.6

*8. Let An be the same as in the preceding problem. Interpret Figure 1.6 as a picture of A³. What geometric object does A³ suggest? Draw a similar picture of A⁴.

Hint. The lines connect n-tuples differing in only one digit.

3. RELATIONS, FUNCTIONS AND MAPPINGS

Now that the notion of a set is available, the logical next step is to consider relationships between two sets. We want to develop a language in which relationships of the most general kind can be described,† quite apart from any attempt to explain them. This point of view may seem a bit fussy at first to theoreticians accustomed to think of relationships in terms of underlying formulas or tables (stemming from the laws of nature), but it is natural enough to experimentalists, who must often be satisfied with merely describing observed relationships, pending the development of suitable theories explaining the recorded data.

Example 1. Let X be the set of all n (eight million or so) residents of New York City. Suppose we want to study New York family life. Then a basic piece of data would be a census showing who is a child of whom. Abstractly stated, we would like to know the set S of ordered pairs (x, y), where x and y both belong to X (i.e., are both New Yorkers), such that y is the child of x. The set S contains no more than 2n ordered pairs (no New Yorker has more than two parents), and in fact fewer than 2n ordered pairs (some New Yorkers have parents who reside elsewhere or are deceased). The set S is called a relation, more exactly a relation from X to X. Note that S contains at least two (distinct) ordered pairs with the same first element (at least one New Yorker has two New Yorkers as children) and at least two ordered pairs with the same second element (at least one New Yorker has two New Yorkers as parents). A relation of this kind is called a many-to-many relation.

Example 2. Pursuing our study of New York family life, we now let S be the set of all ordered pairs (x, y) where x X, y X such that y is the only child of x. (The set X remains the same in Examples 1–4.) Again S is called a relation (from X to X). As before, S contains at least two ordered pairs with the same second element (at least one New Yorker is an only child with two New Yorkers as parents), but this time S does not contain two distinct ordered pairs with the same first element (no New Yorker has two only children). A relation of this kind is called a many-to-one relation.

Example 3. Next let S be the set of ordered pairs (x, y) where x X, y X such that x is the mother of y. This time S is called a one-to-many relation, since it contains at least two ordered pairs with the same first element (at least one New Yorker is the mother of two New Yorkers) but no ordered pairs with the same second element (no New Yorker has two mothers).

Example 4. This time let S be the set of all ordered pairs (x, y) where x X, y X such that y is the legal spouse of x (and conversely). Then S is called a one-to-one relation since it contains no ordered pairs with the same first element (bigamy is illegal) and no ordered pairs with the same second element (same reason).

Example 5. Finally let X be the set of New Yorkers of age forty or more, and let Y be the set of New Yorkers under forty. Then Examples 1–4 have obvious analogues for ordered pairs (x, y) where this time x belongs to X and y belongs to Y, except that now the relations are said to be from X to Y. Thus we see that there is no reason for the first and second elements of a relation to belong to the same set.

The above examples should have convinced you that a relation S from a set X to a set Y is nothing more or less than a subset of the Cartesian product X × Y. Sometimes we already know what lies behind a relation, as in Examples 1–5 stemming from properties of family life in New York. In other cases, hard work may eventually explain a relation in terms of an underlying rule of formation or interdependence, often involving a complicated scientific theory. However, from a purely abstract point of view, there is nothing to prevent the statement "(x, y) belongs to a relation S" from meaning just that and nothing more. In other words, there may be no way of telling whether or not (x, y) belongs to S short of consulting the complete list of elements in S.

Of the four kinds of relations just studied, only two have the property that no two distinct ordered pairs have the same first element. We refer of course to many-to-one and one-to-one relations. Thinking of the first element of an ordered pair as the data in a scientific problem and the second element as the unique answer to the problem, we see at once why these relations play a dominant role in science and applied mathematics, i.e., well-posed scientific problems should have unique answers.

Example 6. Let x be the elapsed time since the launching of a space vehicle, and let y be the vehicle’s subsequent position. Then there is a unique value of y corresponding to every value of x, but the converse is not true since the vehicle may return to an earlier position.

Example 7. Let x be an electrical signal fed into a filter f (i.e., an electrical device with specified characteristics), and let y be the corresponding output, as shown schematically in Figure 1.7’. Then, provided the properties of f do not change, a given input produces a unique output, but a given output may also be produced by several inputs.

FIGURE 1.7

These considerations lead to the following key definition (note that here convention favors the use of a small letter for the set f):

DEFINITION 1.2. A many-to-one or one-to-one relation from a set X to a set Y is called a function from X to Y. In other words, a function f from X to Y is a subset of X × Y such that if (x, yf and (x, zf then y − z. The unique element y associated with x is called the value of f at x, written f(x) and read f of x. The symbol x appearing in f(x) is called the argument of f(x) or of f.

WARNING. Do not confuse f(x), the value of the function f at x, with the function f itself, which is a set of ordered pairs.

REMARK 1. The arguments and values of f need not be numbers.

REMARK 2. In view of Definition 1.2, the terms many-to-one relation and many-to-one function are synonymous, and so are one-to-one relation and one-to-one function.

Example 8. Let I = {1, 2, . . .} be the set of all positive integers and let f be the subset of I × I consisting of all ordered pairs (n, p) such that p is the square of n (n and p are better symbols for integers than x and y). Then f is most simply specified by the formula

f(n) = n²

for all n I.

Example 9. The rule associating f(x) with x may involve two or more formulas. Thus let X = R be the set of all real numbers and consider the function from R to R such that

or more concisely,

In particular, f(0) = 0. This function is important enough to have a special name. It is called the absolute value of x, denoted by |x|.

Example 10. Let I = {1, 2, . . .} be the set of all positive integers as in Example 8, and let Y be an arbitrary set. Then a function f from I to Y defined for every n I is called an infinite sequence or simply a sequence. The values of f at 1, 2, . . . are called the terms of the sequence, and we usually write y = yn instead of y = f(n). A sequence is often specified by listing its terms

y1, y2, . . . , yn, . . .

in order of increasing n, where yn is called the general term, or by writing the general term yn together with an indication, inside parentheses, of the set over which n varies:

yn   (n = 1, 2, . . .).

A more concise way of specifying a sequence is to write its general term inside curly brackets:

{yn}.

Do not confuse {yn} in this context with the set whose only element is yn.

Example 11. By a real sequence we mean a sequence whose terms are real numbers. In other words, a real sequence is a function from I = {1, 2, . . .} to R, the set of all real numbers, defined for every n I. Such a sequence is often specified by giving the law of formation of its general term. For example, let

yn = n!,

where n! is defined in Example 1, p. 6. Then {yn} is the sequence

1, 2, 6, 24, 120, . . .

More generally, yn may be given recursively, i.e., as an expression involving terms of the sequence with lower subscripts. Thus suppose

Then

where the dots mean and so on indefinitely. It follows that

{yn} = 1, 3, 6, 10, . . .

Rules like (1) are called recursion formulas.

Let f be a function from X to Y. Then the set of all arguments of f i.e., the set of all first elements of ordered pairs (x, y) belonging to f is called the domain of f denoted by Dom f and f is said to be defined in any subset of Dom f. Similarly, the set of all values of f i.e., the set of all second elements of ordered pairs (x, y) belonging to f is called the range of f denoted by Rng f. More concisely,

Dom f = {x | (x, yf for some y},    Rng f = {y | (x, yf for some x}.

It is clear that

Dom f ⊂ X, Rng f ⊂ Y.

Moreover (x, yf implies

X Dom f,     y = f(xRng f,

and hence

f ⊂ Dom f × Rng f ⊂ X × Y.

Obviously there is no loss of generality in regarding f as a function from Dom f to Rng f instead of from X to Y, since f cannot contain a pair (x, y) without x being an argument of f and y a value of f.

Example 12. Let f(x) be the age of the wife of x, where x is a New Yorker. Then Dom f is the set of all married male New Yorkers, while Rng f is some subset of the set of positive integers.

Example 13. In Example 9, Dom f is the set R of all real numbers, while Rng f is the set of all nonnegative real numbers.

Example 14. If f is a real sequence, then Dom f = f = {1, 2, . . .}, while Rng f is some subset of R.

The way of looking at functions just presented is the final product of a long period of historical evolution. To complete the picture we must now describe a number of alternative ways of thinking and talking about functional dependence. These alternatives say nothing essentially new, and in fact in some cases say considerably less than our previous approach, which we call the master approach. However, they must be learned if you are to avoid vocabulary problems when reading scientific books. The whole situation is summarized in the following table, where the entries on each line give various ways of saying the same thing:

Of the three alternatives, only the first is as flexible as the master approach, and hence will be used freely on the same footing. In fact, Alternative 1 actually has two ways of saying "(x, y) belongs to f," one emphasizing so to speak the destination of x ("f maps x into y"), the other the source of y ("y is the image of x under f"). In the language of Example 7, these two assertions become "the filter f transforms the input x into the output y and y is the result of feeding x into the filter f." Alternative 2 is less flexible, and in fact it obscures the meaning of f (who thinks of a rule as a set of ordered pairs?). It is also somewhat deficient symbolically, as shown by the missing entry in the third column of the table.

Alternative 3 is the oldest and vaguest of them all. It introduces a new concept, that of the variable (not unlike an unknown in algebra), i.e., a symbol which varies over some set of admissible values. One such symbol, namely x, is allowed to take any of its values and hence is called the independent variable. Then, once a value of x is chosen, the other variable, namely y, takes a value uniquely determined by the value of x, and hence is called the dependent variable. This fact is expressed by saying that "y is a function of x," but no symbol is assigned to the function itself, which at this unsophisticated level is hardly thought of as a set of ordered pairs. The symbolic deficiencies of Alternative 3 are even worse than those of Alternative 2, as shown by the fact that there are two missing entries in the last column of the table. Nevertheless, we can safely use this language, secure in the knowledge that it can always be made more precise. In fact, its very vagueness is often an asset, since it enables us to say things quite simply.

It is often helpful to think of functions and mappings in terms of crude pictures like those shown in Figures 1.8 and 1.9. Suppose f is a function from a set X to a set Y (we also say that "f maps X into Y or f carries X into Y"). Representing X and Y by amorphous blobs and typical elements of X and Y by points inside the blobs, we indicate an ordered pair (x, yX × Y in Figure 1.8 by tying one end of a short piece of string to x and the other end to y.

FIGURE 1.8

If this is done for every ordered pair in f we get a collection of pieces of string representing f some of which are shown in Figure 1.8. We do not insist that X coincide with Dom f or that Y coincide with Rng f.

In the mapping approach, f is regarded as an operation actively carrying x into y. In Figure 1.9 this point of view is expressed in two ways. First we equip the right end of each piece of string with an arrowhead. Secondly, we write the letter f on a typical piece of string to emphasize the role of f in carrying x into y.

FIGURE 1.9

Problem Set 3

1. Let X be the set of all baseball players in the American League and Y the set of all players in the National League. Suppose S is the subset of all ordered pairs in X × Y such that "x is a better hitter than y." What kind of relation is S?

2. In what sense is the empty set a relation?

3. Let Z be the set of all integers (positive, negative and zero). Which of the following subsets of Z × Z are functions:

a) The set of ordered pairs (m, n) such that m is less than n;

b) The set such that m = n²;

c) The set such that m² = n;

d) The set such that m = n + 1?

4. Let r be the radius of a circle and A its area. Is r a function of A? If so, what is the domain of the function?

5. Is the area of a triangle a function of its perimeter?

6. Is the area of a square a function of its perimeter?

7. Let τ be the number of divisors of a positive integer n (including 1 and n itself). Thus τ = 6 if n = 12, since 12 has the divisors 1, 2, 3, 4, 6 and 12. Is n a function of τ?

8. Consider the set S of ordered pairs of real numbers (x, y) such that x² + y² = 1. What kind of relation is S? Is S a function?

9. Let f be a function. Prove that the relation f−1 obtained by writing every ordered pair (x, yf in reverse order is a function if and only if f is one-to-one.

Comment. f−1 is called the inverse of f.

10. According to James and James (Mathematics Dictionary), a constant is a quantity whose value does not change, or is regarded as fixed, during a given sequence of mathematical operations. Find a definition more in keeping with our way of saying things.

11. It is known that if a body falls freely for t seconds under the influence of gravity, the distance fallen equals

where the constant g is the acceleration due to gravity. Describe circumstances under which g becomes a variable.

12. All other things being equal, the way in which the amount of postage paid on a letter depends on the weight of the letter is a function from the set of positive real numbers to the set of positive integers. Justify this assertion, disregarding units. Explain the phrase all other things being equal.

13. Given any real number x, prove that |x| equals the positive square root of x².

14. Write the first ten terms of the sequence of all odd integers arranged in increasing order. What is the general term of this sequence?

15. Write the first ten terms of the sequence {yn}, where yn is the sum of the first n odd integers arranged in increasing order. Can you guess the general term of this sequence?

*16. A sequence is specified by the following rule: Its first two terms equal 1, and the remaining terms are given by the recursion formula

yn = yn−1 + yn−2.

Write the first ten terms of the sequence. Verify that yn is given by the explicit formula

Comment. {yn} is the Fibonacci sequence, with surprising applications to botany, art and architecture (see Life Science Library, Mathematics, pp. 93–94).

17. Can you guess a simple rule of formation of the following sequence

3, 1, 4, 1, 5, 9, 2, 6, 5, 3, . . .?

*18. Let S be a relation from X to Y, i.e., a subset of X × Y. Then, just as in the case of a function, the set of all first elements of ordered pairs in S is called the domain of S and the set of all second elements of ordered pairs in S is called the range of S. Find the domain and range of S in Examples 1–4. Find a New Yorker who does not belong to the domain or range of any of the four relations S.

4. REAL FUNCTIONS

The functions studied in the preceding section are abstract, i.e., they involve elements x and y of arbitrary sets X and Y. The following special choices of X and Y are particularly important:

1) A function from an arbitrary set X to R, the set of all real numbers, is called a real-valued function, or simply a real function.†

2) A function from R to R is called a real function of one real variable.

3) A function from R to Rn, where n exceeds 1, is called a point-valued function, or more often a vector function (see Sec. 72), of one real variable.

4) A function from Rn to R, where n exceeds 1, is called a real function of several real variables (in fact, n variables), and each coordinate of the point x = (x1, x2, . . . , xnRn is called an independent variable.‡

5) A function from Rn to Rp, where n and p both exceed 1, is called a coordinate transformation (see p. 424), and each coordinate of the point y = (y1, y2, . . . , ypRp is called a dependent variable.

This book is primarily concerned with real functions of one or several variables. Therefore we shall henceforth usually omit the adjective real and talk simply about functions of one variable and functions of several variables.

REMARK 1. There is nothing sacred about the use of the letter f to denote a function (apart from its being the first letter of the word function). Other letters would do just as well, and common choices are g, h, φ, ψ, etc. Sometimes the letter is chosen to suggest a geometrical or physical quantity under discussion. Thus A is often used for area, V for volume, p for pressure, T for temperature, and so on. By the same token, any letter at all can be used to denote a variable, whether independent or dependent. In particular, sequences can be written as {an), {bn}, {xn}, etc.

REMARK 2. Despite the warning on p. 11, we endorse the widespread slight abuse of terminology entailed in talking about the "function f(x) instead of the function f" whenever it seems appropriate (for example, whenever a reminder of the symbol used for the independent variable seems helpful). Once the distinction between a function and its values is clearly understood, there is no need to belabor it at the risk of being pedantic.

Example 1. Let

where x is a real number and the radical denotes the positive square root. Find f(0) and f(2a). Does f(−2) exist?

Solution. To find f(0) we merely substitute x = 0 into (1), obtaining)

Similarly,

On the other hand, the quantity

is meaningless, since we cannot extract the square root of a negative number (at least not in this course). Whenever a function is specified by an explicit formula like (1), we shall understand the domain of f to be the largest set of values for which the formula makes sense. In the present example, this set consists of all real numbers from −1 to 1 (more exactly, all real x such that −1 ≤ x ≤ 1).† Note that any smaller set, e.g., the set {−1, 0, 1} can serve as the domain of a function whose values are given by the same value (1), but in such cases we will write

say, explicitly indicating the domain {−1, 0, 1}.

Example 2. The domain of the function

consists of all real numbers except t = 1. In fact,

and the expression on the right is meaningless since division of a nonzero number by zero makes no sense. To see this, let a ≠ 0. Then a/0 must mean a number b such that 0 · b = a. But 0 · b = 0 and hence there is no such number.

Example 3. If two functions f(x) and g(x) have the same domain X and have the same value for every x X, we say that the functions are identical. Thus the functions

defined for all real x, are identical functions. A function like f(x) = 1 which takes the same value for all x is called a constant function. It is a perfectly acceptable function, as we see by considering the set of all ordered pairs (x, y) where x is real and y = 1. If f(x) and g(x) have the same domain X and if f(x) = g(x) for x X, we often write

f(x) ≡ g(x),

where the symbol ≡ means is identically equal to. In particular, f(x) ≡ 1 means the same thing as f(x) = 1 for all real x.

Example 4. Turning to functions of several variables, let

where x and y are both real. Find f(1, 2), f(a, 1/a) and f(y, x).

Solution. Easy substitutions give

Note that f(y, x) ≡ −f(x, y).

Example 5. Find the domain of the function

(here x and y are independent variables and z is the dependent variable).

Solution. The function (2) is defined for all real x and y if x ≠ y, but if x = y it reduces to the meaningless expression 1/0.

Example 6. If x = 0, the function

takes the value

The expression on the right is said to be indeterminate rather than meaningless. In fact, 0/0 must mean a number b such that 0 · 6 = 0. But every number b has this property! If x ≠ 0 we can divide both the numerator and denominator of f(x) by x, obtaining the simpler function

Clearly φ(x) = f(x) for all x ≠ 0. However φ(x), unlike f(x), is defined for x = 0, since

Much of calculus involves making a sensible choice for the value of indeterminate forms like 0/0 (the derivative itself is such a form).

Problem Set 4

1. Functions of a single variable are often specified by a table listing values of the dependent variable corresponding to given values of the independent variable. The function y = f(x) in the following table is familiar from everyday life. What is it? Fill in the missing entries in the table. Find a formula relating y to x and one relating x to y.

2. Functions of two variables are often specified by a table listing values of the dependent variable corresponding to given values of the independent variables. For example, the following table shows how a certain function z = f(x, y) depends on x and y. Find an explicit formula for f(x, y).

Hint. Note that consecutive entries increase by 2 in going from left to right and decrease by 3 in going from top to bottom.

Comment. As an intellectual exercise, you might try imagining such a table in three dimensions.

Problems 3–11 involve functions from R to R.

3. Given

and |f(−2)|.

4. Given

find φ(−1) and φ(1 + aexist?

5. If

prove that ψ(−x) = −ψ(x).

6. Let f(n) = an, where an is determined from the formula

Find f(1), f(3) and f(4).

7. If

find f(0), f(3), g(2), g(−1) and f(−1) + g(1).

8. If

f(x) = x² − 2x + 3,

find all the roots (i.e., solutions) of the equation

a) f(x) = f(0);

b) f(x) = f(−1).

9. Are any of the following pairs of functions identical:

;

;

?

(As always, the radical denotes the positive square root.)

10. Find the values of a and b in the formula

such that

11. Let

Find all the roots of the equation

Problems 12–16 involve functions from R² to R.

12. Let

Find f(3, 1), f(0, 1), f(1, 0), f(a, a) and f(a, −a).

13. If

show that

14. Find the domain of the function

15. If

and s ≠ t, show that

16. If

show that

Are there any exceptional values of x and y, i.e., values such that (3) or (4) becomes meaningless or indeterminate?

Problems 17 and 18 involve functions from R³ to R.

17. Find the domain of the function

18. If

prove that

Discuss possible exceptional values of x, y and z.

Problems 19 and 20 involve functions from Rn to R.

19. Find the domain of the function

20. If

find f(1, 1, . . . , 1). Discuss possible exceptional values of x1, x2, . . . , xn.

5. OPERATIONS ON FUNCTIONS

We now discuss ways of combining two or more functions to get new functions:

Example 1 (Composition of functions). This procedure is perfectly general and can be defined for abstract functions as well as for real functions. Given two functions f and g, suppose Rng g ⊂ Dom f. Then by the function f º g, called the composition of f and g (by the same token, f º g is said to be a composite function),† we mean the function whose value at every x Dom g is equal to the value of f at g(x). In other words,

The interpretation of f º g as a mapping and the way f º g is related to the mappings f and g is shown schematically in Figure 1.10. Note that the condition Rng g ⊂ Dom f guarantees that f is defined at g(x) for all x Dom g. For example, suppose

FIGURE 1.10

denotes the positive square root of x. Then

while

(note that here the conditions Rng g ⊂ Dom f and Rng f ⊂ Dom g both hold). In this case, Dom g º f ⊂ Dom f º g and the functions coincide on Dom g º f the set of all nonnegative real numbers. As another example, suppose

where x is an arbitrary real number. Then the functions

are both defined for all real x, but (f º g)(x) ≠ (g º f)(x) unless x = 0. Thus the operation of composition is noncommutative, i.e., in general f º g ≠ g º f.

Example 2. Find f(f(f(2))) if

Solution.. Since

we have

and

Example 3. Just as in Example 1, p. 17, whenever f and g are given by explicit formulas, we shall understand the domain of f º g to be the largest set of values of x for which (f º g)(x) = f(g(x)) makes sense. For example, suppose

where x is real. Then the domain of f º g is meaningless for other values of x, while the domain of g º f is the set of all real numbers not exceeding 1.

Example 4. In the case of functions of several real variables, composite functions are most simply indicated by suitable use of parentheses. Thus, to find f(g(2), h(2)) where

we note that

and hence

Similarly, to find f(g(1, 2), h(1, 2)) where

we note that

and hence

Example 5 (Algebraic operations on functions). Let f and g be two real functions of a single variable with the same domain X. Then by the sum f + g we mean the function whose value at x X equals the sum of the value of f at x and the value of g at x. In other words,

for all x X. Differences and products of functions are defined similarly:

In particular, if c is a constant, then

As for quotients, we must stipulate that g is nonzero for all x X, and then

Example 6. Find the values of f + g, f − g, fg, f³ and f/g at x = 5 if

Solution. Since

we have

Problem Set 5

1. Find f º f, f º g, g º f and g º g if

2. if

3. What is the domain of the function

4. Prove that the operation º figuring in the definition of the composition of two functions is associative, i.e., that (f º g) º h = f º (g º h), so that the parentheses can be dropped, giving just f º g º h. Generalize this result to any number of functions.

5. Find the value of f º g º h .

6. Find f ± g, fg, g² and f/g for x = 3 if f(x) = 2x, g(x) = x².

7. By the zero function is meant the function identically equal to zero. Prove that the cancellation law of multiplication fails for products of real functions, i.e., that fg ≡ fh and f 0 does not imply g ≡ h.

8. Define algebraic operations on real functions of several variables.

9. Given a function f with domain X, let A be a subset of X. Then by the image of A under f denoted by f(A), is meant the set of all f(x) such that x A. Given that

find f(A) if

;

b) A = {x | x² + 2x + 1 = 0};

c) A is the set of all positive real numbers;

d) A is the set of all real numbers.

*10. Let S and T be two relations. Then by the relation S º T, called the composition of S and T, we mean the set of all ordered pairs (x, z) such that (x, yT and (y, zS for some y. Prove that if S and T are both functions, then S º T is the composition of S and T as defined in Example 1.

*11. Let X be the set of all New Yorkers, let S be the set of all ordered pairs (x, yX × X such that y is the brother of x, and let T be the set of all ordered pairs (x, yX × X such that x is the child of y. Define the relation S º T in simple English.

6. COUNTING AND INDUCTION

At a theatrical performance we can easily tell whether there are more people than seats (some people are forced to stand) or more seats than people (some seats are empty). Similarly, if the number of people present is exactly equal to the number of seats, this can be deduced from the fact that no seats are empty and no people are standing without bothering to count either the number of people present or the number of seats. In other words, the existence of a one-to-one correspondence between the people and the seats proves that the size of the audience is exactly equal to the number of seats.

Transcribing this situation to the case of general sets, we have

DEFINITION 1.3. Two sets A and B are said to be in one-to-one correspondence if there exists a one-to-one function f with domain A and range B. Two sets are said to have the same number of elements if there is a one-to-one correspondence between them.

Example 1. When we say that a set A contains (exactly) n elements, we mean that A has the same number of elements as the set {1, 2, . . . , n} consisting of the first n positive integers, i.e., that there exists a one-to-one function f with domain {1, 2, . . . , n} and range A. To count the elements of A, we call f(1) the first element, f(2) the second element and so on, up to the "nth element" f(n).

Example 2. If a set A contains n elements, where n is some positive integer, we say that A is finite; otherwise A is said to be infinite. (The empty set is regarded as finite.) There are many kinds of infinite sets, and the simplest kind can still be counted. In fact, a set A is said to be countably infinite if it has the same number of elements as the set I = {1, 2, . . .} of all positive integers. For example, the set of all positive fractions is countably infinite. One way of counting them is to write

. To avoid repetition of the same number written in different ways, we need only reduce all fractions to lowest terms and then delete any number which has already been counted.

Example 3. A set A is said to be countable if it is finite or countably infinite. Otherwise A is said to be uncountable. Thus the set of all odd numbers is countable, and so is the set of all prime numbers. The set of all integers (positive, negative and zero) is also countable, as seen by writing

0, 1, −1, 2, −2, 3, −3, . . .

Example 4. The set D of all decimals between 0 and 1 is uncountable. To see this, suppose D is countable. Then there is a list like

(read from top to bottom) which allegedly contains all decimals in D. (The last row of dots stands for the infinitely many decimals yet to be listed, and the dots at the end of each decimal indicate the infinitely many unwritten digits.) But it is easy to find a decimal in D which is missing from the list! In fact, consider

d = 0.αβγ . . . ,

where α ≠ a1, β ≠ b2, γ ≠ c3, etc., and we are careful to avoid writing an infinite run of nines.† For example, if the first three items in the list are 0.834 . . ., 0.721 . . . , 0.313 . . ., then d can begin like 0.631 . . ., 0.752 . . ., etc., or even like 0.999 . . . , but in the last case we must promise to eventually choose one of the digits of d to be a number other than nine. Clearly d cannot appear in the list, since it differs in at least one decimal place from every decimal in the list. Therefore the list cannot contain all decimals between 0 and 1, i.e., D is uncountable. The proof just given is called a proof by contradiction, and is a standard mathematical gambit. The sentence ‘if my being outside this room is false, then I am inside the room" is another such proof.

A key property of the set I = {1, 2, . . .} of all positive integers is that it is well-ordered, i.e., every nonempty subset of I contains a smallest element. For example, the smallest element of I itself is 1, while the smallest element of the set {999, 47, 711} is 47. This fact has an important consequence, which we prove as our first theorem.

THEOREM 1.1 (Principle of mathematical induction). Suppose an assertion involving an arbitrary positive integer n is true for n = k + 1 if it is true for n = k, and moreover suppose the assertion is true for n = 1. Then it is true for all n.

Proof. We will prove the theorem in a moment, after first illustrating its meaning. The sample calculations

suggest the truth of the assertion that the sum of the first n odd integers is n². Is this really true? Yes, according to Theorem 1.1. For suppose the assertion is true for n = k, so that

(justify writing 2k − 1 as the last term on the left, given that the dots indicate the odd numbers between 3 and 2k − 1 written in increasing order). Then, adding the next odd number 2k + 1 to both sides of (1), we obtain

But the left-hand side of (2) is the sum of the first k + 1 odd integers, while the right side is just (k + 1)² written out in full. In other words, the assertion is true for n = k + 1 if it is true for n = k. But it is obviously true for n = 1, since 1 = 1². Therefore, according to Theorem 1.1, it is true for all n.

We now give the promised proof of Theorem 1.1. Again it will be by contradiction. Let A be the set of all positive integers for which the given assertion is false. Then, since the positive integers are well-ordered, A has a smallest element, which we call n0. Clearly n0 exceeds 1, since the assertion is assumed to be true for n = 1. Therefore n0 − 1 is also a positive integer, in fact one for which the assertion is true (otherwise n0 cannot be the smallest positive integer for which the assertion is false). But then the assertion is true for n0 − 1 and false for n0 = (n

signals the end of a proof. It is read Q.E.D. (from the Latin quod erat demonstrandum) or as was to be proved.

REMARK 2. Do not confuse mathematical induction with the principle of inductive reasoning in the physical sciences. For example, the sun is expected to rise tomorrow, since it has never failed to do so in the recorded history of the human race (forget long eclipses). This is a good argument, but not a totally convincing one (hence since appears in quotes). After all, the solar system might conceivably be destroyed tonight! Mathematical induction, on the other hand, is really foolproof, but unfortunately we cannot use it to prove that the sun will rise tomorrow, which is not a mathematical statement.

Example 5. Mathematical induction has already appeared in disguise in Example 11, p. 12, where we used a recursion formula to generate a real sequence {yn}. It